# RECENT TRENDS IN COMPUTATIONAL FLUID DYNAMICS, 2nd Edition

EDITED BY : Muhammad Mubashir Bhatti, Marin I. Marin, Ahmed Zeeshan and Sara I. Abdelsalam PUBLISHED IN : Frontiers in Physics

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ISSN 1664-8714 ISBN 978-2-88966-900-4 DOI 10.3389/978-2-88966-900-4

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## RECENT TRENDS IN COMPUTATIONAL FLUID DYNAMICS, 2nd Edition

Topic Editors:

Muhammad Mubashir Bhatti, Shanghai University, China Marin I. Marin, Transilvania University of Brașov, Romania Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan Sara I. Abdelsalam, National Autonomous University of Mexico, Mexico

Publisher's note: This is a 2nd edition due to an article retraction.

Citation: Bhatti, M. M., Marin, M. I., Zeeshan, A., Abdelsalam, S. I., eds. (2021). Recent Trends in Computational Fluid Dynamics, 2nd Edition. Lausanne: Frontiers Media SA. doi: 10.3389/978-2-88966-900-4

# Table of Contents

*05 Editorial: Recent Trends in Computational Fluid Dynamics* M. M. Bhatti, M. Marin, A. Zeeshan and Sara I. Abdelsalam

*09 Numerical Solution of Casson Nanofluid Flow Over a Non-linear Inclined Surface With Soret and Dufour Effects by Keller-Box Method* Khuram Rafique, Muhammad Imran Anwar, Masnita Misiran, Ilyas Khan, S. O. Alharbi, Phatiphat Thounthong and K. S. Nisar

*22 Study of the Couple Stress Convective Micropolar Fluid Flow in a Hall MHD Generator System*

Zahir Shah, Poom Kumam, Abdullah Dawar, Ebraheem O. Alzahrani and Phatiphat Thounthong

*34 Numerical Simulation of Magnetohydrodynamic Nanofluids Under the Influence of Shape Factor and Thermal Transport in a Porous Media Using CVFEM*

Zahir Shah, Houman Babazadeh, Poom Kumam, Ahmad Shafee and Phatiphat Thounthong

*44 Magnetohydrodynamic Free Stream and Heat Transfer of Nanofluid Flow Over an Exponentially Radiating Stretching Sheet With Variable Fluid Properties*

Muhammad Irfan, Muhammad Asif Farooq and Tousif Iqra

*55 Computational and Physical Examination About the Aspects of Fluid Flow Between Two Coaxially Rotated Disks by Capitalizing Non-fourier Heat Flux Theory: Finite Difference Approach*

Sardar Bilal, Asifa Tassaddiq, A. H. Majeed, Kottakkaran Sooppy Nisar, Farhad Ali and M. Y. Malik

*70 A Numerical Simulation for Darcy-Forchheimer Flow of Nanofluid by a Rotating Disk With Partial Slip Effects*

Malik Zaka Ullah, Stefano Serra-Capizzano and Dumitru Baleanu


Anum Shafiq, Islam Zari, Ilyas Khan, Tahir Saeed Khan, Asiful H. Seikh and El-Sayed M. Sherif


# Editorial: Recent Trends in Computational Fluid Dynamics

M. M. Bhatti <sup>1</sup> \*, M. Marin<sup>2</sup> , A. Zeeshan<sup>3</sup> and Sara I. Abdelsalam<sup>4</sup>

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China, 2Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania, 3Department of Mathematics and Statistics, FBAS, IIU, Islamabad, Pakistan, 4Basic Science, Faculty of Engineering, The British University in Egypt, Al-Shorouk City, Egypt

Keywords: computational fluid dynamics, non-Newtonian/Newtonian fluids, heat and mass transfer, multiphase flow simulations, thermodynamics, nanofluids

Editorial on the Research Topic

#### Recent Trends in Computational Fluid Dynamics

Computational fluid dynamics (CFD) [1] can be described as the set of techniques that assist the computer to provide the numerical simulation of the fluid flows. The three basic principles that can determine the physical aspects of any fluid are the i) energy conservation, ii) Newton's second law, and the iii) mass conservation. These flow problem can be described in terms of these basic laws. Mathematical equations, which are usually in the form of partial differential equations, portrayed the fluid behavior in the flow domain.

The solutions and interactive behavior of solid boundaries with fluid or interaction between the layers of the fluid while flowing are visualized using some CFD techniques. CFD helps replace these differential equations of fluid flow into numbers, and these numbers are beneficial in time and/or space which enable a numerical picture of the complete fluid flow. CFD is powerful in examining a system's behavior, beneficial, and more innovative in designing a system [2]. Also, It is efficient in exploring the system's performance metrics, whether it is for the yielding higher profit margins or in enhancing operational safety, and in various advantageous features [3].

Nowadays, CFD techniques are usually applied in various fields [4–8] i.e. car design, turbomachinery, ship design, and aircraft manufacturing. Moreover, it is beneficial in astrophysics, biology, oceanography, oil recovery, architecture, and meteorology. Numerous numerical Algorithm and software have been developed to perform CFD analysis. Due to the recent advancement in computer technology, numerical simulation for physically and geometrically complex systems can also be evaluated using PC clusters. Large scale simulations in different fluid flow on grids containing millions and trillions of elements can be achieved within a few hours via supercomputers. However, it is completely incorrect to think that CFD describes a mature technology, there are numerous open questions related to heat transfer, combustion modeling, turbulence, and efficient solution methods or discretization methods, etc. The coupling between CFD and other disciplines required further research, therefore, the main goal of this issue is to fill an essential gap that is greatly missed in this field. We sincerely hope that this issue will be beneficial to the readers to present the recent findings in the field and shed some light on the industrial sector.

Rafique et al. [9] used Buongiorno model to discuss the Casson nanofluid boundary layer flow through an inclined surface under the impact of Dufour and Soret. This nonlinear model is beneficial to understand the mechanism of heat and mass transfer by contemplating various essential features of the proposed boundary layer. Further, the Keller-box technique has been used to simulate the results. The results show that the Dufour effect has a strong impact on the temperature profile and

Edited and reviewed by: José S. Andrade Jr, Federal University of Ceara, Brazil

#### \*Correspondence:

M. M. Bhatti mmbhatti@sdust.edu.cn; mubashirme@yahoo.com

#### Specialty section:

This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics

> Received: 09 August 2020 Accepted: 11 September 2020 Published: 01 October 2020

#### Citation:

Bhatti M.M, Marin M, Zeeshan A and Abdelsalam SI (2020) Editorial: Recent Trends in Computational Fluid Dynamics. Front. Phys. 8:593111. doi: 10.3389/fphy.2020.593111

**5**

that the thermophoresis produces an inverse impact on the concentration profile as compared with the temperature profile.

Shah et al. [10] investigated the CVFEM simulation to determine the nanoparticle's migration toward a permeable domain. The considered fluid model contains aluminum oxide nanoparticles. Darcy law, thermal radiation, Lorentz force, and shape factor. The proposed approach is beneficial for the two common schemes of CFD. In the proposed study, it was found that higher convection occurs due to the great influence of shape factor. According to the authors' simulation, it was shown that the magnetic field and temperature gradient have an inverse relationship. Later, Shah et al. [11] studied the behavior of couple stress fluid and nonisothermal convection with magnetic effects over a nonlinear sheet. Analytical simulation with the help of homotopy analysis method has been proposed for the solutions. According to their study, they found that the primary velocity faces significant resistance during the flow. In their proposed simulation, they noticed that magnetic effects produce resistance in the angular velocity, but enhances the temperature profile. Also, the Grashof number and Hall effects show a positive response to the temperature profile.

Shah et al. [12] contemplated the Mohand decomposition scheme to examine the Kortewege–De Vries equations. The fractional derivatives are expressed by Caputo fractional derivative operator. The validation and effectiveness of this scheme have been determined using numerical examples for integer order and fractional problems. According to their results, they concluded that the proposed scheme is easily adaptable, straightforward, and beneficial to solve nonlinear problems.

Irfan et al. [13] investigated the magnetized nanofluid motion with variable features propagating through a radiatively stretching sheet. Their proposed scheme was a numerical shooting method and the bvp4c built-in command in MATLAB. It was noticed that the thermophoresis, thermal conductivity, radiation parameter, and Brownian motion boost the thermal boundary layer. Further, in the proposed simulation, it was found that the Prandtl number suppresses the thermal profile. On the other hand, Brownian motion and Lewis numbers were seen to cause a strong influence on concentration profile, whereas the thermophoretic force was seen to produce and opposite effects. Later, Irfan et al. [14] used computational formulation, i.e., simplified finite difference scheme to establish and discuss the effects of porosity, thermal radiation, a magnetic and electric field with heat generation and absorption. A comparative study is also given using the simplified finite difference scheme and bvp4c where it was noticed that the model has a higher convergence rate.

Shafiq et al. [15] examined and discussed the motion of carbon nanotubes (CNTs) (single- and multi-walled) over a Riga plate. The Riga plate is filled with water as a base fluid. They used the Marangoni model for the fully developed electromagnetohydrodynamics flow. They proposed homotopy analysis method for the graphical and numerical outcomes. They noticed that multi-walled CNTs have higher velocity as compared with single-walled CNTs. They found similar outcomes of the magnetic field on temperature as already done by Shah et al. [11].

Bilal et al. [16] used a similar scheme used by Rafique et al. [9] to examine flow behavior betwixt a pair of rotating disks. They used the theory of the Cattaneo–Christov and Darcy model to formulate the proposed formulation. Further, Karman transformations have been used to model the mathematical modeling and numerical outcomes presented using the finite difference approach. They found that a higher Reynolds number produces resistance in the radial and axial velocities at the lower disk as compared with the upper disk. Further, the thermal profile was reduced due to the strong impact of the Prandtl number. At the lower disk, the shear drag coefficient diminishes while at the upper disk, the wall shear coefficient increases. Later, Ullah et al. [17] considered a similar geometry [16] with a three-dimensional Darcy–Forchheimer model and nanofluid flow. A computational shooting scheme was used to operate the proposed formulation. They found that the Darcy–Forchheimer model effects are negligible on the concentration and temperature profile.

Ahmed et al. [18] analyzed the concealed behavior of thermally radiative and magnetically influenced <sup>c</sup>Al2O3–H2O and Al2O3–H2O nanofluid flow through a wedge. Combined simulation of shooting and RK scheme was used to evaluate the numerical outcomes. Their simulation shows that the Hartree pressure gradient significantly enhances the nanofluids velocity. The proposed composition of <sup>c</sup>Al2O3–H2O and Al2O3–H2O becomes denser due to the strong impact of volume fraction and accordingly opposes the velocity field. The thermal profile <sup>c</sup>Al2O3–H2O and Al2O3–H2O rises for higher volume fraction.

Ahmed and Khan [19] examined the mechanism of sodiumalginate (C6H9NaO7) through a vertical heated plate with acceleration. Further, they contemplated the effects of convection and discussed the entropy generation. Laplace transforms with a combination of integral transforms that were used to generate the exact results. It was concluded that the maximal entropy can be achieved by taking higher values of Brinkmann number, fluid parameter, and Grashof number. It was also noticed that the Bejan number can also be maximal if the Prandtl number is high. The proposed fluid model reveals a dual impact.

Bhatti et al. [20] performed a theoretical analysis of the blood flow under the suspension of nanoparticles and microorganisms through an anisotropic artery in a sinusoidal form. The authors investigated a nonlinear Sutterby fluid model as blood to examine the rheological effects. A perturbation approach was used to elaborate on the series solutions. In their analysis, it was found that the non-Newtonian effects are in favor to resist the flow. Further, they noticed that the wall shear stress diminishes due to the stenosis, nanoparticle, and thermal Grashof number. Moreover, The Peclet number was found to create resistance in the microorganism profile. The results of this study play a significant role in biomedical engineering. Riaz et al. [21] presented a study that is beneficial for the urinary tract infections when the flow is sinusoidal. This analysis is essential to examine white particles occurring in the urine. They investigated the flow in a curved configuration with flexible walls and filled with particles in a fluid. A lubrication theory and perturbation approach was used to formulate the governing equations. Further, they also carried out the numerical results for the pressure along the whole channel.

Alzahrani et al. [22] investigated the magnetohydrodynamics of a 3D flow through a rotating permeable conduit under the effect of Dufour and Soret and viscous dissipation. A viscous electrically-conducting fluid is considered upon which applied a magnetic field. Suitable transformations are used to transform from a nonlinear partial differential system of equations to an ordinary system of equations after which results were computed numerically using the shooting method. Then the pertinent parameters affecting the physical variables of the flow field have been thoroughly investigated.

Sanni et al. [23] studied the MHD flow of an incompressible Maxwell fluid flow induced by a quadratic stretching sheet through a 2D boundary layer. A variable magnetic field was applied to the flow with heat transfer, thermal radiation, and viscous dissipation. The system of partial differential equations has been transformed into ordinary differential equations (ODEs) by using some similarity variables. Numerical results have been achieved to find solutions to the energy and momentum equations in a closed-form.

Ahmed et al. [24] studied the peristaltic micropolar fluid flow influenced upon by heat and mass transfer with the magnetic field. The system of governing equations has been presented using a curvilinear coordinate system where they were further reduced using a lubrication approximation. Solutions were then derived by implementing the finite difference method.

Khan et al. [25] explored the thermal Eyring–Powell nanoliquid with triple diffusion via a periodic-moving system. A combination of some important parameters, such as the porosity parameter and magnetic effect, was also discussed.

### REFERENCES


The Buongiorno's nanofluid theory was investigated through the thermophoretic and Brownian motion effects. Further, the homotopy algorithm was used in order to analyze the fluid flow in a non-dimensional form.

Karuppusamy et al. [26] examined an entropy generation of a nanofluid of third-order with slip effect. The flow investigated was caused by a stretchable sheet through a porous plate under the influence of thermal radiation. Several other influential effects were taken into accounts such as the non-Fourier heat flux, convective surface boundary, and nanoparticle concentration on zero mass flux conditions. Similarity variables have been used in order to solve the governing physical system of equations and modify it into a nonlinear system of ODEs. Results were obtained using the usual homotopy algorithm to discuss the outcomes of the analysis.

### AUTHOR CONTRIBUTIONS

MMB and MM drafted the first version of the editorial. AZ and SIA revised the first draft and made contributions about papers they edited.

### FUNDING

MMB was supported by the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province (Nonlinear Sciences Research Team).

shape factor and thermal transport in a porous media using CVFEM. Front Phys (2019) 7:2396. doi:10.3389/fphy.2019.00164


variable thermal features. Front Phys (2020) 8:246. doi:10.3389/fphy. 2020.00246

26. Karuppusamy L, Muhiuddin G, Alanazi AM, Alkhaldi AH, Rajan S. Entropy optimization of 3rd-grade nanofluid slip flow embedded in a porous sheet with zero mass flux and non-Fourier heat flux model. Front Phys (2020) 8:250.

Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Bhatti, Marin, Zeeshan and Abdelsalam. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Numerical Solution of Casson Nanofluid Flow Over a Non-linear Inclined Surface With Soret and Dufour Effects by Keller-Box Method

Khuram Rafique<sup>1</sup> , Muhammad Imran Anwar 1,2,3, Masnita Misiran<sup>1</sup> , Ilyas Khan<sup>4</sup> \*, S. O. Alharbi <sup>5</sup> , Phatiphat Thounthong<sup>6</sup> and K. S. Nisar <sup>7</sup>

*<sup>1</sup> School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Malaysia, <sup>2</sup> Department of Mathematics, Faculty of Science, University of Sargodha, Sargodha, Pakistan, <sup>3</sup> Higher Education Department (HED) Punjab, Lahore, Pakistan, <sup>4</sup> Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>5</sup> Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al Majmaah, Saudi Arabia, <sup>6</sup> Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, Renewable Energy Research Centre, King Mongkut's University of Technology North Bangkok, Bang Sue, Thailand, <sup>7</sup> Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Al Dawasir, Saudi Arabia*

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Devendra Kumar, University of Rajasthan, India Mohammad Rahimi Gorji, Ghent University, Belgium Mohammad Mehdi Rashidi, Tongji University, China*

> \*Correspondence: *Ilyas Khan ilyaskhan@tdtu.edu.vn*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *08 July 2019* Accepted: *06 September 2019* Published: *11 October 2019*

#### Citation:

*Rafique K, Anwar MI, Misiran M, Khan I, Alharbi SO, Thounthong P and Nisar KS (2019) Numerical Solution of Casson Nanofluid Flow Over a Non-linear Inclined Surface With Soret and Dufour Effects by Keller-Box Method. Front. Phys. 7:139. doi: 10.3389/fphy.2019.00139* In this article, the effects of a Casson Nanofluid boundary layer flow, over an inclined extending surface with Soret and Dufour, is scrutinized. The model used in this study is based on the Buongiorno model of the thermal efficiencies of the fluid flows in the presence of Brownian motion and thermophoresis properties. The non-linear problem for Casson Nanofluid flow over an inclined channel is modeled to gain knowledge on the heat and mass exchange phenomenon, by considering important flow parameters of the intensified boundary layer. The governing non-linear partial differential equations are changed to non-linear ordinary differential equations and are afterward illustrated numerically by the Keller-Box scheme. A comparison of the established results, if the incorporated effects are lacking, is performed with the available outcomes of Khan and Pop [1] and recognized in a nice settlement. Numerical and graphical results are also presented in tables and graphs.

Keywords: casson nanofluid, MHD, power law fluid, soret effect, dufour effect, inclined surface

### INTRODUCTION

In recent times, nanofluid has accomplished an incredible position among scientists because of its dynamic thermal performance and notable potential in the number of heat transfers without any pressure drops. Nanofluid is a formula of various nanoparticles, containing Al2O3, Cu, CuO, in a base liquid, for example, oil, water, ethylene glycol, and so forth. It is investigated through examination that the thermal conductivity of base fluid is usually not exactly the same as the nanofluid Choi and Eastman [2]. Nanofluid is used as a working fluid (base fluid) due to its high thermal conductivity. Buongiorno [3] examined the causes that perform a key job in the advancement of nanofluid's thermal conductivity. He perceived that the Brownian movement and thermophoresis effects in conventional fluid play an important role to enhance the thermal conductivity of the fluid. Nield and Kuznetsov [4] considered the result of thermophoresis and Brownian movement on the boundary layer stream. The steady flow of nanofluid on an extending sheet was examined by Khan and Pop [1]. Anwar et al. [5] presented the Brownian movement and thermophoresis impact on the heat and mass exchange of nanofluids over a non-linear extending sheet. Suriyakumar and Devi [6] examined the nanofluid flow over a slanted sheet. Ziaei-Rad et al. [7] investigated a similar solution of nanofluid stream on a slanted surface. Thumma et al. [8] discussed the nanofluid flow on a slanted plate by incorporating the heat source. Govindarajan [9] discussed the flow of nanofluid over a slanted sheet by incorporating a non-uniform temperature. Khan et al. [10] illustrated the heat and mass transfer of MHD Jeffery nanofluid flow over an inclined sheet. Nanofluid flow with radiation effects on a slanted surface was examined by Chakraborty [11]. Recently, different scholars investigated the nanofluid flow on different models, as some of them are given in references [12–18].

The investigation of boundary layer flow and heat exchange on a stretching surface has been considered by various experts due to its immense mechanical and designing applications in the field of industry, and engineering, for example, strengthening and tinning of copper wires, assembling plastic and elastic sheets, non-stop cooling and fiber turning, expulsion of polymer, wire drawing, food processing, and paper, and so forth. Boundary layer flow on a steady surface was first investigated by Sakiadis [19]. Additionally, Crane [20] considered the closed structure solution of boundary layer flow on an extending sheet. Ali et al. [21] investigated the conjugate effects of heat and mass exchange on MHD free convection flow over an inclined plate. Ramesh et al. [22] investigated the boundary layer flow over the slanted sheet with convective boundaries. MHD free convection dissipative fluid stream past over an inclined sheet was investigated by Malik [23]. The boundary layer flow on a slanted sheet through convective boundaries was discussed by Ramesh et al. [24]. Griffiths [25] investigated the non-Newtonian boundary layer flow over a slanted sheet. Soret and Dufour effect over a slanted plate was discussed by Pal and Mondal [26]. Pandya and Shukla [27] investigated the unsteady MHD flow over a slanted surface by taking viscous dissipations. Thermal radiation impacts are important in solar plants [28].

In 1959, Casson offered the Casson fluid model for the flow of viscoelastic liquids. Casson fluid is a shear thinning fluid which should have zero viscosity at an infinite rate of shear and infinite viscosity at zero rates of shear, yielding stress under which no flow takes place. Some examples of Casson fluid are, honey, jelly, sauce, soup etc. [29] Ali et al. [30] examined the Casson fluid flow on a slanted sheet by incorporating the Soret-Dufour effects. Manideep et al. [31] studied the Casson fluid flow on vertically inclined sheets. Shamshuddin et al. [32] numerically investigated the effect of chemical reaction on Casson fluid flow on a slanted plate. Casson fluid flow over a slanted plate calculated by Vijayaragavan and Kavitha [33]. Prasad et al. [34] investigated the Casson fluid flow over an inclined sheet by considering the hall current. Jain and Parmar [35] studied the inclined Casson fluid flow on a permeable sheet. Sailaja et al. [36] studied the Casson fluid flow on a vertical sheet by incorporating the angle effect. Rawi et al. [37] discussed the Casson fluid flow over a slanted sheet by considering nanoparticles. Rauju et al. [38] discussed the Casson fluid flow on a vertically slanted sheet. The Casson fluid model is more compatible with blood flow simulation [39, 40].

Persuaded by the above referred literature review, and due to the growing needs of non-Newtonian nanofluid flows in industry and engineering areas, the present work focuses on the Casson nanofluid flow over a non-linear inclined stretching surface with Soret and Dufour effects. Casson nanofluid is more helpful for cooling and friction reducing agents compared to Newtonian based nanofluid flow [15]. To the best of the author's knowledge, the solution of the Casson nanofluid flow over a non-linear inclined stretching surface with radiation, as well as Soret and Dufour effects with the Keller-Box method, has not yet been reported. The model under consideration is newly developed from Khan and Pop [1] and results obtained from the current study are new. In this work, we found that the Dufour effect reduces the Nusselt and Sherwood number due to Soret impact. A non-linear form of radiative heat exchange also enhances the fluid temperature. This study is very useful in nuclear reactors, MHD generators, and in geothermal energy. In the future, it can be extended on an exponentially inclined stretching surface.

#### PROBLEM FORMULATION

A steady, two-dimensional boundary layer flow of Casson Nano fluid over a non-linear slanted extending surface on angle γ is considered. The extending and free stream velocities are taken as, u<sup>w</sup> (x) = ax<sup>m</sup> and u<sup>∞</sup> (x) = 0. Where, "x" is the coordinate dignified in the direction of the extending surface with "a" supposed constant. An external transverse magnetic field is assumed normal to the flow path. The Brownian motion and thermophoresis effects are considered. The temperature T and Nano particle fraction C at the wall take the constant values T<sup>w</sup> and Cw, while the ambient forms for the nanofluid mass and temperature fractions C∞ and T∞ are accomplished as y approaches to immensity, as shown in **Figure 1**.

The flow equations for this study [1] are given by:

$$\begin{aligned} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} + \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{y}} &= \mathbf{0} \\ \boldsymbol{u} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} + \nu \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{y}} &= \nu (1 + \frac{1}{\beta}) \frac{\partial^2 \boldsymbol{u}}{\partial \boldsymbol{y}^2} + \mathbf{g} \left[ \boldsymbol{\beta}\_t \left( T - T\_\infty \right) \right] \end{aligned} \tag{1}$$

$$+\beta\_{\rm c}(C-C\_{\infty})\big]\cos\chi-\frac{\sigma B\_{0}^{-2}(\chi)\mu}{\rho}\tag{2}$$

$$
\mu \frac{\partial T}{\partial \boldsymbol{\omega}} + \nu \frac{\partial T}{\partial \boldsymbol{\chi}} = \alpha \frac{\partial^2 T}{\partial \boldsymbol{\chi}^2} - \frac{1}{(\delta \boldsymbol{\varepsilon})\_f} \frac{\partial q\_r}{\partial \boldsymbol{\chi}} + \pi \left[ D\_B \frac{\partial C}{\partial \boldsymbol{\chi}} \frac{\partial T}{\partial \boldsymbol{\chi}} \right]
$$

$$
+ \frac{D\_T}{T\_{\infty}} \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right)^2 \Bigg[ + \frac{D\_T K\_T}{C\_s C\_p} \frac{\partial^2 C}{\partial \boldsymbol{\chi}^2},
\tag{3}
$$

$$
\mu \frac{\partial \mathcal{C}}{\partial \boldsymbol{\omega}} + \nu \frac{\partial \mathcal{C}}{\partial \boldsymbol{\chi}} = D\_B \frac{\partial^2 \mathcal{C}}{\partial \boldsymbol{\chi}^2} + \frac{D\_T K\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial \boldsymbol{\chi}^2}, \tag{4}
$$

Here, the Rosseland approximation (for radiation flux) is defined as:

$$qr = -\frac{4\sigma^\*}{3k^\*}^\* \frac{\partial T^4}{\partial \nu},\tag{5}$$

Where, σ ∗ is the Stefan-Boltzmann constant and k ∗ is the mean absorption coefficient. It is assumed that the temperature difference between the free steam T∞ and local temperature T is small enough, expanding T 4 in Taylor series about T∞ and neglecting higher order terms for:

$$T^4 \cong 4T\_{\infty}^3 T - 3T\_{\infty}^4,\tag{6}$$

Using Equations (5) and (6) the Equation (3) converts into:

$$
\mu \frac{\partial T}{\partial \boldsymbol{x}} + \nu \frac{\partial T}{\partial \boldsymbol{y}} = (\alpha + \frac{16\boldsymbol{\sigma}^{\ast}}{3k^{\ast} \left(\boldsymbol{\delta}\boldsymbol{c}\right)\_{f}}) \frac{\partial^{2} T}{\partial \boldsymbol{y}^{2}} + \pi \left[D\_{B} \frac{\partial C}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{y}}\right]
$$

$$
+ \frac{D\_{T}}{T\_{\infty}} \left(\frac{\partial T}{\partial \boldsymbol{y}}\right)^{2}\Big] + \frac{D\_{T}K\_{T}}{C\_{\text{s}}C\_{p}} \frac{\partial^{2} C}{\partial \boldsymbol{y}^{2}},\tag{7}
$$

Where u and v are the velocity components in the x and y directions, respectively, g is the acceleration due to gravity, B<sup>0</sup> is the uniform magnetic field strength, σ is the electrical conductivity, u is the viscosity, δ<sup>f</sup> is the density of the base fluid, δ<sup>p</sup> is the density of the nanoparticle, β is the Casson parameter, β<sup>t</sup> is the coefficient of thermal expansion, β<sup>c</sup> is the coefficient of concentration expansion, D<sup>B</sup> is the Brownian diffusion coefficient and D<sup>T</sup> is the thermophoresis diffusion coefficient, k is the thermal conductivity, (δc)<sup>p</sup> is the heat capacitance of the nanoparticles, (δc)<sup>f</sup> is the heat capacitance of the base fluid, α = k (δc)<sup>f</sup> is the thermal diffusivity parameter, τ = (δc)<sup>p</sup> (δc)<sup>f</sup> is the ratio between the effective heat capacity of the

nanoparticle and heat capacity of the fluid.

The subjected boundary conditions are:

$$\begin{aligned} \boldsymbol{u} &= \boldsymbol{u}\_{\boldsymbol{w}} \left( \boldsymbol{\varepsilon} \right) = a \boldsymbol{x}^{\boldsymbol{m}}, \ \boldsymbol{\nu} = \boldsymbol{0}, \ \boldsymbol{T} &= T\_{\boldsymbol{w}}, \ \boldsymbol{C} = \boldsymbol{C}\_{\boldsymbol{w}} \quad \text{at} \ \boldsymbol{y} = \boldsymbol{0}, \\ \boldsymbol{u} &\to \boldsymbol{u}\_{\infty} \left( \boldsymbol{x} \right) = \boldsymbol{0}, \ \boldsymbol{\nu} &\to \boldsymbol{0}, \ \boldsymbol{T} &\to \boldsymbol{T}\_{\infty}, \ \boldsymbol{C} &\to \boldsymbol{C}\_{\infty} \quad \text{at} \ \boldsymbol{y} &\to \infty, \ \{ \boldsymbol{8} \} \end{aligned}$$

The non-linear partial differential equations are reduced into non-linear ordinary differential equations. For that purpose, the stream function ψ = ψ(x, y) is defined as:

$$
\mu = \frac{\partial \Psi}{\partial y}, \nu = -\frac{\partial \Psi}{\partial x}, \tag{9}
$$

continuity (Equation 1) is satisfied identically.

The similarity transformations are defined as:

$$\psi \,\,=\sqrt{\frac{2\nu a \varkappa^{m+1}}{m+1}} f(\eta), \,\,\theta(\eta) = \frac{T - T\_{\infty}}{T\_w - T\_{\infty}},$$

$$\phi(\eta) = \frac{C - C\_{\infty}}{C\_{\text{w}} - C\_{\infty}}, \eta = \chi \sqrt{\frac{(m+1)a\omega^{m-1}}{2\nu}}.\tag{10}$$

On substituting (Equation 7), system of Equations (2), (3), and (7) reduce to the following non-linear ordinary differential equations:

$$\begin{aligned} (1+\frac{1}{\beta})f^{\prime\prime} + f^{\prime\prime} - \left(\frac{2m}{m+1}\right)f^2 + \frac{2}{m+1}\left(\lambda\theta - \delta\phi\right)\cos\chi\\ -\left(\frac{2M}{m+1}\right)f^{\prime} = 0 \end{aligned} \tag{11}$$

$$PrN\theta'' + f\theta' + N\_b\phi'\theta' + N\_l\theta'^2 + D\_f\phi'' = 0\tag{12}$$

$$
\phi'' + \mathcal{L}ef\phi' + \mathcal{S}r\mathcal{L}e\theta'' = 0\tag{13}
$$

Where,

$$\lambda = \frac{Gr\_{\mathbf{x}}}{Re}, \delta = \frac{Gc}{Re}, M = \frac{\sigma \, B\_0^{\; 2} \, (\mathbf{x})}{a\rho}, Le = \frac{\nu}{D\_B} \, Pr = \frac{\nu}{\alpha},$$

$$N\_b = \frac{\tau D\_B \, (C\_\mathbf{w} - C\_\infty)}{\nu}, N\_t = \frac{\tau D\_t \, (T\_\mathbf{w} - T\_\infty)}{\nu \, T\_\infty},\tag{14}$$

$$Gr\_{\mathbf{x}} = \frac{\mathbf{g}\beta\_{\mathbf{f}}\left(T\_{\mathbf{w}} - T\_{\infty}\right)\mathbf{x}}{a\upsilon}, \text{Re} = \frac{\mu\_{\mathbf{w}}\mathbf{x}}{\upsilon}, \text{G}\mathbf{c}\_{\mathbf{x}} = \frac{\mathbf{g}\beta\_{\mathbf{c}}\left(\mathbf{C}\_{\mathbf{w}} - \mathbf{C}\_{\infty}\right)\mathbf{x}}{a\upsilon}, \text{g}$$

$$\begin{split} Pr\_{N} &= \frac{1}{Pr} (1 + \frac{4}{3}N), N = \frac{4\sigma^{"\prime} T\_{\infty}^{3}}{\alpha k^{\*}}, \\ D\_{f} &= \frac{D\_{T} K\_{T} \left(C\_{\text{w}} - C\_{\infty}\right)}{\nu C\_{\text{s}} C\_{p} \left(T\_{\text{w}} - T\_{\infty}\right)}, Sr = \frac{D\_{T} K\_{T} \left(T\_{\text{w}} - T\_{\infty}\right)}{\nu T\_{\infty} \left(C\_{\text{w}} - C\_{\infty}\right)}, \end{split}$$

Here, primes denote the differentiation with respect to η, λ Buoyancy parameter, δ Solutal buoyancy parameter, M is the magnetic parameter called Hartmann number, ν is the kinematic viscosity of the liquid, Pr denotes the Prandtl number, Le denotes the Lewis number, N<sup>b</sup> denotes the Brownian motion parameter, N<sup>t</sup> indicates thermophoresis parameter, and N is the radiation parameter.

The corresponding boundary conditions are transformed to:

$$\begin{aligned} f\left(\eta\right) &= 0, \quad f'\left(\eta\right) = 1, \; \theta\left(\eta\right) = 1, \; \phi\left(\eta\right) = 1 \quad \text{at } \eta &= 0, \\\ f'\left(\eta\right) &\to 0, \quad \theta\left(\eta\right) \to 0, \; \phi\left(\eta\right) &\to 0 \quad \text{as } \ \eta &\to \infty \text{(15)} \end{aligned}$$

The skin friction, Sherwood number and Nusselt number for the present problem are defined as:

$$Nu\_{\mathbf{x}} = \frac{\varkappa q\_{\mathbf{w}}}{k \left(T\_{\mathbf{w}} - T\_{\infty}\right)}, Sh\_{\mathbf{x}} = \frac{\varkappa q\_{\mathbf{m}}}{D\_{\text{B}} \left(C\_{\mathbf{w}} - C\_{\infty}\right)}, C\_{f} = \frac{t\_{\mathbf{w}}}{\mu\_{\mathbf{w}}^{2} \rho\_{f}} \tag{16}$$

Where,

$$q\_{\rm w} = -k \frac{\partial T}{\partial \boldsymbol{\chi}}, q\_{\rm m} = -D\_{\rm B} \frac{\partial C}{\partial \boldsymbol{\chi}}, \boldsymbol{\tau}\_{\rm w} = \mu (1 + \frac{1}{\beta}) \frac{\partial \boldsymbol{\mu}}{\partial \boldsymbol{\chi}} \text{ at } \boldsymbol{\chi} = 0 \tag{17}$$

The associated expressions of dimensionless reduced Nusselt number −θ ′ (0), reduced Sherwood number −φ ′ (0), and skin friction coefficient Cfx are defined as

$$\left(-\theta'(0) = \frac{N\mu\_{\text{x}}}{\left(1 + \frac{4}{3}N\right)\sqrt{\frac{m+1}{2}R\varepsilon}}, -\phi'(0) = \frac{S\hbar\_{\text{x}}}{\sqrt{\frac{m+1}{2}R\varepsilon}}, C\_{\text{fx}} = C\_{\text{f}}\sqrt{\frac{2}{m+1}R\varepsilon} \tag{18}$$

Where, Re = uwx ν is the local Reynolds number.

The converted non-linear differential (Equations 9–12) with the boundary conditions (14) are elucidated by the Keller box scheme consisting of the steps; finite-differences technique,

TABLE 1 | Comparison of the reduced Nusselt number −θ ′ (0) and the reduced Sherwood number −θ ′ (0) when *M*, *N* = 0, β → ∞, δ, *Sr*, *Df*, λ = 0, *m* = 1, *Pr* = *Le* = 10 and γ = 90◦ .


Newton's scheme, and the block elimination process clearly explained by Anwar et al. [41]. This method has been extensively applied and it looks to be the most flexible compared to common techniques. It has been presented as much quicker, easier to program, more efficient and easier to practice. Currently, there are many alternative techniques to solve such types of problems [42].

#### RESULTS AND DISCUSSION

This portion of study manages the calculated results of converted non-linear ordinary differential (Equations 8–10) with boundary conditions (12) elucidated via Killer-box method. Concerning numerical results of physical parameters, including Brownian motion parameter Nb, radiation parameter N, thermophoresis parameter Nt, magnetic factor M, buoyancy factor λ, solutal buoyancy factor δ, inclination factor γ , Prandtl number Pr, Lewis number Le, Dufour effect Df , Soret effect Sr, non-linear stretching parameter m, and Casson fluid parameter β, several figures and tables are prepared. In **Table 1**, in the deficiency of Dufour effect Df , Soret effect Sr, buoyancy parameter λ, solutal buoyancy constraint δ, magnetic factor M, radiation parameter N and non-linear stretching parameter 'm' K with γ = 90 ◦ when Casson parameter β → ∞ produces a reduced Nusselt number −θ ′ (0), the reduced Sherwood number −φ ′ (0) equate with existing outcomes of Khan and Pop [1]. The consequences establishes a brilliant settlement. The effects of −θ ′ (0), −φ ′ (0), and Cfx(0), against changed values of involved physical parameters Nb, Nt, M, β, λ, δ, γ , m, Le, Df , Sr, N, and Pr are shown in **Table 2**. From **Table 2** is can be clearly seen that −θ ′ (0) declines for growing values of Nb,Nt, M,Le, Df , m, γ , β, N, and increases with enhancing numerical values of λ, δ, Sr, and Pr. Moreover, it is perceived that −φ ′ (0) enhanced with the larger values of Nb, λ, δ, Nt, Le, N, Pr, Df , and drops for bigger values of M, m, β, Sr, and γ . Physically, the thermal boundary layer thickness enhances, as the Brownian


*Bold values show variation in that parameter.*

parameter Nb increases impacting a large extent of the fluid. Moreover, for large values of thermophoresis effects, the Nusselt number decreases and the Sherwood number increases because, the thermal boundary layer becomes thicker due to deeper diffusion penetration into the fluid. On the other hand, Cfx(0) rises with the growing values of Nb, Le, M, β, Pr, γ , m, Df , and drops with the higher values of, λ, δ, Nt, N, and Sr. The current results are novel and show the impact of the buoyancy parameter, solutal buoyancy parameter and inclination parameter impacts on the driven flow in the presence of Soret and Dufour effects on power-law fluid which is currently unavailable in the literature.

An image of the effect of factor M on velocity profile is portrayed in **Figure 2**. According to **Figure 2**, by improving the constraint M, the velocity outline reduces. Since the magnetic field produces Lorentz force, by slowing down the speed of the liquid. On the other hand, the velocity profile slows down for large values of the non-linear stretching parameter m, shown in **Figure 3**. Physically, the momentum boundary layer thickness reduces for higher values of m.

The impact of the buoyancy factor is shown in **Figure 4**. It is observed that the velocity profile rises by improving the buoyancy limit. It is due to the fact that buoyancy effect increases the strength of the fluid flow whereby the boundary layer thickness and velocity enhances. **Figure 5** indicates that the velocity outline increases by enhancing the solutal buoyancy factor. Physically, the buoyancy parameter reduces the viscous forces whereby the velocity upturns. In addition, the opposite impact can be seen in temperature and concentration profiles for large values of δ.

**Figure 6** interprets the significance of inclination factor γ on the velocity outline. It is perceived in **Figure 6** that the velocity outline runs down by enhancing the values of γ . Moreover, the circumstances indicate that the maximum gravitational force acts on flow in the case of γ = 0, because in this state the sheet will be vertical. On the other hand, for γ = 90<sup>0</sup> , the sheet will

be horizontal which causes the decline in velocity profile as the power of the bouncy forces drop.

The effect of the Casson parameter on the velocity parameter is presented in **Figure 7**. It is observed that for large values of the Casson parameter, the velocity profile decreases. The reason behind this behavior is that by increasing the values of the Casson parameter, β increases the fluid viscosity i.e., reducing the yield stress. Therefore, the momentum boundary layer thickness reduces [43].

**Figures 8**, **9** show the effect of the Brownian motion on the temperature and concentration profiles, respectively. The temperature sketch enlarges on enlarging Nb; on the other hand, concentration distribution enlightens a dissimilar style. Physically, the boundary layer heats up due to the development in the Brownian motion which is inclined to transport nanoparticles from the extending sheet to the motionless liquid. Therefore, the absorption nanoparticle lessens. **Figures 10**, **11** present temperature and concentration profiles for altered values

of thermophoresis parameters Nt. It is perceived that both temperature and concentration contours upsurge by growing the thermophoresis parameter because thermophoresis causes the small particles to compel away from a warm surface to the cold one.

**Figure 12** reveals that by growing the values of the Prandtl number parameter Pr, the temperature profile drops because the thermal boundary layer viscosity declines when growing the Prandtl number Pr. In short an upturn in the Prandtl number Pr means a deliberate amount of thermal dispersion.

**Figure 13** shows that the temperature profile becomes large for larger values in parameter D<sup>f</sup> . This can be justified as an increase in the Dufour parameter, causing an increase in the concentration gradient, resulting in a mass diffusion taking place more rapidly. In this way, the rate of energy transfer related to the particles becomes higher. That is why the temperature

profiles enhance. The impact of the Soret number on the concentration profile is observed similar to the impact of the Dufour number on the temperature profile. As parameter Sr increases, the concentration profile increases as displayed in **Figure 14**. Additionally, **Figure 15** indicates a temperature profile enhanced for large values of N.

## CONCLUSIONS

This study explored the heat and mass exchange of Casson nanofluid flow over a non-linear slanted extending sheet. The numerical results are successfully obtained via the Keller-Box method and are finally performed with the resulting outcomes

of already published work [1]. The main findings of the current study are summarized as:


### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

#### AUTHOR CONTRIBUTIONS

The formulation was done by KR and MA. Similarity transformation was done by MM. Problem solved

#### REFERENCES


by IK and SA. Results plotted and discussed by PT and KN. All authors equally contributed in writing this manuscript.

#### ACKNOWLEDGMENTS

The author (SA) would like to thank the Deanship of Scientific Research (DSR) Majmaah University for supporting this work.


peristaltic wave. Therm Sci. (2018) **22:**2439–48. doi: 10.2298/TSCI16010 2287R


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Rafique, Anwar, Misiran, Khan, Alharbi, Thounthong and Nisar. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Study of the Couple Stress Convective Micropolar Fluid Flow in a Hall MHD Generator System

Zahir Shah<sup>1</sup> \*, Poom Kumam2,3,4 \*, Abdullah Dawar <sup>5</sup> , Ebraheem O. Alzahrani <sup>6</sup> and Phatiphat Thounthong<sup>7</sup>

*<sup>1</sup> Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>2</sup> KMUTT Fixed Point Research Laboratory, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>3</sup> KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>4</sup> Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan, <sup>5</sup> Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan, <sup>6</sup> Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia, <sup>7</sup> Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand*

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*R. Ellahi, University of California, Riverside, United States Arash Asadollahi, Southern Illinois University Carbondale, United States*

\*Correspondence:

*Zahir Shah zahir.sha@kmutt.ac.th Poom Kumam poomkumam@kmutt.ac.th*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *25 August 2019* Accepted: *16 October 2019* Published: *05 November 2019*

#### Citation:

*Shah Z, Kumam P, Dawar A, Alzahrani EO and Thounthong P (2019) Study of the Couple Stress Convective Micropolar Fluid Flow in a Hall MHD Generator System. Front. Phys. 7:171. doi: 10.3389/fphy.2019.00171*

The steady non-isothermal convective heat transfer in magnetohydrodynamic micropolar fluid flow over a non-linear extending wall is examined. The fluid flow is treated with strong magnetic field. The influence of magnetic field, Hall current, and couple stress are mainly focused in this work. The fluid flow problem is solved analytically. The impact of developing dimensionless parameters on primary, secondary, and angular velocity components and temperature profile are determined through graphs. The primary velocity component has reduced throughout the flow study. The greater magnetic parameter, Hall parameter and couple stress parameter have increased the secondary velocity component while the local Grashof number has reduced the secondary velocity component. The greater magnetic parameter and Hall parameter have reduced the angular velocity component. The greater magnetic parameter has increased the temperature profile while the Hall parameter and local Grashof number have decreased the temperature profile. The impact of developing dimensionless parameters on skin friction coefficient and local Nusselt number are determined through Tables.

Keywords: hall MHD generator system, convective heat transfer, magnetohydrodynamic, micropolar fluid, couple stress, hall current, HAM

### INTRODUCTION

The flow of non-Newtonian fluids has plentiful importance in industries and modern technology. Recently, the couple stress fluid among non-Newtonian fluid has acquired the exceptional position due to the spin field in the fluid. The elementary concept of couple stress was established by Stokes [1]. Khan et al. [2] deliberated the suggested model of couple stress fluid in a uniformly porous stretching channel. The axial velocity function heightens while the radial velocity function declines for escalating couple stress. The couple stress effect on heat transfer in four different nanofluids flows was determined by Farooq et al. [3]. Srinivasacharya et al. [4] explored the couple stress fluid flow. They originate that the couple stress parameter diminishes the fluid velocity and temperature

**22**

while heightens the concentration. Ramzan et al. [5] deliberated the couple stress fluid flow over extending sheet. It is found that velocity profiles along both directions are declined with the escalation in couple stress parameter. Also the fluid temperature escalated with viscous dissipation effect. Hayat et al. [6] determined the heat transmission rate in the couple stress flow over extending surface and originate that the heat transfer intensifies with the rising estimations of the couple stress. Over an extending sheet, the couple stress fluid flow was determined by Turkyilmazoglu [7]. It is concluded that over a stretching sheet the couple stress gives double solution while over shrinking sheet it gives triple solution. The unsteady couple stress fluid flow was determined by Awad et al. [8]. Here, the fluid velocity and temperature decline with heightened couple stress. Sreenadh et al. [9] examined the fluid flow with couple stress impact. Hayat et al. [10] analyzed the mass transfer in couple stress fluid with chemical reaction. Khan et al. [11] scrutinized the incompressible and unsteady couple stress fluid flow considering three dimensional cylindrical polar coordinate systems. Hayat et al. [12] measured the flow of couple stress nanofluid with convective conditions. The fluid temperature and concentration are increased with escalating couple stress parameter. The dissipation influence on couple stress nanofluid flow was determined by Ramzan [13]. The magnetic field impact on couple stress nanofluid flow was determined by Hayat et al. [14]. In this article, the authors determined that temperature of fluid flow up surged with the large estimation of couple stress parameter. With Cattaneo-Chritov heat flux Hayat et al. [15] deliberated the flow of couple stress nanofluid flow. They found that the velocity components are increased while the temperature is decreased with the couple stress parameter. Umavathi et al. [16] deliberated the laminar flow of couple stress fluid and heat transmission considering horizontal plates. Umavathi et al. [17] scrutinized the fluid flow with couple stress impact in between two infinite porous walls. They concluded that the fluid velocity and temperature are reduced in the boundary layer regime. Srinivasacharyulu et al. [18] observed the couple stress fluids flow over stretching walls. Zueco et al. [19] inspected the couple stress nanofluid in a rigid channel. Zakaria [20] deliberated the couple stress fluid under magnetic field impact. Ellahi et al. [21] determined the couple stress blood flow under the impact of activation energy and chemical reaction.

In recent times, the researchers have got interest in megnetohydrodynamic (MHD) owing to plentiful applications in industrial, engineering, and medical devices. Rudolf et al. [22] briefly reviewed the properties of magnetic field in the universe. The MHD nanofluid flow with chemical reaction was deliberated by Hayat et al. [23]. The fluid flow velocity is reduced with higher estimation of magnetic field, and temperature escalated with chemical reactions and Dufour influences. The heat transmission in the flow of MHD nanofluid over unsteady extending sheet was observed by Lin et al. [24]. The fluid flow velocity is reduced with heightens in magnetic field while the temperature of the fluid escalated. The heat transfer in the flow of MHD incompressible second-grade nanofluid was deliberated by Ramesh et al. [25]. The MHD nanofluid flow in a symmetric channel was probed by Reddy et al. [26]. The elementary study of micropolar fluid was introduced by Eringen [27]. Bég et al. [28] presented the applications of micropolar fluid flow. Uddin et al. [29] probed the MHD micropolar fluid with Hall effect. Here, interesting results are concluded. The velocity of the fluid heightens with the escalation in magnetic field while the temperature of the fluid reduces with higher estimation of magnetic field (i.e. M>2). Khan et al. [30] determined the radiation and inertial coefficient influences on the flow of nanofluid. The higher inertial coefficient, porosity parameter, and coupling parameter reduce the fluid velocity and the temperature heightens with the escalation in thermal radiation. Dawar et al. [31] deliberated the unsteady MHD nanofluid with viscous dissipation effect. Here, the authors originate that the fluid flow velocity reduces with escalation in magnetic field and the fluid flow temperature reduces with viscous dissipation impacts. Kumam et al. [32] probed the MHD Casson nanofluid flow. Shah et al. [33] deliberated the flow of MHD thin film fluid with radiation impact. The MHD Casson nanofluid flow in a cylindrical tube was considered by Ali et al. [34]. The nanofluid flow with Hall effect was studied by Shah et al. [35]. The MHD nanofluid flow with magnetic and electric fields, and Hall impacts was determined by Shah et al. [36]. Kumar et al. [37] investigated the MHD nanofluid with magnetic and heat sink/source impacts. Temple et al. [38] scrutinized the nanoparticles of ferromagnetic for their size and magnetic properties. Ellahi et al. [39] examined the MHD nanofluid flow with thermal conductivity. Asadollahi et al. [40] deliberated the phase change of a fluid in a square microchannel. The most relevant and new studied studies can be reads in Ellahi et al. [41–43], Bhatti et al. [44], Ameen et al. [45], Vo et al. [46], Ahmad et al. [47], Sheikholeslami et al. [48], Ali et al. [49], and Ullah et al. [50].

In view of the above mentioned literature survey, the authors are in position to examine the three-dimensional MHD micropolar fluid flow over an extending wall with couple stress, Hall current and viscous dissipation influences. Section of Problem Formulation agrees with problem formulation. In the section of Solution by HAM, the recommended model is solved by HAM. Results section includes the results of the problem and the section of Discussion of the problem is presented independently. The final observations are obtainable in the section of Conclusion.

#### PROBLEM FORMULATION

We assume the incompressible, steady, and electrically conducting couple stressed flow of micropolar fluid and heat transfer in the near wall zone of MHD Hall generator. The wall is considered as non-linearly stretching and concerned with x−direction (as shown in **Figure 1**). The magnetic field B<sup>0</sup> is functional in y−axis. In the presence of magnetic field, the Hall current influences the electrically conducting fluid. The flow of fluid develops to 3D due to the Hall current, which increases the force in z−direction. All properties of fluid are considered constant and isotropic.

The principal equations for the fluid flow can be written as [27, 28]:

$$\frac{\partial \nu}{\partial y} + \frac{\partial u}{\partial x} = 0,\tag{1}$$

$$\begin{split} \nu \frac{\partial \mu}{\partial \boldsymbol{\nu}} &+ \
u \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} = \nu \frac{\partial^2 \boldsymbol{u}}{\partial \boldsymbol{\nu}^2} - \nu' \frac{\partial^4 \boldsymbol{u}}{\partial \boldsymbol{\nu}^4} - \frac{B\_0}{\rho} \boldsymbol{I}\_z \\ &+ \
\g \boldsymbol{\beta} \,(\boldsymbol{T} - \boldsymbol{T}\_\infty) + \boldsymbol{K}\_1 \frac{\partial \boldsymbol{N}}{\partial \boldsymbol{\nu}}, \end{split} \tag{2}$$

$$\nu \frac{\partial \mathcal{w}}{\partial \boldsymbol{\chi}} + \mu \frac{\partial \mathcal{w}}{\partial \boldsymbol{\chi}} = \nu \frac{\partial^2 \mathcal{w}}{\partial \boldsymbol{\chi}^2} - \nu' \frac{\partial^4 \mathcal{w}}{\partial \boldsymbol{\chi}^4} + \frac{B\_0}{\rho} \mathcal{J}\_{\text{x}} \tag{3}$$

$$\frac{G\_1}{K\_2} \frac{\partial^2 N}{\partial y^2} = 2N + \frac{\partial \mu}{\partial y},\tag{4}$$

$$\begin{split} \nu \frac{\partial T}{\partial \boldsymbol{\nu}} &+ \, \mu \frac{\partial T}{\partial \boldsymbol{\omega}} = \frac{\kappa}{\rho c\_p} \frac{\partial^2 T}{\partial \boldsymbol{\nu}^2} \\ &+ \, \frac{\sigma \mu\_e B\_0^2 \lambda}{\rho c\_p \left(m^2 \dot{\boldsymbol{\kappa}}^2 + 1\right)} \left(\boldsymbol{\nu}^2 + \boldsymbol{\mu}^2\right), \end{split} \tag{5}$$

with

$$\begin{aligned} \mu = U &= \text{Pr}^{\boldsymbol{\eta}}, \; \nu = 0, \; \boldsymbol{w} = 0, \; N &= 0, \; T = T\_{\text{w}} = T\_{\infty} \\ &+ \; A \boldsymbol{x}^{\boldsymbol{\prime}} \text{ at } \boldsymbol{y} = 0, \\ \boldsymbol{u} &\to \; \mathbf{0}, \; \boldsymbol{w} \to \mathbf{0}, \; N &\to \; \mathbf{0}, \; T \to T\_{\infty} \; \text{at } \boldsymbol{y} \to \infty. \end{aligned} \tag{6}$$

Here, the positive n indicates the acceleration of the wall and negative n indicates the deceleration of wall form the origin whereas n = 0 is the case for stationary wall, u, v and w are the velocity components, N is the micro-rotation, T denotes the fluid temperature, J<sup>x</sup> = σµeB0λ 1+m<sup>2</sup>λ 2 (λmu − w) and J<sup>z</sup> = σµeB0λ 1+m<sup>2</sup>λ 2 (u − λmw) are the currents along x− and z−directions correspondingly, also electrical conductivity-σ, fluid viscosityµe, applied uniform magnetic field-B0, Hall parameter-m, couple stress viscosity-ν ′ , λ = cos α where α indicates the angle between the magnetic field and the transverse plane to the plate, thermal expansion volumetric coefficient-β, kinematic viscosity-ν, fluid density-ρ, Eringen vortex viscosity-K2, thermal conductivity-κ, Eringen spin gradient viscosity-G1, specific heatcp, γ , and A-constants.

To transform the coordinate system to a non-dimensional one and this is achieved readily via non-similar transformations, simultaneously eliminating one of the independent variables and reducing the PDEs into ODEs, the following transformation variables are defined.

$$\begin{split} \xi &= \sqrt{\frac{\mathbf{P}\left(n+1\right)}{2\upsilon}} \mathbf{x}^{\frac{n-1}{2}}, \ u = \mathbf{P} \mathbf{x}^{n} f'\left(\xi\right), \\ \nu &= -\sqrt{\mathbf{P}\upsilon\left(\frac{n+1}{2}\right)} \mathbf{x}^{\frac{n-1}{2}} \left(f + \frac{n-1}{n+1} \xi f'\left(\xi\right)\right), \\ \nu &= \mathbf{P} \mathbf{x}^{n} \mathbf{g}\left(\xi\right), \ N = \mathbf{P} \sqrt{\frac{\mathbf{P}\left(n+1\right)}{2\upsilon}} \mathbf{x}^{\frac{3n-1}{2}} h\left(\xi\right), \\ \theta\left(\xi\right) &= \frac{T-T\_{\infty}}{T\_{w} - T\_{\infty}}, \end{split} \tag{7}$$

The transformed equations are defined as:

$$\begin{aligned} &f'''' + f''' - N1h' - \frac{2}{n+1} \left[nf'^2 - Gr\theta \\ &+ \frac{M\hbar}{1+m^2\hbar^2} \left(f' + m\hbar g\right)\right] - \frac{n+1}{2} Kf'^{\prime\prime\prime\prime} = 0 \\ &g'' + fg' - \frac{2}{n+1} \left[nf'g - \frac{M\hbar}{1+m^2\hbar^2} \left(m\hbar f' - g\right)\right] \\ &n+1 \end{aligned} \tag{8}$$

$$-\frac{n+1}{2}\text{Kg}^{\prime\prime\prime} = 0\tag{9}$$

$$G\left(\frac{n+1}{2}\right)h'' - f'' - 2h = 0\tag{10}$$

$$1\Big|\_{\alpha\prime\prime \to -\varepsilon\alpha\prime}$$

$$\begin{split} \frac{1}{\text{Pr}} \theta'' &+ f \theta' \\ &- \frac{2}{n+1} \left[ \gamma f' \theta - \frac{M \mathfrak{k}}{1+m^2 \mathfrak{k}^2} \text{Ec} \left( f'^2 + g^2 \right) \right] = 0, \text{ (11)} \end{split}$$

with transformed boundary conditions:

$$\begin{aligned} f = 0, \; f' = 1, \; g = 0, \; h = 0, \; \theta = 1 \quad \text{at } \; \xi = 0, \\\ f' \to 0, \; g \to 0, \; h \to 0, \; \theta \to 0 \quad \text{as } \; \xi \to \infty. \end{aligned} \tag{12}$$

Here, G<sup>r</sup> = gβ(Tw−T∞)x U<sup>2</sup> symbolizes the Grashof number, M = σµeB 2 0 x ρU characterizes the Hartmann number in which B<sup>0</sup> = <sup>√</sup> P x is the scaled magnetic field strength, G = G1Pxn−<sup>1</sup> K2ν represents the micro-rotation parameter, m Hall parameter, K = ν ′ ν <sup>2</sup>Px2(n−1) represents the dimensionless couple stress parameter, γ indicates the non-isothermal power-law index, N1 = K1 ν characterizes the material parameter, Pr = ρνcp κ embodies the Prandtl number, Ec = U 2 cp(Tw−T∞) epitomizes the Eckert number, and n represents the non-linear wall geometric parameter.

For primary and secondary velocity components, the skin frication are defined as:

τwx = µ ∂u ∂y y=0 = µU √ x s U (n + 1) 2ν f ′′ (0), (13)

$$\pi\_{\le z} = \mu \left. \frac{\partial \, w}{\partial \boldsymbol{\uprho}} \right|\_{\boldsymbol{\uprho}=\boldsymbol{0}} = \frac{\mu \boldsymbol{U}}{\sqrt{\boldsymbol{\uprho}}} \sqrt{\left(\frac{\boldsymbol{U}\left(\boldsymbol{n}+1\right)}{2\boldsymbol{\uprho}}\right)} \boldsymbol{g}'(\boldsymbol{0}),\tag{14}$$

Using Equation (7), the skin fraction coefficients for primary and secondary velocities are reduced as:

$$C\_{\rm fx} = \frac{\tau\_{\rm wx}}{\frac{1}{2}\rho \, U^2} = \sqrt{\frac{2\left(n+1\right)}{\text{Re}}} f''(0),\tag{15}$$

$$\mathcal{C}\_{\sharp} = \frac{\mathfrak{r}\_{wz}}{\frac{1}{2}\rho \, U^2} = \sqrt{\frac{2\,(n+1)}{\mathrm{Re}}} g'(0). \tag{16}$$

The Nusselt number is specified by:

$$Nu\_{\chi} = -\frac{\varkappa}{(T\_{\text{w}} - T\_{\infty})} \left. \frac{\partial T}{\partial \mathcal{Y}} \right|\_{\mathcal{Y} = 0} = -\sqrt{\frac{\text{Re}\,(n+1)}{2}} \theta'(0), \tag{17}$$

#### SOLUTION BY HAM

To solve the Equations (8)–(11) using boundary conditions (12), we proceed HAM with the following manners.

Initial gausses

$$f\_0(\xi) = 1 - e^{\xi}, \; g\_0(\xi) = 0, \; h\_0(\xi) = 0, \; \theta\_0(\xi) = e^{-\xi}. \tag{18}$$

Linear operators

$$L\_f(f) = \frac{d^3f}{d\xi^3} - \frac{df}{d\xi}L\_{\xi}(g) = \frac{d^2g}{d\xi^2} - g,\\ L\_h(h) = \frac{d^2h}{d\xi^2} - h,$$

$$L\_\theta(\theta) = \frac{d^2\theta}{d\xi^2} - \theta,\text{ (19))}$$

with the following properties:

$$\begin{array}{ll} L\_f(s\_1 + s\_2e^{-\xi} + s\_3e^{\xi}) = 0, & L\_\xi(s\_4e^{-\xi} + s\_5e^{\xi}) = 0, \\ L\_h(s\_6e^{-\xi} + s\_7e^{\xi}) = 0, & L\_\theta(s\_8e^{-\xi} + s\_9e^{\xi}) = 0, \end{array} \tag{20}$$

where si(i = 1 − 9) are arbitrary constants.

The consequential non-linear operators N<sup>f</sup> , N<sup>g</sup> , Nh, and N<sup>θ</sup> are specified as:

$$N\_f \left[ f(\xi; ; \Theta), g(\xi; ; \Theta), h(\xi; ; \Theta), \theta(\xi; ; \Theta) \right]$$

$$N\_{\xi}\left[g(\xi;\Theta),f(\xi;\Theta)\right] = \frac{\partial^2 g(\xi;\Theta)}{\partial \xi^2} - f(\xi;\Theta)\frac{\partial g(\xi;\Theta)}{\partial \xi}$$

$$-\frac{2}{n+1}\left[ng(\xi;\Theta)\frac{\partial f(\xi;\Theta)}{\partial \xi} - \frac{M\hbar}{1+m^2\hbar^2}$$

$$\left(m\hbar\frac{\partial f(\xi;\Theta)}{\partial \xi} - g(\xi;\Theta)\right)\right] - \frac{n+1}{2}K\frac{\partial^4 g(\xi;\Theta)}{\partial \xi^4},\tag{22}$$

$$N\_h\left[h(\xi;\Theta), f(\xi;\Theta)\right] = G\left(\frac{n+1}{2}\right) \frac{\partial^2 h(\xi;\Theta)}{\partial \xi^2}$$

$$-\frac{\partial^2 f(\xi;\Theta)}{\partial \xi^2} - 2h(\xi;\Theta),\qquad(23)$$

$$\begin{aligned} N\_{\theta} \left[ \theta(\xi; \Theta), f(\xi; \Theta), g(\xi; \Theta) \right] \\ = \frac{1}{\text{Pr}} \frac{\partial^2 \theta(\xi; \Theta)}{\partial \xi^2} + f(\xi; \Theta) \frac{\partial \theta(\xi; \Theta)}{\partial \xi} \\ - \frac{2}{n+1} \left[ \chi \theta(\xi; \Theta) \frac{\partial f(\xi; \Theta)}{\partial \xi} \right] \end{aligned}$$

$$\frac{M\hbar}{1+m^2\hbar^2}Ec\left(\left(\frac{\partial f(\xi;\,\Theta)}{\partial\xi}\right)^2+\left(g(\xi;\,\Theta)\right)^2\right),\tag{24}$$

The zeroth-order problems from Equations (8)–(11) are:

$$\begin{aligned} &(1 - \Theta) L\_f \left[ f(\xi; \Theta) - f\_0(\xi) \right] \\ &= \Theta \hbar\_f N\_f \left[ f(\xi; \Theta), g(\xi; \Theta), h(\xi; \Theta), \theta(\xi; \Theta) \right], \end{aligned} \tag{25}$$

$$(1 - \Theta)L\_{\mathbb{X}}\left[\mathcal{g}(\xi;\Theta) - \mathcal{g}(\xi)\right] = \Theta \hbar\_{\mathbb{X}} N\_{\mathbb{X}}\left[\mathcal{g}(\xi;\Theta), f(\xi;\Theta)\right],\tag{26}$$

$$(1 - \Theta)L\_{\mathbb{h}}\left[h(\xi;\Theta) - f\_{0}(\xi)\right] = \Theta \hbar\_{\mathbb{h}} N\_{\mathbb{h}}\left[h(\xi;\Theta), f(\xi;\Theta)\right],\tag{27}$$

$$(1 - \Theta)L\_{\theta}\left[\theta(\xi; \Theta) - \theta\_0(\xi)\right] = \Theta \hbar\_{\theta} N\_{\theta}\left[\theta(\xi; \Theta), f(\xi; \Theta), g(\xi; \Theta)\right]. \tag{28}$$

The equivalent boundary conditions are:

−

$$\begin{array}{ll} \frac{\left.\frac{\partial f(\xi;\Theta)}{\partial \xi}\right|\_{\xi=0}}{\left.\frac{\partial}{\partial \xi}\right|\_{\xi=0}} = 1, \left.f(\xi;\Theta)\right|\_{\xi=0} = 0, & \left.g(\xi;\Theta)\right|\_{\xi=0} = 0, \\\left.h(\xi;\Theta)\right|\_{\xi=0} = 0, & \left.\theta(\xi;\Theta)\right|\_{\xi=0} = 1, \\\left.\frac{\partial f(\xi;\mathbf{r})}{\partial \xi}\right|\_{\xi\to\infty} = 0, & \left.g(\xi;\mathbf{r})\right|\_{\xi\to\infty} = 0, & h(\xi;\mathbf{r})\Big|\_{\xi\to\infty} = 0, \\\left.\theta(\xi;\mathbf{r})\right|\_{\xi\to\infty} = 0. \end{array} \tag{29}$$

1.2 −1.081639 1.3 -.1117715 1.4 −1.152781

> 0.3 −1.561382 0.5 −2.936200 0.7 −8.747216


TABLE 1 | Influence of *M*, *Gr*, *G*, *N*1, *m*, *Ec*, Pr, *n*, and *K* on *Cfx* .

When 2 = 0 and 2 = 1 we have:

$$f(\xi; 1) = f(\xi), \,\mathrm{g}(\xi; 1) = \mathrm{g}(\xi), \,\, h(\xi; 1) = h(\xi),$$

$$\theta(\xi; 1) = \theta(\xi). \tag{30}$$

By Taylor's series expansion f(ξ ; 2), g(ξ ; 2), h(ξ ; 2), and θ(ξ ; 2) can be written as:

$$\begin{aligned} f(\xi;\Theta) &= f\_0(\xi) + \sum\_{q=1}^{\infty} f\_q(\xi)\Theta^q, \ g(\xi;\Theta) = g\_0(\xi) + \sum\_{q=1}^{\infty} g\_q(\xi)\Theta^q, \\\ h(\xi;\Theta) &= h\_0(\xi) + \sum\_{q=1}^{\infty} h\_q(\xi)\Theta^q, \ \theta(\xi;\Theta) = \theta\_0(\xi) + \sum\_{q=1}^{\infty} \theta\_q(\xi)\Theta^q, \end{aligned} \tag{31}$$

where

$$\begin{array}{ll} f\_{\boldsymbol{\theta}}(\boldsymbol{\xi}) = \frac{1}{\boldsymbol{q}!} \frac{\partial f(\boldsymbol{\xi}; \boldsymbol{\Theta})}{\partial \boldsymbol{\xi}} \Big|\_{\boldsymbol{\Theta} = \boldsymbol{0}}, \mathcal{g}\_{\boldsymbol{\theta}}(\boldsymbol{\xi}) = \frac{1}{\boldsymbol{q}!} \frac{\partial \mathcal{g}(\boldsymbol{\xi}; \boldsymbol{\Theta})}{\partial \boldsymbol{\xi}} \Big|\_{\boldsymbol{\Theta} = \boldsymbol{0}}.\\\ h\_{\boldsymbol{q}}(\boldsymbol{\xi}) = \frac{1}{\boldsymbol{q}!} \frac{\partial h(\boldsymbol{\xi}; \boldsymbol{\Theta})}{\partial \boldsymbol{\xi}} \Big|\_{\boldsymbol{\Theta} = \boldsymbol{0}}, \partial\_{\boldsymbol{q}}(\boldsymbol{\xi}) = \frac{1}{\boldsymbol{q}!} \frac{\partial f(\boldsymbol{\xi}; \boldsymbol{\Theta})}{\partial \boldsymbol{\xi}} \Big|\_{\boldsymbol{\Theta} = \boldsymbol{0}}.\end{array} \tag{32}$$

The secondary constraints h¯<sup>f</sup> , h¯<sup>g</sup> , h¯<sup>h</sup> and h¯<sup>θ</sup> are nominated in such a way that the series (31) converges at 2 = 1, changing 2 = 1 in Equation (31), we get:

$$\begin{aligned} f(\xi) &= f\_0(\xi) + \sum\_{q=1}^{\infty} f\_q(\xi), \ g(\xi) = g\_0(\xi) + \sum\_{q=1}^{\infty} g\_q(\xi), \\ h(\xi) &= h\_0(\xi) + \sum\_{q=1}^{\infty} h\_q(\xi), \ \theta(\xi) = \theta\_0(\xi) + \sum\_{q=1}^{\infty} \theta\_q(\xi). \end{aligned} \tag{33}$$

The q th−order problem satisfies the following:

$$\begin{array}{l} L\_{f} \left[ f\_{q}(\xi) - \chi\_{q} f\_{q-1}(\xi) \right] = \hbar\_{f} U\_{q}^{f}(\xi), \\ L\_{\xi} \left[ d\_{q}(\xi) - \chi\_{q} d\_{q-1}(\xi) \right] = \hbar\_{\xi} U\_{q}^{\xi}(\xi), \\ L\_{h} \left[ F\_{q}(\xi) - \chi\_{q} F\_{q-1}(\xi) \right] = \hbar\_{h} U\_{q}^{h}(\xi), \\ L\_{\theta} \left[ D\_{q}(\xi) - \chi\_{q} D\_{q-1}(\xi) \right] = \hbar\_{\theta} U\_{q}^{\theta}(\xi). \end{array} \tag{34}$$

The equivalent boundary conditions are:

$$\begin{aligned} f\_q(0) = f'\_q(0) = f'\_q(\infty) = 0, & \text{ } g\_q(0) = g\_q(\infty) = 0, \\ h\_q(0) = h\_q(\infty) = 0, & \theta\_q(0) = \theta'\_q(\infty) = 0. \end{aligned} \tag{35}$$

Here,

$$U\_q^f(\xi) = f^{\prime\prime\prime}\_{\quad q-1} + \sum\_{k=0}^{q-1} f\_{q-1-k} f^{\prime\prime}\_{\quad k} - N1h\_{q-1}^{\prime}$$



$$\begin{aligned} &-\frac{2}{n+1}\left[n\left(\boldsymbol{f}'\_{q-1}\right)^2-Gr\theta\_{q-1}\\ &+\frac{M\dot{\mathsf{x}}}{1+m^2\dot{\mathsf{x}}^2}\left(\boldsymbol{f}'\_{q-1}+m\dot{\mathsf{x}}\mathsf{g}\_{q-1}\right)\right]-\frac{n+1}{2}K\boldsymbol{f}'^{\prime\prime\prime}\_{\ \left.\begin{array}{c} \left.\ . \ \end{array}\right.\end{aligned} \tag{36}$$

$$\begin{split} U\_{q}^{\mathbb{g}}(\xi) &= \mathcal{G}^{\prime\prime}\_{\ \left. \begin{array}{c} q \end{array}} + \sum\_{k=0}^{q-1} f\_{q-1-k} \mathcal{g}^{\prime}\_{\ \ k} - \frac{2}{n+1} \left[ n \sum\_{k=0}^{q-1} f^{\prime}\_{\ \ \ \ -1-k} \mathbb{g}\_{k} \right] \\ &- \frac{M\hbar}{1+m^{2}\dot{\pi}^{2}} \left( m\sharp f^{\prime}\_{\ \ \ \ \ -1} - \mathcal{g}\_{\! \! \ -1} \right) \right] - \frac{n+1}{2} K \mathcal{g}^{\prime\prime\prime}\_{\ \ \ \ \ \ -1}, \end{split} \tag{37}$$

$$U\_q^h(\xi) = G\left(\frac{n+1}{2}\right)h''\_{\
u-1} - f''\_{\
u-1} - 2h\_{\!\!q-1},\tag{38}$$

$$U\_q^{\theta}(\xi) = \frac{1}{\text{Pr}} \theta^{\prime\prime}\_{\quad q-1} + \sum\_{k=0}^{q-1} f\_{q-1-k} \theta^{\prime}\_k - \frac{2}{n+1} \left[ \gamma \sum\_{k=0}^{q-1} f^{\prime}\_{\quad q-1-k} \theta\_k \right.$$

$$-\left. \frac{M\hbar}{1 + m^2 \hbar^2} E \varepsilon \left( \left( f^{\prime}\_{\quad q-1} \right)^2 + \left( g\_{q-1} \right)^2 \right) \right], \text{ (39)}$$

where,

$$\chi\_q = \begin{cases} 0, \text{ if } \Theta \le 1 \\ 1, \text{ if } \Theta > 1. \end{cases}$$

#### RESULTS

Electrically conducting steady non-isothermal convective heat transfer in magnetohydrodynamic micropolar fluid flow over a non-linear extending wall is examined. Modeled equations are solved analytically through HAM. The impact of obtained important parameters M, Gr, m, and K on the fluid flow behavior are displayed in **Figures 2**–**15**.

#### DISCUSSION

In this section we have discussed the effects of obtained parameter which are shown graphically and numerically through tables. The greater Hartmann number strongly reduced the primary and angular velocity profile owing to the Lorentz drag force components as appear in Equations (8) and (9). The components are negative and positive and thus inhibit the fluid flow. According to the secondary Lorentz drag force is truthfully positive and is assistive to secondary


TABLE 3 | Influence of *M*, *Gr*, *G*, *N*1, *m*, *Ec*, Pr, *n*, and *K* on *Nu<sup>x</sup>* .

momentum development when the magnetic field is positive. These impacts are depicted in **Figures 2**, **4**. The opposite impacts of M on secondary velocity and temperature functions are depicted in **Figures 3**, **5**. It is perceived that the strong magnetic field has direct relationship with the secondary velocity and temperature functions. Against the magnetic field, the upsurge in temperature function is an attribute to the dissipation in kinetic energy consumed in dragging the micropolar. In addition, the temperature is always supreme at the wall.

**Figures 6**–**9** display the consequence of Gr on f ′ (ξ ), g (ξ ), h (ξ ), and θ (ξ ). The influence of Gr on f ′ (ξ ) is portrayed in **Figure 6**. Here, the velocity heightens with the acceleration in Grashof number near the wall. However, the free convention current deteriorates at a critical distance from the wall which conserved into the free stream. A similar impact of Grashof number secondary velocity can be seen in **Figure 7**. Near the wall the fluid flow escalates with greater Grashof number but thereafter a deceleration started after some critical distance. Furthermore, the greater proportion of the region is observed for secondary velocity in comparison of primary velocity. **Figure 8** reveals the consequence of Gr on h (ξ ). The angular velocity heightens via Grashof number. A very quick growing behavior in the whole boundary layer regime is observed in the angular velocity. **Figure 9** reveals the impact of Gr on θ (ξ ). The intensifying Grashof number shrinks the boundary layer thickness, consequently the decline in temperature function is depicted.

**Figures 10**–**13** reveal the impact m on f ′ (ξ ), g (ξ ), h (ξ ), and θ (ξ ). **Figure 10** reveals the impact of m on f ′ (ξ ). Acceleration in m escalates the f ′ (ξ ) in the neighborhood of the wall. Further toward the free stream, after some critical points the primary velocity function reduces. The drag force moderates which produce acceleration in f ′ (ξ ) and in conclusion f ′ (ξ ) diminishes. **Figure 11** reveals the impact of m on g (ξ ). Acceleration in m escalates the g (ξ ) throughout the fluid flow. The Hall term in Equation (9) is effectively positive for positive magnetic field parameter. This assists to support the cross flow and demonstrates in significant cross flow spurt. **Figure 12** reveals the impact of m on h (ξ ). The Hall current parameter shows dual behavior in the flow of fluid. An enhancement in h (ξ ) is perceived nearer to the wall and then deceleration to the flow stream is observed at some critical points. Generally, nevertheless the Hall current emboldens the rotary motions of microelements. **Figure 13** reveals the impact of m on θ (ξ ). The temperature function is regularly inhibited with Hall current parameter. Here, the decline in thickness of the boundary layer is perceived.

**Figures 14**, **15** reveal the impact of K on f ′ (ξ ), g (ξ ), h (ξ ), and θ (ξ ). At the point when an extra force added to the fluid which contradicts the fluid stream, this resistance makes a couple forces thus a couple stresses are persuaded in the fluid. This sort of fluid is recognized as couple stress fluid. Generally, the couple stress parameter and couple stress viscosity parameter n ′ has direct relationship. The growing couple stress parameter leads the fluid to be more viscous which reduces the fluid flow. Therefore, the escalation approximations of couple stress parameter reduced the primary and secondary velocity as shown in **Figures 14**, **15**. Additionally, the couples stress parameter is associated with the fluid motion. Therefore, it has no impact on temperature function.

**Tables 1**–**3** are displayed to observe the impact of embedded parameters on velocities and temperature profiles. The impact M, Gr, G, N1, m, Ec, Pr, n, and K on Cfx and Cfz are shown in **Tables 1**, **2**. The rising value of M, N1, n, and K augmented the skin friction along x-axis Cfx where Gr, G, and m have opposite impact on the skin friction along x-axis Cfx. The higher value of M, Gr, G, and Ec augmented skin friction along z-axis Cfz where, m, N1, n, and K reduces the skin friction along z-axis Cfz. The influence of M, Gr, N1, m, Ec, Pr, n, and K on heat flux Nu<sup>x</sup> are presented in **Table 3**. The greater value of Gr, m, Pr, n, and K augmented the heat flux Nu<sup>x</sup> while, remaining parameter reduces the heat flux Nux. It should be noted that G has no impact on Nux.

#### CONCLUSION

In the current paper, the MHD micropolar boundary layer flow and heat transfer over a non-linear extending sheet infused by a strong magnetic field with couple stress, viscous dissipation and Hall impact have been determined.

The final observations are:

• The primary velocity reduces with greater magnetic parameter, local Grashof number, Hall parameter and couples stress parameter.

#### REFERENCES


### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### AUTHOR CONTRIBUTIONS

ZS and PK developed the numerical method and led the manuscript preparation. AD contributed to the code development and to the article preparation. EA and PT contributed to the analysis and discussion of the results.

#### FUNDING

This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT under Grant KMUTNB-61-GOV-A-0.

#### ACKNOWLEDGMENTS

This work was supported by the research program International Research Common Laboratory in cooperation under Renewable Energy Research Centre (RERC)—King Mongkut's University of Technology North Bangkok (KMUTNB), Center of Excellence in Theoretical and Computational Science (TaCS-CoE)—King Mongkut's University of Technology Thonburi (KMUTT), and Groupe de Recherche en Energie Electrique de Nancy (GREEN)—Université de Lorraine (UL) under Grant KMUTNB-61-GOV-A-01.


50. Ullah A, Alzahrani EO, Shah Z, Ayaz M, Islam S. Nanofluids thin film flow of reiner-philippoff fluid over an unstable stretching surface with brownian motion and thermophoresis effects. Coatings. (2019) **9**:21. doi: 10.3390/coatings9010021

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Shah, Kumam, Dawar, Alzahrani and Thounthong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Numerical Simulation of Magnetohydrodynamic Nanofluids Under the Influence of Shape Factor and Thermal Transport in a Porous Media Using CVFEM

#### Zahir Shah<sup>1</sup> \*, Houman Babazadeh2,3, Poom Kumam4,5,6 \*, Ahmad Shafee<sup>7</sup> and Phatiphat Thounthong<sup>8</sup>

*<sup>1</sup> SCL 802 Fixed Point Laboratory, Science Laboratory Building, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>2</sup> Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>3</sup> Faculty of Environment and Labour Safety, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>4</sup> KMUTT-Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>5</sup> KMUTT-Fixed Point Theory and Applications Research Group, Science Laboratory Building, Faculty of Science, Theoretical and Computational Science Center (TaCS), King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>6</sup> Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan, <sup>7</sup> Public Authority of Applied Education and Training, Applied Science Department, College of Technological Studies, Shuwaikh, Kuwait, <sup>8</sup> Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, Renewable Energy Research Centre, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand*

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Tehseen Abbas, University of Education Lahore, Pakistan Titan Chandra Paul, University of South Carolina Aiken, United States*

#### \*Correspondence:

*Zahir Shah zahir.sha@kmutt.ac.th Poom Kumam poom.kum@kmutt.ac.th*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *19 August 2019* Accepted: *11 October 2019* Published: *06 November 2019*

#### Citation:

*Shah Z, Babazadeh H, Kumam P, Shafee A and Thounthong P (2019) Numerical Simulation of Magnetohydrodynamic Nanofluids Under the Influence of Shape Factor and Thermal Transport in a Porous Media Using CVFEM. Front. Phys. 7:164. doi: 10.3389/fphy.2019.00164* In this article, the migration of nanomaterials through a permeable domain was modeled numerically. Aluminum oxide was dispersed into testing fluid which was selected water in the current paper. Utilizing Darcy LAW for a porous medium helps us to find simpler form of equations. Influences of shape factor and radiation on the thermal conduct of nanoparticles within a porous region were scrutinized. Nanomaterial within such region is applied under the Lorentz force. CVFEM approach for simulation goals has been applied. This approach provides the advantages of two common CFD methods. Impacts of radiation, magnetic, buoyancy parameters on the treatment of nanomaterials were demonstrated. Outcomes showed that greater amounts of shape factor cause stronger convection. Reverse relationships exist between the Hartmann number and temperature gradient.

Keywords: nanoparticle's shape, porous space, magnetic force, darcy LAW, radiation, nanofluid, CVFEM

### INTRODUCTION

Nanotechnology is one of the most interesting fields nowadays. It is interesting due to its vast applications in solar cells, food, fuel cells, batteries, and fuel, etc. In simple, nanotechnology has made its way to each and every branch. Investigators started interest in this field and developed a new sub-branch of nanotechnology, nanofluids. Nanofluids were utilized by Choi [1] for the first time. In real-world fluids exist in abundance, among all, nanopowders can be offered as the most applicable fluids both from its use and its unique nature. Nanofluids are two-phase nanometer-size

**34**

fluids in which base fluid ranging up to 100 nm. Nanofluids are used in metal oxides, oxides ceramics, and allotropes of carbon and in other chemical stable elements. Nanoparticles nowadays play a key role in thermal analysis. Pak and Cho [2] used titanium dioxide and, and found an improvement in the heat flux. Nanofluids in which the nanoparticles size range less than are considered more ideal [3]. Radiation impact on nanomaterial flow was performed by Zeeshan et al. [4] and they added the impact of MHD on titanium dioxide transportation. Impose of nanomaterial into usual carrier fluid leads to greater conductivity [5–10]. Copper oxide migration within an absorptive medium with the use of Lorentz force in the actuality of magnetic force has been demonstrated by Sheikholeslami [11]. A numerical survey is performed by Sheikholeslami [12] for CuO-H2O nanofluid in a penetrable medium with the help of a microscopic technique. Shah et al. [13] have worked on the 3-D nanofluid flow of third-grade fluid with physical properties inside a rotating frame. An analytical investigation is performed by Dawar et al. [14] for Casson fluid with MHD carbon nanotubes (CNT's) inside a rotating channel. A numerical survey is presented by Sheikholeslami and Shehzad [15] by analyzing Fe3O4–H2O nanofluid flow with inside a permeable channel. To depict the changes in flow style in the appearance of Kelvin forces, Sheikholeslami and Vajravelu [16] examined the FHD impact on nanomaterial flow. CNT migration in a time-dependent problem has been analyzed by Ahmed et al. [17] and they supposed the plate is porous and Lorentz force was added in momentum equations. The transfer of heat due to convection of ferrofluid is described by Yimin et al. [18]. In recent years, Thermal irreversibility in nanofluid through a pipe with a turbulator by means of FVM was analyzed by Sheikholeslami et al. [19]. For a detailed survey, interested readers are referred to Sheikholeslami et al. [20], Dat et al. [21], Bhatti et al. [22], Sheikholeslami [23], Cattaneo [24], Sheikholeslami and Shehzad [25] for more detail and related study of nanofluids flow. Cattaneo [24] made a modification in the thermal relaxation time to improve the heat transfer effects. Cattaneo attempt made for a specific material and obtained some interesting results in the heat transmission investigation by presenting an innovative flux approach. A Maxwell fluid was realized to this model by Mustafa [26] for the study of upper convection. A numerical investigation is performed by Ai and Sandeep [27] by considering this model for MHD Casson-ferrofluid for heat transfer analysis. Previous articles on Nanomaterials for dissimilar phenomena and their usages can be found [28–33]. Sheikholeslami et al. [34] recently presented the application of electric and magnetic field of nanofluid and ferrofluid and with transfer in an enclosure walls. Jawad et al. [35, 36] studied nanofluid thin film and their applications. Nasir et al. [37, 38] have studied 3-D nanomaterial flow CNTs and thermal analysis along a stretching surface. Entropy generation in nanofluid flow can be studied in Alharbi et al. [39]. The studied of nanofluids are further extended to liquid film due to its abundant uses in various sciences [40–48]. Nanomaterial transportation over a wedge was scrutinized by Hassan et al. [49]. An experimental approach was performed by Sheikholeslami et al. [50] to study the boiling of refrigerant with the use of nanoparticles.

TABLE 1 | Coefficient of carrier fluid.


Though there is intense research in the literature about nanofluid modeling and the MHD effect in different categories, there is still limited information about the complex geometries and Darcy model. Additionally, the radiative effect is an important source. The determination of this article is to study the migration of nanopowder within a porous space. The effects of shape factor and radiation on the thermal conduct of nanomaterials within a porous space were scrutinized.

#### PROBLEM EXPLANATION

In this modeling and simulation, water-based nanofluid exists through permeable geometry has considered. Impact of Lorentz force and thermal behavior are taken on nanofluid. Sketch of the porous tank is depicted in **Figure 1**. Nanofluid is thermally conducting and impact of Lorentz force was involved. Control Volume finite element technique with a triangular element has been used (see **Figure 1**). Needed boundary constraints were established in **Figure 1**. The Darcy LAW [15] is involved for porous terms.

TABLE 2 | Some physical thermal features.


TABLE 3 | Structure of *m* at dissimilar values.

TABLE 4 | Deviation of *Nuave* at different mesh size when *Ra* = 600, *Ha* = 0, *Rd* = 0.8 and φ = 0.04.


#### GOVERNING EQUATIONS, FORMULATION, AND CVFEM

Nanopowder migration through a permeable domain with the help of Darcy model was considered in the current article and involving single-phase model results in below equations:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial \nu}{\partial y} = 0 \tag{1}
$$

$$\frac{\mu\_{\rm nf}}{K}u = -\frac{\partial P}{\partial x} + \sigma\_{\rm nf}B\_0^2 \left[ (\sin \chi) \,\nu \left( \cos \chi \right) - \mu (\sin \chi)^2 \right] \tag{2}$$

$$\begin{split} \frac{\mu\_{\eta f}}{K} \nu &= -\frac{\partial P}{\partial \boldsymbol{\eta}} + (T - T\_c) \grho\_{\eta f} \boldsymbol{\beta}\_{\eta f} \\ &+ \,\,\sigma\_{\eta f} \boldsymbol{B}\_0^2 \left( \cos \boldsymbol{\eta} \right) \left[ \left( \sin \boldsymbol{\eta} \right) \boldsymbol{u} - \left( \cos \boldsymbol{\eta} \right) \boldsymbol{\nu} \right] \end{split} \tag{3}$$

$$
\frac{\partial q\_r}{\partial \mathbf{y}} \left( \rho \mathbf{C}\_p \right)^{-1}\_{\eta f} + \left( \frac{\partial T}{\partial \mathbf{y}} \nu + u \frac{\partial T}{\partial \mathbf{x}} \right) = k\_{\eta f} \left( \frac{\partial^2 T}{\partial \mathbf{y}^2} + \frac{\partial^2 T}{\partial \mathbf{x}^2} \right) \left( \rho \mathbf{C}\_p \right)^{-1}\_{\eta f},
$$

$$
\quad , \quad \left[ T^4 \cong 4T\_c^3 T - 3T\_c^4, q\_r = -\frac{4 \sigma\_c}{3 \beta\_R} \frac{\partial T}{\partial \mathbf{y}} \right] \tag{4}
$$

The fundamental characteristics of nanofluid are estimated as:

$$\begin{aligned} \mathfrak{R}\_{nf} &= \mathfrak{R}\_f + \left( \mathfrak{R}\_s - \mathfrak{R}\_f \right) \phi \\ \mathfrak{R} &= \mathcal{C}\_{\mathcal{P}} \rho \end{aligned} \tag{5}$$

$$(\rho \beta)\_{\eta \zeta} + (\rho \beta)\_{\zeta} (\phi - 1) = \phi (\rho \beta)\_{\varsigma} \tag{6}$$

$$
\rho\_{\eta f} = (1 - \phi)\rho\_f + \phi\rho\_s \tag{7}
$$

$$\frac{\sigma\_{\eta f}}{\sigma\_f} - 1 = \frac{\left(-1 + \frac{\sigma\_i}{\sigma\_f}\right) (3\phi)}{\left(2 + \frac{\sigma\_i}{\sigma\_f}\right) + \phi \left(1 - \frac{\sigma\_i}{\sigma\_f}\right)} \tag{8}$$

To apply the effects of shape factor and kBrownian, the following correlations were examined:

$$
\mu\_{\rm eff} = \mu\_{\rm static} + \frac{k\_{\rm Brownian}}{k\_f} \times \frac{\mu\_f}{\text{Pr}\_f}
$$

$$
k\_{\rm Brownian} = \left. 5 \times 10^4 c\_{p,f} \rho\_f \text{g}'(d\_p, \phi, T) \phi \sqrt{\frac{\kappa\_b T}{\rho\_p d\_p}} \right. \tag{9}
$$

$$\begin{aligned} \text{g } \left(d\_p, \phi, T\right) &= \left(a\_1 + a\_3 \text{Ln}(d\_p)^2 + \text{Ln}\left(\phi\right)a\_4 \text{Ln}\left(d\_p\right) + a\_2 \text{Ln}\left(d\_p\right)\right) \\ &+ a\_3 \text{Ln}\left(\phi\right)\left(\text{Ln}\left(T\right) + \left(a\_6 + a\_{10} \text{Ln}\left(d\_p\right)^2 + a\_8 \text{Ln}\left(\phi\right)\right) \\ &+ a\_7 \text{Ln}\left(d\_p\right) + a\_9 \text{Ln}\left(d\_p\right) \text{Ln}\left(\phi\right)\right) \\ \frac{k\_{\text{uf}}}{k\_f} &= \frac{-\Im m\phi + k\_p + \Im\phi + k\_f + mk\_f + k\_f}{k\_f m + k\_p + \Im\phi + k\_f}, \\ \Im &= k\_p - k\_f \end{aligned} \tag{10}$$

Equation (11) presents a dimensionless form:

$$\Psi = \psi / \alpha\_{\eta f}, \theta = \frac{T - T\_c}{\Delta T}, \Delta T = L \frac{q^{\prime \prime}}{k\_f}, (X, Y) = L^{-1} \left( \mathbf{x}, \mathbf{y} \right) \tag{11}$$

So, the last format of equations is:

$$\begin{cases} \frac{\partial^2 \Psi}{\partial Y^2} + \frac{\partial^2 \Psi}{\partial X^2} = -Ha \frac{A\_6}{A\_5} \\ \frac{2}{3} \left( \sin \chi \right) \frac{\partial^2 \Psi}{\partial X \partial Y} \left( \cos \chi \right) + \frac{\partial^2 \Psi}{\partial Y^2} \left( \sin^2 \chi \right) + \left( \cos \chi \right) \frac{\partial^2 \Psi}{\partial X^2} \left( \cos \chi \right) \\ - \frac{A\_3}{A\_4} \frac{A\_2}{A\_5} \frac{\partial \vartheta}{\partial X} Ra \end{cases} \tag{12}$$

$$\left(1+\frac{4}{3}RdA\_4\right)\frac{\partial^2 \theta}{\partial Y^2} + \left(\frac{\partial^2 \theta}{\partial X^2}\right) = \frac{\partial \theta}{\partial X}\frac{\partial \Psi}{\partial Y} - \frac{\partial \Psi}{\partial X}\frac{\partial \theta}{\partial Y} \tag{13}$$

The mentioned variables in Equation (13) are:

$$\begin{array}{l}Ha = \frac{\sigma\_{f}K\_{0}^{2}}{\mu\_{f}}, \; Ra = \frac{g^{K}(\rho\beta)\_{f}L\,\Delta T}{\mu\_{f}\,\alpha\_{f}}, \; Rd = 4\sigma\_{e}T\_{c}^{3}/\left(\beta\_{R}k\_{f}\right) \\\ A\_{1} = \frac{\rho\_{\eta f}}{\rho\_{f}}, \; A\_{2} = \frac{(\rho C\_{\mathrm{P}})\_{\mathrm{uf}}}{(\rho\,\mathrm{C}\_{\mathrm{P}})\_{f}}, \; A\_{5} = \frac{\mu\_{\mathrm{nf}}}{\mu\_{f}}, \\\ A\_{3} = \frac{(\rho\beta)\_{\eta f}}{(\rho\beta)\_{f}}, \; A\_{6} = \frac{\sigma\_{\eta f}}{\sigma\_{f}}, \; A\_{4} = \frac{k\_{\mathrm{nf}}}{k\_{f}} \end{array} \tag{14}$$

Besides, summarizations of boundaries are:

$$\begin{aligned} \theta &= 0.0 \text{ on outer surfaces} \\ \Psi &= 0.0 \text{ on all walls} \\ \frac{\partial \theta}{\partial n} &= 1.0 \text{ on inner wall} \end{aligned} \tag{15}$$

Nuloc and Nuave are:

$$Nu\_{loc} = \frac{1}{\theta} \left( 1 + \frac{4}{3} \left( \frac{k\_{\eta f}}{k\_f} \right)^{-1} Rd \right) \left( \frac{k\_{\eta f}}{k\_f} \right) \tag{16}$$

$$Nu\_{\text{ave}} = \frac{1}{S} \int\_0^s Nu\_{\text{loc}} \, ds \tag{17}$$

#### Simulation Technique, Grid and Verification

Sheikholeslami [29] has been discovered a new approach namely CVFEM for analyzing thermal problems. This technique utilizes a triangular element and the Gauss-Seidel approach uses for the final step of calculating scalars. **Tables 1**–**3** illustrate the properties of carrier fluid. Grid size must be independent of outcomes and we present special cases in **Table 4**. Validation for presents study for nanofluid [5] are presented in **Figure 2** and provide nice accuracy.

#### RESULTS AND DISCUSSION

In this article transportation of electrically and thermally conducting nanomaterial with different shapes were modeled numerically. Aluminum oxide was dispersed into testing fluid which was selected water in current paper. Utilizing Darcy low for porous medium helps us to find simpler form of equations. Impacts of shape factor and the radiation on thermal conduct of nanoparticles inside a permeable space were investigated. Impacts of Radiation parameter, shape factor, magnetic force, and fraction of alumina have been demonstrated. The Darcy Law is involved for a permeable term in geometry.

Impacts of imposing nanopowders into H2O by selecting other parameters are shown in **Figure 3**. Actually this is the nanofluid scattering rule. It is observed that nanofluid motion augmented with the imposing of nanoparticles. The impacts of Hartmann for different cases were plotted in **Figures 4**, **5**. Impose of the Lorentz effect declines the motion of nanoparticles.

Actually, with the augmentation of the magnetic parameter, the top two eddies were amalgamate together and the thermal spiral disappear. It is observed that adding magnetic impact, stronger conduction occurs. Reverse relationships exist between the Hartmann number and temperature gradient. Impacts of scrutinized variables on Nusselt number were displayed in **Figure 6**. Variations for different cases are presented here.

Distortion of isotherms augments in consequence of augment in buoyancy and makes stronger vortex which indicates the growth of free convection. With the domination of convective mode, isotherms become more complex with generating plume. Therefore, increasing permeability and buoyancy term makes the Nusselt number to augment. Resistance against the nanomaterial migration reduces with augment of Lorentz forces and in turn, Nusselt number can reduce. Temperature distribution becomes less complex with involving magnetic field and higher Lorentz force can eliminate the plumes. Shear stress among nanoparticles declines with augment of permeability of the region. So, the power of the flow augments with rise of permeability which indicates greater convective flow. The influence of permeability on the style of nanofluid flow reduces with decreasing buoyancy forces. Greater nanofluid mixing occurs within the domain with

the rise of buoyancy forces and this influence reduces with imposing magnetic field. Resistance against the nanomaterial migration declines with augment of Darcy number but opposite phenomena appear with augment of the Hartmann number. Magnetic forces work against buoyancy forces, which can reduce the strength of streamline and imposing greater magnetic force, leads to conduction domination. The temperature gradient becomes independent on the Lorentz forces again, owing to the weakening of the buoyancy.

Changes of Nusselt number respect to variables are presented in **Figure 6**. The mathematical relationship has presented in Equation (18).

$$Nu\_{ave} = 3.34 + 0.087m + 1.04Rd + 0.19Ra - 0.14Ha$$

$$\begin{split} +1.1 &\times 10^{-2}m \, Ha - 0.092Rd \, Ha - 0.19RaHa \\ +1.359 &\times 10^{-4}m^2 \end{split} \tag{18}$$

It is concluded that the augment in distortion of temperature with buoyancy terms and permeability enhances the gradient of temperature. Moreover, transmission mode improves with a boost of the Lorentz force. Thus, convection diminishes with the

escalation of the magnetic field. It is found from **Figure 6** that Nusselt number is augmenting function for radiation parameter.

### CONCLUSIONS

In current CVFEM simulation, nanomaterial was offered as a feasible way to more augmentation of convection in permeable tank and various shapes of powder ware involved. To manage the migration of particles, magnetic forces was employed, and the influence of radiation has been imposed in the energy equation. Outcomes prove that augmenting Lorentz force declines the convection and make isotherms to lower dense near the wall. An indirect relationship was reported for temperature gradient and Lorentz forces. Furthered distortion was observed in isotherms with the rise of buoyancy force.

### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### REFERENCES


### AUTHOR CONTRIBUTIONS

ZS developed the numerical method and led the manuscript preparation. HB and PK contributed to the code development and to the article preparation. AS and PT contributed to the analysis and discussion of the results.

### FUNDING

This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

### ACKNOWLEDGMENTS

This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Shah, Babazadeh, Kumam, Shafee and Thounthong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# Magnetohydrodynamic Free Stream and Heat Transfer of Nanofluid Flow Over an Exponentially Radiating Stretching Sheet With Variable Fluid Properties

#### Muhammad Irfan\*, Muhammad Asif Farooq and Tousif Iqra

*Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan*

This article deals with the nanofluid flow and heat transfer of the MHD free stream over an exponentially radiating stretching sheet accompanied by constant and variable fluid characteristics together. The underlying governing partial differential equations (PDEs) have been translated into nonlinear ordinary differential equations (ODEs) by incorporating adequate similarity transformations. By using the shooting method and the MATLAB built-in solver *bvp4c*, the corresponding ODEs are effectively solved. The impact on the skin friction coefficient (quantifying resistance), the local Nusselt number (heat transfer rate) and the local Sherwood number (mass transfer rate) on the surface due to the flow field variables has been computed against various parameters i.e., magnetic parameter M, Prandtl number *Pro*, Lewis number Le, thermophoresis parameter Nt, Brownian motion parameter Nb, velocity parameter λ, radiation parameter Rd and thermal conductivity parameter ǫ. Graphs are also plotted to study the impact of distinct parameters on velocity, temperature and concentration profiles. It has been noted by raising the values of ǫ, the heat transfer rate reduces for variable fluid properties. On the other hand, raising *Pr<sup>o</sup>* increases the heat transfer rate.

Keywords: magnetohydrodynamics (MHD), exponentially stretching sheet, nanofluid, shooting method, constant and variable fluid properties

### 1. INTRODUCTION

Because of a stretching surface, studying fluid dynamics is essential as it has many practical and industrial applications. In a number of industrial and manufacturing processes, material production occurs and involves sheets of metal, and polymer. For instance, cooling an infinite metal plate in a cooling bath, the boundary layer along material handling conveyors, plastic sheet aerodynamic extrusion, the boundary layer along a liquid film in condensation procedures, paper manufacturing, glass blowing, steel spinning and plastic film drawing.

Boundary layer for incompressible flow on a moving flat plate was studied by Sakiadis [1]. The study focused on the flow through a moving flat plate while considered static fluid contrary to the work by Blasius [2] who considered flow over a fixed plate. The study carried out by Crane [3] diverted to the study of boundary layer flow of a fluid with high viscosity and uniform density on

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Tehseen Abbas, University of Education Lahore, Pakistan Anwar Shahid, International Islamic University, Islamabad, Pakistan*

> \*Correspondence: *Muhammad Irfan irfan.akbar30@gmail.com*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *11 September 2019* Accepted: *29 October 2019* Published: *15 November 2019*

#### Citation:

*Irfan M, Farooq MA and Iqra T (2019) Magnetohydrodynamic Free Stream and Heat Transfer of Nanofluid Flow Over an Exponentially Radiating Stretching Sheet With Variable Fluid Properties. Front. Phys. 7:186. doi: 10.3389/fphy.2019.00186*

**44**

a plate being stretched linearly. Magyari and Keller [4] conducted the research on an exponentially stretching steady surface to explore heat and mass transfer in the boundary layers but without variable fluid properties and MHD consideration. Elbashbeshy [5], who researched the features of flow and heat transfer over an exponentially stretching permeable sheet, adds a different dimension to this inquiry. The considered research is without the characteristics of MHD and varying fluid properties. Many researchers have extended the work for different flow model. But most of those studies have been focusing on constant fluid properties. The analysis of boundary layer flow with variable fluid properties on a moving flat plate in a parallel free stream was studied by Bachok et al. [6]. They computed solution numerically. Andersson and Aarseth [7] investigated the properties of fluid under the influence of temperature.

Magnetohydrodynamics (MHD) is the study of the flow of electrically conducting fluids in an electro-magnetic-fields. MHD flow research is of significant concern in contemporary processes of metallurgy and metalworking. Makinde et al. [8] examined the MHD flow of variable viscosity of nanofluid over a radially stretching sheet. They indicated that Brownian motion enhances the rate of mass transfer. Mukhopadhyay et al. [9] carried out the study of investigation of magnetic field effects on a fluid flow with variable viscosity on heated surface. They reported that the fluid velocity reduces as the viscosity declines. The influence of temperature on viscosity during heating surface was investigated by Elbashbeshay and Bazid [10] and evaluated solution with the help of a shooting method. The effect of variable fluid properties on the hydro-magnetic flow and heat transfer over a nonlinearly stretching sheet was discussed by Popley et al. [11]. They have numerically addressed their problem. Similarly, the influence of a study of temperature-dependent fluid properties on MHD free stream flow and heat transfer over a nonlinearly stretching sheet was studied by Prasad et al. [12].

Some important applications for radiative heat transfer are the MHD accelerator, high temperature plasmas, power generation

devices and cooling of nuclear reactors. Many procedures occur in engineering areas at higher temperatures and understanding the transfer of radiative heat becomes very crucial for the design of appropriate equipment. Heat transfer assessment of boundary layer flow with radiation is also vital in electrical power generation, astrophysical flows, solar power technology, and other industrial areas. Raptis et al. [13] recorded the impact of thermal radiation over a semi-infinite stationary plate on the MHD flow of a viscous fluid. Devi and Reddy [14] presented analysis of the radiation and mass transfer effects on MHD boundary layer flow due to an exponentially stretching sheet with heat source. Mukhopadhyay [15] discussed the slip effects on MHD flow over an radiating exponentially stretching sheet with suction/blowing. The influence of radiation effect over an exponentially stretching sheet was studied by Ishak [16] and Mabood et al. [17]. Bidin and Nazar [18] carried out a numerical study to investigate the effect of thermal radiation on boundary layer flow over an exponentially stretching sheet. Poornima and Reddy [19] presented an analysis of the radiation effects on MHD free convective boundary layer flow of nanofluids over a nonlinear stretching sheet. Most of the above studies have not discussed variable fluid properties and radiation simultaneously.

Because of the unique physical and chemical properties of nanometer-sized products, nanofluids have many applications in the industrial sector. Nanofluids are composites of solidliquid materials, typically 1–100 nm, consisting of powerful nanoparticles or liquid-suspended nanofibers. The term nanofluid was suggested by Choi [20]. He revealed that supplying a tiny quantity of nanoparticles to conventional fluids (<1 percent by volume fraction) improved the heat conductivity of the fluid by ∼2 times. Nield and Kuznestov [21] studied convected boundary layer flow of nanofluid in a porous medium. They considered natural convection past a vertical flat plate. Khan et al. [22] presented non-aligned MHD stagnation point flow of nanofluid with variable viscosity over a stretching sheet with radiation effect. They found that non-alignment of the reattachment point decreases with an increase in magnetic parameter M. Bachok et al. [23] discussed stagnation-point and heat transfer flow over an exponentially stretching/shrinking sheet in a nanofluid. They discovered that the solution obtained for shrinking sheet is not unique. Nada et al. [24] examined the effect of nanofluid while variable properties are taken into account. They considered enclosures for the studies. Malik et al. [25] studied Casson nanofluid's boundary layer flow over a cylinder that stretches exponentially and found solution numerically. Eid [26] addressed the impact of chemical reaction over an exponentially stretching sheet on the MHD boundary layer flow of two-phase nanofluid. They found that thermal boundary layer is dependent on the reaction and source parameter. Gangaiah et al. [27] examined the MHD flow of nanofluid in the presence of viscous dissipation and chemical reaction over an exponentially stretching sheet. They showed that thermal boundary layer depends on viscous dissipation parameter. The effect of different variables like variable viscosity, buoyancy and variable thermal conductivity on mixed convection heat transfer due to an exponentially stretching sheet was discussed by Abel et al. [28]. They obtained solution numerically. In Yousif et al. [29] and Ellahi et al. [30], they have discussed MHD Carreau and non-Newtonian nanofluid flow over an exponentially and slippery walls, respectively. Unsteady flow with CNT-based MHD nanofluid, variable viscosity and a permeable shrinking surface have been discussed in Ahmed et al. [31]. See also Thoi et al. [32] for a different fluid flow aspect in a Y-shaped fin. Previous studies mostly concern with nanofluid with variable viscosity but these are devoid of variable thermal conductivity.

There exists a very extensive literature with and without nanofluid on the topic of a constant fluid properties. But not many studies were dedicated to explore the effects of variable fluid properties on nanofluid flow. To bridge that gap, the present research focuses on the effects of variable viscosity and variable thermal conductivity on the boundary layer nanofluid flow. The structure of the paper is as follows. In section 2, we formulate the fundamental physical problem's mathematical model. The constant and variable fluid characteristics are discussed in section 3. The numerical methods are outlined in section 4. Results and analysis are presented in section 5. Conclusion of the current work is drawn at the end in section 6.

#### 2. PROBLEM FORMULATION

We consider a laminar, MHD nanofluid flow over an exponentially stretching sheet with thermal radiation. The sheet is situated at y = 0. A variable magnetic field B(x) = Boe x 2L has been applied normal to the sheet. **Figure 1** is the geometry of the flow, in which x-axis is along and y-axis is taken as normal to the sheet.

Let U<sup>w</sup> = ae x <sup>L</sup> is the wall velocity, whereas U<sup>∞</sup> = be <sup>x</sup> <sup>L</sup> is a free stream velocity, in which stretching parameters a, b > 0. The sheet has been kept at constant wall temperature T<sup>w</sup> and T∞ refers to the ambient temperature. Under the hypothesis of a low magnetic Reynolds number, the induced magnetic field is ignored. The boundary layer equations with Buongiorno model [33] which regulate the above flow are:

$$\frac{\partial \mu}{\partial x} + \frac{\partial \nu}{\partial y} = 0,\tag{1}$$

$$\mu \frac{\partial \mu}{\partial \boldsymbol{x}} + \nu \frac{\partial \nu}{\partial \boldsymbol{y}} = U\_{\infty} \frac{dU\_{\infty}}{d\boldsymbol{x}} + \frac{1}{\rho} \frac{\partial}{\partial \boldsymbol{y}} (\frac{\mu \partial \boldsymbol{u}}{\partial \boldsymbol{y}}) - \frac{\sigma \boldsymbol{B}^{2}}{\rho} (\boldsymbol{u} - U\_{\infty}), \tag{2}$$

$$u\frac{\partial T}{\partial \chi} + \nu \frac{\partial T}{\partial \chi} = \frac{1}{\rho c\_p} \frac{\partial}{\partial \chi} (\frac{k \partial T}{\partial \chi}) + $$

$$\pi (D\_B \frac{\partial T}{\partial \chi} \frac{\partial C}{\partial \chi} + \frac{D\_T}{T\_\infty} (\frac{\partial T}{\partial \chi})^2) - \frac{1}{\rho c\_p} \frac{\partial q\_r}{\partial \chi}, \tag{3}$$

$$
\mu \frac{\partial C}{\partial x} + \nu \frac{\partial C}{\partial y} = D\_B \frac{\partial^2 C}{\partial y^2} + \frac{D\_T}{T\_\infty} \frac{\partial^2 T}{\partial y^2}, \tag{4}
$$

where the coordinates of velocities (u, v) are along x− and y− axes, respectively. µ is a fluid viscosity coefficient, B(x) is a variable magnetic field along the y− axis. Here T is the temperature, C is the nanoparticles concentration, c<sup>p</sup> is the specific heat constant, D<sup>B</sup> is the Brownian motion coefficient, D<sup>T</sup> is the thermophoretic diffusion coefficient, τ = (ρc)p (ρc)<sup>f</sup> is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the fluid and q<sup>r</sup> is the radiative heat flux. Appropriate boundary conditions complete the above system by:

$$\begin{aligned} \mu &= \ U\_{\text{w}}\left(\mathbf{x}\right) = ae^{\frac{\mathbf{y}}{L}}, \nu = 0, T = T\_{\text{w}}, \mathbf{C} = \mathbf{C}\_{\text{w}} \text{ at } \boldsymbol{\nu} = \mathbf{0} \\ \mu &\to \ U\_{\infty} = be^{\mathbf{x}/L}, T \to T\_{\infty}, \mathbf{C} \to \mathbf{C}\_{\infty} \text{ as } \boldsymbol{\nu} \to \infty \end{aligned} \tag{5}$$

Using the following similarity transformation on above equations which are defined as:

$$\eta = \sqrt{\frac{a}{2\upsilon L}} e^{\frac{\chi}{2L}} \jmath, \qquad \psi = \sqrt{2a\upsilon L} e^{\frac{\chi}{2L}} f(\eta), \qquad \theta = \frac{T - T\_{\infty}}{T\_{\text{w}} - T\_{\infty}},$$

$$u = a e^{\frac{\chi}{L}} f'(\eta), \qquad \nu = -\sqrt{\frac{\nu a}{2L}} e^{\frac{\chi}{2L}} (f(\eta) + \eta f'(\eta)). \tag{6}$$

Equation (1) is identically satisfied. Moreover, when above similarity variables used in Equations (2), (3), and (4) which yields:

$$\begin{aligned} \left(\frac{\mu}{\mu\_o}\boldsymbol{\tilde{f}}\,\boldsymbol{\tilde{\up{\{}}}}\right)' + 2(\boldsymbol{\lambda}^2 - \left(\boldsymbol{\tilde{f}}\,\boldsymbol{\tilde{\up{\{}}}}\right)^2) + \boldsymbol{\tilde{f}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{\{}}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{}}}\,\boldsymbol{\tilde{\up{$$

$$(1 + \frac{4}{3}R\_d)(\frac{k}{k\_o}\boldsymbol{\theta}^\prime)^\prime + Pr\_\boldsymbol{\theta}(\boldsymbol{\theta}^\prime - \boldsymbol{f}^\prime \boldsymbol{\theta} + N\_b \boldsymbol{\theta}^\prime \boldsymbol{\phi}^\prime + N\_t(\boldsymbol{\theta}^\prime)^2) = 0,\tag{8}$$

$$\boldsymbol{\phi}^\prime + \frac{N\_t}{N\_b}\boldsymbol{\theta}^\prime + L\_t(\boldsymbol{f}\boldsymbol{\phi}^\prime - \boldsymbol{f}^\prime\boldsymbol{\phi}) = 0.$$

The boundary conditions transformed into:

$$f(0) = 0, f'(0) = 1, \theta(0) = 1,\\ f'(\infty) = \lambda, \theta(\infty) = 0, \phi(0) = 1, \phi(\infty) = 0,\tag{10}$$

where M = 2σB 2 0 L ρa is a magnetic parameter, λ = b a is a ratio of the free stream velocity to the velocity of the stretching sheet, Pr<sup>o</sup> = µocpo ko is the Prandtl number, N<sup>b</sup> = τDB(Cw−C∞) ν is the Brownian motion parameter, N<sup>t</sup> = τDT(Tw−T∞) T∞ν is the thermophoresis parameter, R<sup>d</sup> = 4σ ∗T 3 ∞ k0k <sup>∗</sup> denotes the radiation parameter and L<sup>e</sup> = ν DB is the Lewis number.

#### 3. ANALYSIS ON FLUID PROPERTIES

This section comprises of two subsections. Firstly, an overview of the constant fluid properties will be presented followed by the discussion on variable fluid properties.

(9)

#### 3.1. Case A: Constant Fluid Properties

For this case, Equations (7), (8), and (9) can be adjusted as follows to incorporate constant fluid properties:

$$\left(f'' + 2(\lambda^2 - (\not f)^2) + \not f'' - M(\not f' - \lambda) = 0\right)$$

$$\left(1+\frac{4}{3}R\_d\right)\boldsymbol{\theta}^{\prime}+\boldsymbol{Pr}\_o(\boldsymbol{f}\boldsymbol{\theta}^{\prime}-\boldsymbol{f}^{\prime}\boldsymbol{\theta}+\boldsymbol{N}\_b\boldsymbol{\theta}^{\prime}\boldsymbol{\phi}^{\prime}+\boldsymbol{N}\_l(\boldsymbol{\theta}^{\prime})^2)=\boldsymbol{0}\tag{12}$$

$$
\boldsymbol{\phi}^{\prime} + \frac{N\_t}{N\_b} \boldsymbol{\theta}^{\prime} + L\_t(\boldsymbol{f} \boldsymbol{\phi}^{\prime} - \boldsymbol{f}^{\prime} \boldsymbol{\phi}) = 0 \quad \text{(13)}
$$

### 3.2. Case B: Variable Fluid Properties

For this case, viscosity and thermal conductivity in Equations (7), (8), and (9) is considered variable and taken as a function of a temperature. For viscosity we write:

$$
\mu(T) = \frac{\mu\_{ref}}{1 + \gamma(T - T\_{ref})},
\tag{14}
$$

where we follow Andersson and Aarseth [7] and reference within to write above expression (14). In above γ is a fluid property. If

TABLE 1 | (For Case A) Comparison of −θ ′ (0) for different values of *M*, *R<sup>d</sup>* and *Pr*0, when λ = *N<sup>b</sup>* = *N<sup>t</sup>* = *L<sup>e</sup>* = 0.


TABLE 2 | (For Case B) Comparison of the values of *f* ′′(0), θ ′ (0) and φ ′ (0) for different values of ǫ and λ when *M* = λ = *R<sup>d</sup>* = 0, *Pr*<sup>0</sup> = 1, θ*<sup>r</sup>* = −5, *L<sup>e</sup>* = 1.3.


TABLE 3 | (For Case B) Comparison of the values of *f* ′′(0) and θ ′ (0) for different values of θ*<sup>r</sup>* and λ when *M* = 0, *Pr*<sup>0</sup> = 10, ǫ = 0.


Nb

5

T<sup>o</sup> ≈ Tref then above formula (14) becomes:

$$\mu = \frac{\mu\_o}{1 - \frac{T - T\_o}{\theta\_r (T\_w - T\_o)}} = \frac{\mu\_o}{1 - \frac{\theta \{\eta\}}{\theta\_r}},\tag{15}$$

here θ<sup>r</sup> = −1 γ(Tw−To) . If the above viscosity relation is incorporated in the Equation (7), then it can be rewritten as:

$$\frac{\theta\_r}{\{\theta r - \theta\}}{\{\theta r - \theta\}}^{\prime\prime} + \frac{f^{\prime}\theta^{\prime}\theta\_r}{\{\theta\_r - \theta\}^2} + 2(\lambda^2 - (f^{\prime})^2) + \underline{f}^{\prime} - M(f^{\prime} - \lambda) = 0. \tag{16}$$

The variable thermal conductivity is expressed in terms of temperature by following Prasad et al. [12] as:

$$k(T) = k\_o(1 + \epsilon \theta) \tag{17}$$

Under this above relation the mathematical form of Equation (8) can be described as:

$$(1 + \frac{4}{3}R\_d)((1 + \epsilon\theta)\boldsymbol{\theta}^\prime + \epsilon(\boldsymbol{\theta}^\prime)^2)$$

$$+ P\_{r\_\diamond}(\boldsymbol{f}\boldsymbol{\theta}^\prime - \boldsymbol{f}^\prime\boldsymbol{\theta} + N\_b\boldsymbol{\theta}^\prime\boldsymbol{\phi}^\prime + N\_t(\boldsymbol{\theta}^\prime)^2) = 0.\tag{18}$$

To measure the roughness, heat transfer rate and mass transfer rate onto the surface, we calculate the skin friction coefficient C<sup>f</sup> the local Nusselt number Nu<sup>x</sup> and the local Sherwood number Shx, respectively, i.e.,

$$\mathcal{C}\_{\!f} = \frac{\mathfrak{r}\_{\mathcal{W}}}{\rho U\_{\mathcal{W}}^2} = \frac{f''(0)}{\sqrt{2Re\_{\mathcal{X}}}},\qquad \text{(19)}$$

$$Nu\_{\mathbf{x}} = -\frac{\mathbf{x}q\_{\mathbf{w}}}{T\_{\mathbf{w}} - T\_{\infty}} = -\sqrt{\frac{\mathbf{x}\mathbf{Re}\_{\mathbf{x}}}{2L}}\boldsymbol{\theta}'(\mathbf{0})\,,\tag{20}$$

$$\text{Sh}\_{\text{x}} = -\frac{\text{x}\_{\text{W}}}{\text{C}\_{\text{w}} - \text{C}\_{\text{\infty}}} = -\sqrt{\frac{\text{x} \text{Re}\_{\text{x}}}{2L}} \phi'(0). \tag{21}$$

#### 4. NUMERICAL PROCEDURE

#### 4.1. Shooting Method

To apply the shooting technique to Cases A and B together with the boundary conditions, we transformed boundary value problem (BVP) into an initial value problem (IVP) and convert higher order ODEs into a system of first order ODEs. The Newton-Raphson technique was used to locate the root. After that, the order five Runge-Kutta method was implemented to determine the IVP solution. The shooting method is implemented in MATLAB. For Cases A and B, the system of first order ODEs are written as,

(a) Case A:

$$\begin{aligned} f &= \boldsymbol{\jmath}\_1, \boldsymbol{\jmath}' = \boldsymbol{\jmath}\_2, \boldsymbol{\jmath}'' = \boldsymbol{\jmath}\_3, \boldsymbol{\jmath}''' = \boldsymbol{\jmath}\_3' = -2(\boldsymbol{\lambda}^2 - \boldsymbol{\jmath}\_2^2) - \boldsymbol{\jmath}\_1 \boldsymbol{\jmath}\_3 \\ &+ M(\boldsymbol{\jmath}\_2 - \boldsymbol{\lambda}), \\ \boldsymbol{\jmath}\_4 = \boldsymbol{\theta}, \boldsymbol{\jmath}\_5 = \boldsymbol{\theta}', \boldsymbol{\theta}'' = \boldsymbol{\jmath}\_5' = -\frac{\text{Pr}\_o}{(1 + \frac{4}{3})\text{R}\_d} \{\boldsymbol{\jmath}\_1 \boldsymbol{\jmath}\_5 - \boldsymbol{\jmath}\_2 \boldsymbol{\jmath}\_4 + N\_b \boldsymbol{\jmath}\_5 \boldsymbol{\jmath}\_7 \\ &+ N\_l \boldsymbol{\jmath}\_5^2 \}, \\ \boldsymbol{\jmath}\_6 = \boldsymbol{\phi}, \boldsymbol{\wp}\_7 = \boldsymbol{\phi}', \boldsymbol{\phi}'' = \boldsymbol{\jmath}\_7' = -L\_e \{\boldsymbol{\jmath}\_1 \boldsymbol{\jmath}\_7 - \boldsymbol{\jmath}\_2 \boldsymbol{\jmath}\_8 \} - \frac{N\_l}{N\_l} \boldsymbol{\jmath}\_5'. \end{aligned}$$

(b) Case B:

$$f = \mathcal{y}\_1, f' = \mathcal{y}\_2, f'' = \mathcal{y}\_3, f''' = \mathcal{y}\_3' = \frac{(\mathcal{y}\mathcal{y}\mathcal{y})}{(\mathcal{y}\_4 - \theta\_7)}$$

$$+ \frac{(\mathcal{y}\_4 - \theta\_7)}{\theta\_r} (2(\lambda^2 - \mathcal{y}\_2^2) + \mathcal{y}\_1\mathcal{y}\_3 - M(\mathcal{y}\_2 - \lambda)),$$

$$\begin{split} \mathcal{y}\_4 &= \theta, \mathcal{y}\_5 = \boldsymbol{\theta}', \boldsymbol{\theta}'' = \boldsymbol{\mathcal{y}}\_5' = \frac{-\epsilon\mathcal{y}\_5^2}{1 + \epsilon\mathcal{y}\_4} \\ &- \frac{P\_{r\_0}}{(1 + \epsilon\mathcal{y}\_4)(1 + \frac{4}{3}R\_d)} (\mathcal{y}\_1\mathcal{y}\_5 - \mathcal{y}\_2\mathcal{y}\_4 + N\_b\mathcal{y}\_5\mathcal{y}\_7 + N\_t\mathcal{y}\_5^2), \end{split}$$

$$\begin{split} \mathcal{y}\_6 &= \boldsymbol{\phi}, \boldsymbol{\eta}\boldsymbol{\tau} = \boldsymbol{\phi}', \boldsymbol{\phi}'' = \boldsymbol{\upmu}' = -L\boldsymbol{\epsilon} (\boldsymbol{\eta}\boldsymbol{\iota}\boldsymbol{\gamma} - \boldsymbol{\chi}\boldsymbol{\varrho}\boldsymbol{\epsilon}) - \frac{N\_t}{N\_b}\boldsymbol{\upmu}', \end{split}$$

TABLE 4 | (For Case B) Comparison of the values of *f* ′′ (0) and θ ′ (0) for different values of *R<sup>d</sup>* and *Pr*<sup>0</sup> when *M* = λ = ǫ = 0, θ*<sup>r</sup>* = −5, *N<sup>b</sup>* = 0.8, *N<sup>t</sup>* = 0.5, *L<sup>e</sup>* = 1.3.


FIGURE 2 | Velocity profile *f*

Using MATLAB bvp4c algorithm, BVP can even be solved. bvp4c solver employs the collocation technique in the background. It manages to find a solution after supplying initial guess, domain size and the number of points. Please see reference [34] for more detail and examples.

(η) for different *M* (For Case B).

### 5. RESULTS AND DISCUSSION

′

In **Table 1**, we compute the local Nusselt number and compared its values with published results for distinct parameters Prandtl number Pr0, radiation parameter Rd and magnetic parameter M.

**Table 2** illustrates that the skin friction coefficient is not significantly changed whereas the local Nusselt number drops for ǫ and increases for the values of λ. The local Sherwood number grows with the rise of λ and ǫ. It is observed in **Table 3** that the local Nusselt and the Sherwood numbers rises with a rise of λ but the skin friction coefficient held opposite behavior. For fixed values of λ = 0, 0.5 and an increase in viscosity parameter θ<sup>r</sup> brings the increasing change in the skin friction coefficient but the local Nusselt and Sherwood numbers has shown decreasing behavior. **Table 4** demonstrates that as Pr<sup>o</sup> and R<sup>d</sup> rises, there is a negligible change in the skin friction coefficient. But the local Nusselt numbers decreases and local Sherwood number increases by increasing the values of radiation parameter Rd. Moreover, the local Nusselt number increases by increasing Prandtl number but the local Sherwood number decreases.

**Figure 2** shows that the momentum boundary layer thickness is reduced with the increase in M. It happens because of a

transverse magnetic field as it opposes the phenomenon of transport. The Lorentz force generates resistance to the fluid flow with a rise of M and slows down the velocity.

In **Figure 3**, we observe that by rising the viscosity parameter θr , a momentum boundary layer thins. **Figure 4** shows that there is a rise in temperature profile with an increase in thermal conductivity parameter ǫ.

**Figures 5**, **6** are plotted for different values of Brownian motion parameter N<sup>b</sup> and we observe that by increasing N<sup>b</sup> thermal boundary layer thickness increases while concentration boundary layer decrease by increasing N<sup>b</sup> .

As seen in **Figures 7**, **8** that by increasing thermophoresis parameter N<sup>t</sup> , temperature and concentration profiles increases.

FIGURE 11 | Concentration profile φ(η) for different *Le* (For Case B).

**Figure 9** indicates that by increasing Pr<sup>o</sup> the thermal boundary layer thickness decreases. This is because, when Pr<sup>o</sup> increases, the thermal diffusivity decreases and thus the heat is diffused away from the heated surface more slowly and in consequence increase the temperature gradient at surface.

**Figure 10** shows that temperature and thermal boundary layer thickness increases when the radiation parameter intensifies. **Figure 11** describe the influence of the Lewis number Le on concentration profile. We observe that by increasing Le there is decrease in concentration profile. Lewis number is the ratio of Prandtl number and Schmidt number, so with the increase in Lewis number Le, molecular diffusivity decreases. As a result, increase in Le the nanoparticle fraction is lowered.

#### 6. CONCLUSIONS

The current study offers the findings of a two-dimensional MHD flow of an incompressible fluid through an exponentially stretched sheet whereas treating viscosity and thermal conductivity constant in Case A and variable for Case B. The significance of various parameters on velocity, temperature and concentration is examined. The study's main results for Case B are as follows:

• Momentum boundary layer thickness decrease by increasing fluid viscosity parameter θ<sup>r</sup> and magnetic parameter M.


#### REFERENCES


### DATA AVAILABILITY STATEMENT

All datasets generated for this study are included in the article/supplementary material.

### AUTHOR CONTRIBUTIONS

MI collected the data and wrote the paper. MF made the analysis of the paper. TI made the geometry of problem and arrange the setting of the paper.


Methods Heat Fluid Flow. (2019). doi: 10.1108/HFF-04-2019-0346. [Epub ahead of print].


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Irfan, Farooq and Iqra. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# Computational and Physical Examination About the Aspects of Fluid Flow Between Two Coaxially Rotated Disks by Capitalizing Non-fourier Heat Flux Theory: Finite Difference Approach

Sardar Bilal <sup>1</sup> \*, Asifa Tassaddiq<sup>2</sup> , A. H. Majeed<sup>1</sup> , Kottakkaran Sooppy Nisar <sup>3</sup> , Farhad Ali <sup>4</sup> and M. Y. Malik <sup>5</sup>

*<sup>1</sup> Department of Mathematics, Air University, Islamabad, Pakistan, <sup>2</sup> College of Computer and Information Sciences, Majmaah University, Al Majma'ah, Saudi Arabia, <sup>3</sup> Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia, <sup>4</sup> Department of Mathematics, City University of Science and Information Technology, Peshawar, Pakistan, <sup>5</sup> Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia*

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Anwar Shahid, Nanjing University of Aeronautics and Astronautics, China Ilyas Khan, Ton Duc Thang University, Vietnam Kh S. Mekheimer, Al-Azhar University, Egypt*

> \*Correspondence: *Sardar Bilal sardarbilal@mail.au.edu.pk*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *21 September 2019* Accepted: *21 November 2019* Published: *09 January 2020*

#### Citation:

*Bilal S, Tassaddiq A, Majeed AH, Nisar KS, Ali F and Malik MY (2020) Computational and Physical Examination About the Aspects of Fluid Flow Between Two Coaxially Rotated Disks by Capitalizing Non-fourier Heat Flux Theory: Finite Difference Approach. Front. Phys. 7:209. doi: 10.3389/fphy.2019.00209* This pagination is executed to exemplify flow features exhibited by viscous fluid between two coaxially rotated disks. Thermal analysis is performed by using Cattaneo-Christov heat flux theory. Porosity aspects are also taken into account. Mathematically structured non-linear PDEs are transmuted into non-linear ODEs by employing Karman transformations. Afterward, solution is heeded by applying implicit finite difference scheme renowned as Keller box method. Interpretation of flow controlling parameters on axial, tangential, and radial components of velocity, thermal distribution is exhibited. Assurance of computed data is done by managing comparison for skin friction coefficients at walls of disks. From the attained outcomes, it is addressed that the magnitude of axial and radial velocities diminishes at lower disk contrary to upper disk for intensifying magnitude of Reynolds number. Increment in tangential component of velocity is also demonstrated for uplifts values of Reynolds number. It is also concluded that thermal field decrements for increasing of *Pr* and thermal relaxation parameter. It is worthy to mention that shear drag coefficient at wall of lower disk decreases conversely to the wall shear coefficient magnitude at wall of upper disk.

Keywords: Cattaneo Christov heat flux model, permeable medium, fluid flow with coaxially rotated disks, implicit finite difference scheme, coaxially rotated disks, viscous fluid

### INTRODUCTION

Rotational fluid flow generated by coaxial disks is one of the classical problems of fluid mechanics. In recent years, it has become a popular research area and has persuaded researchers due to magnificent theoretical and practical significance in engineering and applied sciences. Some important practical fields in which rotatory flow is capitalized are rotor–stator system, gas turbine engineering, air rotational cleaners, medical equipment, chemical engineering, and thermal power–generating systems. In view of its capitalization in various processes, researcher fraternity is

**55**

examining such type of flows in current days. Inaugurated work on flow induced due to rotating disk is performed by Karman [1]. He introduced transformations and provided a mathematical framework for construction of ordinary differential systems of rotational flows from Navier Stokes theory. Cochran [2] also used these transformations to scrutinize rotating disk flow by using numerical integration scheme. Batchelor [3] validated that Karman transformation can be evenly used for fluid flow between two coaxial rotating disks. Rotating flow by two coaxial disks is primarily examined by Stewartson [4]. Chapple and Stokes [5] elucidated the flow features of fluid between two coaxially rotated disks. Mellor et al. [6] bestowed comprehensive treatment of fluid flow restricted between two coaxial infinite disks, one rotating, and other stationary. Thermal aspects of fluid between rotational disks were discussed by Arora and Stokes [7]. Interpretation of flow phenomenon between porous stationary disk and solid rotating disk was manipulated by Kumar et al. [8]. Xun et al. [9] considered rotating disk of variable thickness and adumbrated the flow features of Power law fluid. Hall effects on an unsteady MHD (magneto hydrodynamics) flow of viscous incompressible electrically conducting fluid between two rotating disks with noncoincident parallel axes embedded in a porous viscous medium were accorded by Das et al. [10]. Asgher et al. [11] conducted Lie group analysis on the thermal features of fluid manifested by rotating disks. Elmaboud et al. [12] discussed peristaltic flow induced by sinusoidal wave propagating with constant speed on the walls of two-dimensional infinite rotating channel by heeding semi-analytical solutions.

In the most recent couple of decades, researcher fraternity has shown fantastic energy in exploring the heat propagation by means of a wave mechanism rather than essentially by diffusion. Late studies affirm that this is not just a lowtemperature phenomenon but heat transfer mechanism also occurs at high temperature through diffusion. Just about 200 years prior, thermal features in various circumstances and especially in flowing fluid environment were interpreted by Fourier law of heat conduction [13]. However, this law is inadequate in comprehending complete description about the heat exchange procedure among multiple connected surfaces in various conditions because of its disablement to fulfill the principle of causality. Later on, in 1948, Cattaneo [14] modified Fourier law by viewing the inadequateness generated by Fourier law of heat conduction and explored that this law explains the thermal attribute at low temperature because it generates parabolic heat equation in which initial disturbance are felt throughout the domain. After getting thorough analysis about Fourier law and viewing vector field aspect of heat flux, he included thermal relaxation time term to control generated thermal inertia, which is known as Maxwell-Cattaneo law. Afterward, Christov [15] proposed that objective time derivative instead of material time derivative is used for exact fulfillment of causality principle. He changed the time derivative in Maxwell-Cattaneo model by Oldryod upper convective derivative, which has successfully preserved the material invariant formulation and famously known as Cattaneo-Christov heat flux law. Cattaneo-Christov heat flux model has bounteous applications in engineering and modern industrial procedures like in skin burns and nanofluids, cooling of electronic devices, food technology, nuclear reactor cooling, power generation, heat exchangers, heat propagation in tissues, and so many. The uniqueness and stability of the solution for governing temperature equations by Cattaneo-Christov model in some initial and boundary value problems were proven by Straughan [16]. Additionally, steadiness of structure of Cattaneo-Christov heat flux model with uniqueness was revealed by Ciarletta and Straughan [17]. Tibullo and Zampoli [18] explicated the behavior of Cattaneo-Christov heat flux model in incompressible fluid flows. Aqsa [19] and Haddad [20] heeded numerical solution for thermal convection of an incompressible viscous fluid by obliging Cattaneo-Christov heat flux model. Mekheimer and Elmaboud [21] interpreted the aspects of temperature-dependent viscosity and thermal conductivity on peristaltic flow of a Newtonian fluid in a vertical asymmetric channel. Mekneimer [22] addressed heat transfer features of peristaltic couple stress fluid in asymmetric channel generated by wave with different phase and amplitudes. All of the abovementioned thought-provoking investigations have generated prodigious interest of researchers toward the analysis of flow in the presence of thermal aspects [23–26].

Transport procedures through porous space are commonly encountered in various chemical, mechanical, geophysical, electrochemical, and metallurgical routines. The theory about macroscopic movement of fluid in porous medium comprises differential equation that expresses linear relation between velocity and pressure gradient. Initially, Henry Darcy [27] (in 1856) presented a law to explicate the dynamic phenomenon in porous medium by working on the flow of sandy water through pebbles. Several technological processes depend on porous media theory, such as hydrology, oil exploration, solar collectors, porous insulations, packed beds, chromatography, heterogeneous catalysis, control of shear stresses at the seabed bottom, and oscillatory flow through seabed ripples. Darcy theory has promising applications in the field of biomedicine and the development of biological clogging and flow through tissues [28]. Granular material [29] where significant amount of pore structures exist has application in manufacturing, paper, ceramic products, and textiles. Taseer et al. [30] addressed the flow behavior of Maxwell nanofluid in porous medium by implementing zero mass flux condition. They capitalized on Darcy-Forchheimer law to depict the flow pattern. They found that porosity parameter mounts the magnitude of temperature and concentration of particles. Seddeek [31] probed convective heat transfer in fluid immersed in porous medium. Analytical results for Darcy flow was described by Jha and Kaurangini [32]. Aziz et al. [33] computed the traveling wave solution for the time-dependent viscoelastic fluid by way of a porous flat plate.

Magnetohydrodynamics is the study of the interaction between magnetic field and conductive fluid. The essence about magnetization is that the external magnetic field controls the turbulence in flow field. In addition, the magnetized flows differ from ordinary fluids because the generated current in the bulk fluid produces volumetric Lorentz force that extensively modifies the features. In recent years, magnetization and its impact on flow features have attained pervasive focus due to its extraordinary industrial applications, such as magnetized materials processes, manufacturing of glass, and MHD controlled electric generators. So the analysis of application of magnetic field has experienced great development and diversity. Andersson [34] performed exclusive study by manipulating electromagnet hydrodynamic waves mathematically. The stretched flow of two-dimensional Newtonian fluid under the effects of applied magnetic field was contemplated by Andersson [34]. Liu [35] extended the work of Andersson [34] and described the heat and mass transfer of MHD viscous fluid flow over stretching surface. He computed exact solution of the problem by following the procedure of Andersson [34]. The impact of normally impinging magnetic field on boundary layer flow of Newtonian fluid over permeable stretching sheet was analyzed by Kumaran et al. [36]. Yirga and Tesfay [37] developed the numerical simulations for MHD viscous fluid flow over non-linear stretching sheet. The fluid flow equations were solved via Keller-Box method, and variations in physical quantities were presented regarding different parametric conditions. Recently, Yasin et al. [38] simulated the problem of two-dimensional MHD viscous nanofluid flow over porous stretched sheet. The formulated equations were solved by implementing well-known shooting technique. Mabood et al. [39] developed the approximate analytic solution of MHD boundary layer fluid flow over exponentially stretching surface. Some of the literature regarding the mentioned aspects is accessed through the references [40–43].

Present disquisition is addressed to excogitate thermophysical features exhibited in viscous fluid flow between two coaxially rotating disks embedded in permeable medium by obliging Cattaneo-Christov heat flux law. According to author's knowledge and available literature survey, it is found that very concise work is done so far in this direction. Tremendous engineering and practical application generated by rotating disk flows make present analysis highly potential. The authors hope that this manuscript will serve as a reference study for future researches. The article is strategized in such a way that the literature assessment is presented in section Introduction, whereas the mathematical structuring is provided in section Mathematical Model. The explanation about the solution methodology is debated in section Numerical Procedure. Comprehensive analysis and interpretation of flow controlling parameters are disclosed in section Results and Discussion. Last, the outcome norms are listed in section Conclusions.

#### MATHEMATICAL MODEL

Consider a steady, incompressible flow of viscous fluid between two coaxially rotated disks. The lower disk is placed at z = 0, whereas the distance between the disks is h units. Lower and upper disks possess angular velocities <sup>1</sup> and 2, respectively, and a<sup>1</sup> and a<sup>2</sup> are corresponding stretching rates (**Figure 1**). Porous medium between disks is considered, and Cattaneo-Christov heat flux model is obliged to analyze thermal features of fluid flow model.

We have used cylindrical coordinates (r, θ, z) with velocity components (uˆ, vˆ,wˆ) to the velocity profile and temperature equations as follows:

uˆ

uˆ

uˆ

$$\begin{aligned} \frac{\partial \hat{u}}{\partial r} + \frac{\hat{u}}{r} + \frac{\partial \hat{w}}{\partial z} &= 0, \\ \frac{\partial \hat{u}}{\partial r} + \hat{w} \frac{\partial \hat{u}}{\partial z} - \frac{\hat{v}^2}{r} &= -\frac{1}{\rho} \frac{\partial \hat{p}}{\partial r} + \nu \left( \frac{\partial^2 \hat{u}}{\partial r^2} + \frac{1}{r} \frac{\partial \hat{u}}{\partial r} + \frac{\partial^2 \hat{u}}{\partial z^2} - \frac{\hat{u}}{r^2} \right) \end{aligned} \tag{1}$$

$$-\frac{\sigma\beta\_0^2}{\rho}\hat{u} - \frac{\mu}{k\_0}\hat{u},\tag{2}$$

$$\varepsilon\_{\alpha}\partial\hat{\nu}\_{\alpha\beta}\hat{u}\hat{\nu}\_{\alpha\beta}\hat{u}\hat{\nu}\_{\alpha\beta} + \varepsilon\_{\alpha}\left(\partial^2\hat{\nu}\_{\alpha\beta}\hat{u}^2\hat{u}^2\hat{\nu}\_{\alpha\beta}\hat{u}^2\hat{\nu}\_{\alpha\beta}\hat{u}^2\hat{\nu}\_{\alpha\beta}\hat{u}\hat{\nu}\_{\alpha\beta}\right)$$

$$\frac{\partial\hat{\boldsymbol{\nu}}}{\partial r} + \hat{\boldsymbol{\nu}}\frac{\partial\hat{\boldsymbol{\nu}}}{\partial z} + \frac{\hat{\boldsymbol{\mu}}\hat{\boldsymbol{\nu}}}{r} = \boldsymbol{\nu} \left( \frac{\partial^2\hat{\boldsymbol{\nu}}}{\partial r^2} + \frac{1}{r}\frac{\partial\hat{\boldsymbol{\nu}}}{\partial r} + \frac{\partial^2\hat{\boldsymbol{\nu}}}{\partial z^2} - \frac{\hat{\boldsymbol{\nu}}}{r^2} \right)$$

$$-\frac{\sigma\beta\_0^2}{r}\hat{\boldsymbol{\nu}} - \frac{\mu}{\hbar}\hat{\boldsymbol{\nu}}\tag{3}$$

$$\begin{aligned} \frac{\partial \hat{\boldsymbol{w}}}{\partial r} + \hat{\boldsymbol{w}} \frac{\partial \hat{\boldsymbol{w}}}{\partial z} &= -\frac{1}{\rho} \frac{\partial \hat{\boldsymbol{p}}}{\partial z} + \nu \left( \frac{\partial^2 \hat{\boldsymbol{w}}}{\partial r^2} + \frac{1}{r} \frac{\partial \hat{\boldsymbol{w}}}{\partial r} + \frac{\partial^2 \hat{\boldsymbol{w}}}{\partial z^2} \right) \\ &- \frac{\mu}{k\_0} \hat{\boldsymbol{w}}, \end{aligned} \tag{4}$$

$$
\rho \mathcal{C}\_p \left( \hat{\boldsymbol{\mu}} \frac{\partial \hat{T}}{\partial r} + \hat{\boldsymbol{w}} \frac{\partial \hat{T}}{\partial z} \right) = -\nabla \cdot \overrightarrow{\boldsymbol{q}} \,, \tag{5}
$$

with boundary conditions:

$$\begin{aligned} \hat{u} &= ra\_1, \,\hat{\nu} = r\Omega\_1, \,\hat{\nu} = 0, \,\hat{T} = \hat{T}\_1 \,\text{at} \, z = 0, \\ \hat{u} &= ra\_2, \,\hat{\nu} = r\Omega\_2, \,\hat{\nu} = 0, \,\hat{T} = \hat{T}\_2 \,\text{at} \, z = h, \end{aligned} \tag{6}$$

where Equations (3–5) are referred to Hayat et al. [25], also pressure is expressed as pˆ, Tˆ <sup>1</sup>and <sup>T</sup><sup>ˆ</sup> <sup>2</sup> are the temperatures of upper and lower disks, and flux of heat −→<sup>q</sup> satisfies:

$$\overrightarrow{\overline{q}} + \overline{\chi} \left( \frac{\partial \overrightarrow{\overline{q}}}{\partial t} + \mathbf{V}.\overrightarrow{\nabla \overline{q}} - \overrightarrow{\overline{q}}.\nabla \mathbf{V} + (\nabla.\mathbf{V})\overrightarrow{\overline{q}} \right) = -k\nabla \hat{T}, \text{ (7)}$$

where γ is the thermal relaxation parameter (It is defined as the parameter that controls the speed of heat waves produced within the system and makes them move with finite speed to follow the principle of causality), and k is the thermal conductivity. Now, we omit q from the Equations (5, 7) and obtain:

$$\left(\hat{u}\frac{\partial\hat{T}}{\partial r} + \hat{w}\frac{\partial\hat{T}}{\partial z}\right) = \frac{k}{\rho C\_p} \left(\frac{\partial^2\hat{T}}{\partial r^2} + \frac{1}{r}\frac{\partial\hat{T}}{\partial r} + \frac{\partial^2\hat{T}}{\partial z^2}\right)$$

$$-\gamma \left(\hat{u}^2\frac{\partial^2\hat{T}}{\partial r^2} + \hat{w}^2\frac{\partial^2\hat{T}}{\partial z^2} + 2\hat{u}\hat{v}\frac{\partial^2\hat{T}}{\partial r\partial z}\right)$$

$$+ \left(\hat{u}\frac{\partial\hat{u}}{\partial r} + \hat{w}\frac{\partial\hat{u}}{\partial z}\right)\frac{\partial\hat{T}}{\partial r} + \left(\hat{u}\frac{\partial\hat{w}}{\partial r} + \hat{w}\frac{\partial\hat{w}}{\partial z}\right)\frac{\partial\hat{T}}{\partial z}\right). \tag{8}$$

Equations (2–5) and Equation (8) are transformed into ordinary differential equations by obliging Von Karman transformations [1]:

$$
\hat{u} = r\Omega\_1 f'\left(\zeta\right), \hat{v} = r\Omega\_1 g\left(\zeta\right), \hat{w} = -2h\Omega\_1 f\left(\zeta\right), \dots
$$

θ =

Tˆ − Tˆ 2

Tˆ <sup>1</sup> <sup>−</sup> <sup>T</sup><sup>ˆ</sup> 2 ,

Mass conservation is identically satisfied, and Equations (2–4, 6, 8) take the following form:

$$f^{\prime\prime\prime} + \text{Re}\left(2f^{\prime\prime} - f^{\prime2} + g^2 - \frac{1}{\beta}f^{\prime} + Mf^{\prime}\right) - \epsilon = 0,\tag{10}$$

$$\operatorname{Re}\left(2f'\mathbf{g} - 2\mathfrak{f}\mathbf{\hat{g}}' + \frac{1}{\beta}\mathbf{g} + \operatorname{Mg}\right) - \mathbf{g}'' = 0,\qquad(11)$$

$$P' = \operatorname{Re}\left(\frac{2}{\beta}f - 4f'\right) - 2f'',\qquad \text{(12)}$$

$$\frac{1}{Pr}\theta^{\prime\prime} + 2\text{Ref}\theta^{\prime} - 4\lambda \text{Re}\left(f^2 \theta^{\prime\prime} + f f^{\prime} \theta^{\prime}\right) = 0,\tag{13}$$

$$f\left(0\right) = 0, f\left(1\right) = 0, f'\left(0\right) = A\_{1,}f'\left(1\right) = A\_2,\text{ g}\left(0\right) = 1,$$

$$g\left(1\right) = \text{r}, \theta\left(0\right) = 1, \theta\left(1\right) = 0, P\left(0\right) = 0,\text{ (14)}$$

where

$$\begin{aligned} \text{Re} &= \frac{\Omega\_1 h^2}{\nu\_f}, \Pr = \frac{\left(\rho \mathcal{C}\_p\right)\_f \nu\_f}{k\_f}, \lambda = \mathcal{V}\Omega\_1, \\\ A\_1 &= \frac{a\_1}{\Omega\_1}, \ A\_2 = \frac{a\_2}{\Omega\_2}, \ \mathfrak{r} = \frac{\Omega\_2}{\Omega\_1}, \beta = \frac{k\_0 \Omega\_1}{\nu}, \end{aligned} \tag{15}$$

where Re denotes Reynolds number, Pr is the Prandtl number, A<sup>1</sup> and A<sup>2</sup> are scaled stretching parameters, λ is the thermal relaxation, τ and β are rotational number and porosity parameter.

with

FIGURE 5 | Behavior of *f*(ξ ) for different *A*1.

To make a simpler form of Equation (10), we removed ǫ.

$$f^{\{i\nu\}} + \text{Re}\left(2f^{\prime\prime\prime} + 2\text{gg}^{\prime} - \frac{1}{\beta}f^{\prime\prime} - Mf^{\prime\prime}\right) = 0.\tag{16}$$

The pressure parameter ǫ can be found by using Equations (10 and 14) as:

$$\epsilon = f''\left(0\right) - \operatorname{Re}\left(\left(f'\left(0\right)\right)^2 - \left(\operatorname{g}\left(0\right)\right)^2 + \frac{1}{\beta}f'\left(0\right) - Mf'\left(0\right)\right). \tag{17}$$

Equation (17) vanishes due to the given initial conditions in Equation (14).

The radial and tangential components of shear stress at lower disk are τzr and τz<sup>θ</sup>

$$\mathfrak{r}\_{zr} = \mu \left. \frac{\partial \hat{u}}{\partial z} \right|\_{z=0} = \frac{\mu r \Omega\_1 f''(0)}{h}, \ \mathfrak{r}\_{z\theta} = \mu \left. \frac{\partial \hat{\nu}}{\partial z} \right|\_{z=0}$$

$$=\frac{\mu r \Omega\_1 g'(0)}{h}.\qquad(18)$$

where τ<sup>w</sup> is the total shear stress, which is defined as:

$$
\pi\_{\mathcal{W}} = \sqrt{\mathfrak{r}\_{zr}^2 + \mathfrak{r}\_{z\theta}^2}. \tag{19}
$$

C<sup>f</sup> <sup>1</sup> and Cf2 are the skin friction coefficients at lower and upper disks defined as:

$$\mathcal{C}\_{\!\!\!1} = \frac{\mathfrak{r}\_{\text{w}}|\_{z=0}}{\rho \left(r \mathfrak{Q}\_1\right)^2} = \frac{1}{Re\_r} \left( \left(f^{\prime\prime}(0)\right)^2 + \left(g^{\prime}(0)\right)^2 \right)^{1/2},\tag{20}$$

$$\mathcal{C}\_{\mathcal{f}2} = \frac{\mathfrak{r}\_{\text{w}}\vert\_{z=h}}{\rho \left(r\mathfrak{Q}\_{1}\right)^{2}} = \frac{1}{Re\_{r}}\left(\left(\mathcal{f}''\left(1\right)\right)^{2} + \left(\mathcal{g}'\left(1\right)\right)^{2}\right)^{1/2},\tag{21}$$

where Re<sup>r</sup> = r1h v is the local Reynolds number.

#### NUMERICAL PROCEDURE

Manipulation of accurate solution is necessary for physical interpretation of current work. Initially, equations are modeled by using Karman approximation and afterward, we have attained an intricate system of ordinary differential Equations (10–13) along with boundary conditions in Equation (14). We have applied Keller-box scheme referred to [40, 44– 46], that is, the implicit finite difference scheme. For the implementation of this technique, first, we have to transform it into a system of first-order equations and define new variables u y, ζ , v y, ζ ,

$$\begin{aligned} \, \, \_W \left( \wp, \xi \right), \, \_s \left( \wp, \xi \right), \, t \left( \wp, \xi \right) \, \_and \, \, \theta \left( \wp, \xi \right) = q(\wp, \xi) \, \, \text{are} \\\ f' = \, \_t \,, \, \iota' = \nu, \, \nu' = \nu, \, \text{g}' = s \, \text{and} \, q' = t, \end{aligned} \tag{22}$$

and Equations (11–13, 16) are reduced to

t

$$\mathbf{s}' - \mathrm{Re}\left(2\imath\mathfrak{g} - 2\mathfrak{f}\mathfrak{s} + \mathrm{Mg} + \frac{1}{\beta}\mathfrak{g}\right) = \mathbf{0},\tag{23}$$

$$f' + 2Pr\text{Ref}f - 4Pr\lambda \text{Re} \left(f^2 t' + fut\right) = 0,\tag{24}$$

$$\mathcal{W}' + \operatorname{Re}\left(2f\nu + 2\operatorname{gs} - M\nu - \frac{1}{\beta}\nu\right) = 0.\tag{25}$$

Similarly, the boundary conditions are converted into the following forms

$$\begin{array}{ll} f\ (0) = 0, \ u\ (0) = A\_1, \ g\ (0) = 1, q\ (0) = 1, \\ f\ (1) = 0, \ u\ (1) = A\_2, \ g\ (1) = \text{\textquotedblleft}, \ q\ (1) = 0. \end{array} \tag{26}$$

Average and center difference gradients at the point of net derivatives are demarcated in **Figure 2** and mathematically

as below

$$
\eta\_o = 0, \eta\_k = \eta\_{k-1} + \eta\_k, \quad k = 1, 2, 3, \dots, k \quad \eta\_k = \eta\_{\infty}.
$$

Applying the Newton iteration fk+<sup>1</sup> = f<sup>k</sup> + δf<sup>k</sup> , for all dependent variables involved in linearized non-linear algebraic equations and substituting these expressions in non-linear equations and neglecting quadratic and higher order terms in δ, a linear tridiagonal system is presented as follows:

$$
\begin{aligned}
\delta f\_k - \delta f\_{k-1} - \frac{h\_k}{2} \left(\delta u\_k + \delta u\_{k-1}\right) &= (r\_1) \underset{k-\ \begin{array}{c} 1 \ \text{ $k$ } \\ 2 \end{array}}{ $k - \delta u\_k - \frac{h\_k}{2}$ }, \\
\delta u\_k - \delta u\_{k-1} - \frac{h\_k}{2} \left(\delta \nu\_k + \delta \nu\_{k-1}\right) &= (r\_5) \underset{k-\ \begin{array}{c} 1 \ \text{ $k$ } \\ 2 \end{array}}{ $k - \frac{1}{2}$ }.
\end{aligned}
$$

δv<sup>k</sup> − δvk−<sup>1</sup> − hk 2 δw<sup>k</sup> + δwk−<sup>1</sup> = (r6) k− 1 2 , δg<sup>k</sup> − δgk−<sup>1</sup> − hk 2 δs<sup>k</sup> + δsk−<sup>1</sup> = (r7) k− 1 2 , δq<sup>k</sup> − δqk−<sup>1</sup> − hk 2 δt<sup>k</sup> + δtk−<sup>1</sup> = (r8) k− 1 2 , (a1)<sup>k</sup> δf<sup>k</sup> + (a2)<sup>k</sup> δfk−<sup>1</sup> + (a3)<sup>k</sup> δu<sup>k</sup> + (a4)<sup>k</sup> δuk−<sup>1</sup> + (a5)<sup>k</sup> δg<sup>k</sup> + (a6)<sup>k</sup> δgk−<sup>1</sup> + (a7)<sup>k</sup> δs<sup>k</sup> + (a8)<sup>k</sup> δsk−<sup>1</sup> = (r2)k<sup>−</sup> <sup>1</sup>/<sup>2</sup> , b1 k δf<sup>k</sup> + b2 k δfk−<sup>1</sup> + b3 k δu<sup>k</sup> + b4 k δuk−<sup>1</sup> + b5 k δt<sup>k</sup> + b6 k δtk−<sup>1</sup> = (r3)k<sup>−</sup> <sup>1</sup>/<sup>2</sup> ,

$$\begin{aligned} \left( (\mathfrak{c}\_{1})\_{k} \, \delta f\_{k} + (\mathfrak{c}\_{2})\_{k} \, \delta f\_{k-1} + (\mathfrak{c}\_{3})\_{k} \, \delta \nu\_{k} + (\mathfrak{c}\_{4})\_{k} \, \delta \nu\_{k-1} \\ + (\mathfrak{c}\_{5})\_{k} \, \delta \nu\_{k} + (\mathfrak{c}\_{6})\_{k} \, \delta \nu\_{k-1} + (\mathfrak{c}\gamma)\_{k} \, \delta g\_{k} + (\mathfrak{c}\mathfrak{s})\_{k} \, \delta g\_{k-1} \\ + (\mathfrak{c}\_{7})\_{k} \, \delta s\_{k} + (\mathfrak{c}\_{8})\_{k} \, \delta s\_{k-1} = (\mathfrak{r}\_{4})\_{k-1/2} \, \end{aligned}$$

with boundary conditions are:

$$\delta f\_o = 0, \delta u\_o = A\_1, \delta g\_o = 1, \delta q\_o = 1,\\ \delta f\_k = 0, \delta u\_k = A\_2, \delta g\_k = \mathfrak{r}, \delta q\_k = 1,$$

where

$$\begin{aligned} (a\_1)\_k &= (a\_2)\_k = h \text{Re}\left(s\_{k - 1/2}\right), \\ (a\_3)\_k &= (a\_4)\_k = h \text{Re}\left(g\_{k - 1/2}\right), \\ (a\_5)\_k &= (a\_6)\_k = h \text{Re}\left(u\_{k - 1/2} + \frac{1}{\beta} + M\right), \end{aligned}$$

$$\begin{aligned} (a\_{\mathbb{Z}})\_k &= 1 + 2h \text{Re}\left(f\_{k-1/2}\right), \\ (as)\_k &= -1 + 2h \text{Re}\left(f\_{k-1/2}\right), \\ (b\_1)\_k &= \left(b\_2\right)\_k = h \text{Re}\left(w\_{k-1/2}\right), \\ (b\_3)\_k &= \left(b\_4\right)\_k = h \text{Re}\left(M + \frac{1}{\beta}\right), \\ \left(b\_5\right)\_k &= 1 + 2h \text{Re}\left(f\_{k-1/2}\right), \\ \left(b\_6\right)\_k &= -1 + 2h \text{Re}\left(f\_{k-1/2}\right), \\ \left(b\_7\right)\_k &= \left(b\_8\right)\_k = 2h \text{Re}\left(s\_{k-1/2}\right), \\ \left(b\_9\right)\_k &= \left(b\_{10}\right)\_k = 2h \text{Re}\left(g\_{k-1/2}\right), \\ (c\_1)\_k &= (c\_2)\_k = 2h \text{Re}\text{Pr}\left(t\_{k-1/2} - 2\lambda f\_{k-1/2}\right), \\ (c\_3)\_k &= (c\_4)\_k = 2h \text{Re}\text{Pr}\left(t\_{k-1/2}\right)\left(f\_{k-1/2}\right), \\ (c\_5)\_k &= 1 + 2h \text{Re}\text{Pr}\left(t\_{k-1/2} - 2\lambda \left(f\_{k-1/2}\right)^2\right) \end{aligned}$$

,

$$+\left(u\_{k-1/2}\right)\left(f\_{k-1/2}\right),$$

$$(c\_6)\_k = -1 + 2h \text{Re}\text{Pr}\left(t\_{k-1/2} - 2\lambda \left(f\_{k-1/2}\right)^2\right)$$

$$+\left(u\_{k-1/2}\right)\left(f\_{k-1/2}\right),$$

and

$$\begin{aligned} (r\_1)\_k &= \hat{f}\_{k-1} - \hat{f}\_k + h\left(u\_{k-1/2}\right), \\ (r\_5)\_k &= u\_{k-1} - u\_k + h\left(v\_{k-1/2}\right), \\ (r\_6)\_k &= \nu\_{k-1} - \nu\_k + h\left(w\_{k-1/2}\right), \\ (r\_7)\_k &= \xi\_{k-1} - \xi\_k + h\left(s\_{k-1/2}\right), \\ (r\_8)\_k &= q\_{k-1} - q\_k + h\left(t\_{k-1/2}\right), \\ \vdots \\ (r\_2)\_k &= s\_{k-1} - s\_k + h\text{Re}\left(2u\_{k-1/2}\xi\_{k-1/2} - 2f\_{k-1}\frac{1}{2}s\_{-1}\frac{1}{2} \\ &+ \left(M + \frac{1}{\beta}\right)\xi k^{-1/2}\frac{1}{2}s\_{-1}\frac{1}{2}\right), \\ (r\_3)\_k &= w\_{k-1} - w\_k + h\text{Re}\left(2f\_{k-1/2}w\_{k-1/2} + 2\xi\_{k-1/2}s\_{k-1/2} \\ &- \left(M + \frac{1}{\beta}\right)\nu\_{k-1/2}), \\ (r\_4)\_k &= t\_{k-1} - t\_k + h\text{Re}\Pr\left(2f\_{k-1/2}t\_{k-1/2} - 4\lambda\left(f\_{k-1/2}f\_{k-1/2}\right) \\ &+ f\_{k-1/2}u\_{k-1/2}t\_{k-1/2}\right)). \end{aligned}$$

Now, we consist the tridiagonal block matrices of given linearized equations in the form:

$$\mathbf{A}\boldsymbol{\delta} = \mathbf{r} \tag{27}$$

where

$$\begin{pmatrix} \begin{bmatrix} [A\_1] \ [\mathbf{C}\_1] \\ \begin{bmatrix} [B\_1] \ [\mathbf{A}\_2] \end{bmatrix} \\\\ \begin{bmatrix} \mathbf{C}\_2 \end{bmatrix} \end{pmatrix} \cdot \begin{bmatrix} \ddots & \ddots & \ddots & & & & \\ & \ddots & \ddots & \ddots & & & \\ & & \ddots & \ddots & \ddots & & \\ & & & & [B\_{k-1}] \end{bmatrix} \begin{bmatrix} \mathbf{C}\_{k-1} \\\\ \begin{bmatrix} A\_{k-1} \end{bmatrix} \ [\mathbf{C}\_{k-1}] \end{bmatrix} \\\\ \begin{bmatrix} [\delta\_1] \\ \begin{bmatrix} [\delta\_2] \\\\ \vdots \\ [\delta\_{k-1}] \\\\ [\delta\_k] \end{bmatrix} = \begin{pmatrix} [r\_1] \\ \vdots \\ [r\_{k-1}] \\\\ [r\_k] \end{bmatrix} \end{pmatrix} \tag{28}$$

In Equation (28), the elements are defined as:

$$[A\_1] = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & -1 & -\varepsilon\_k & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & -\varepsilon\_k & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & -\varepsilon\_k & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -\varepsilon\_k \end{pmatrix}, \varepsilon\_k = \frac{1}{2}h\_k$$

$$[\alpha\_k] = \begin{pmatrix} 1 & -e\_k & 0 & 0 & 0 & 0 & 0 & 0\\ (a\_1)\_k & 0 & (a\_3)\_k & (a\_5)\_k & (a\_7)\_k & (a\_9)\_k & 0 & 0\\ \left(b\_1\right)\_k & \left(b\_3\right)\_k & 0 & 0 & \left(b\_5\right)\_k & \left(b\_7\right)\_k & 0 & 0\\ (c\_1)\_k & \left(c\_3\right)\_k & 0 & 0 & 0 & 0 & 0 & (c\_5)\_k\\ 0 & -1 & -e\_k & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & -e\_k & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & -e\_k & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & -e\_k \end{pmatrix},$$

$$2 < k < K - 1.$$

$$
\begin{split}
\begin{pmatrix}
\alpha\_{K}\end{pmatrix} &= \begin{pmatrix}
1 & -\varepsilon\_{k} & 0 & 0 & 0 & 0 & 0 & 0 \\
(a\_{1})\_{k} & 0 & (a\_{3})\_{k} & (a\_{5})\_{k} & (a\gamma)\_{k} & (a\gamma)\_{k} & 0 & 0 \\
(b\_{1})\_{k} & \{b\_{3}\}\_{k} & 0 & 0 & \left(b\_{5}\right)\_{k} & \{b\_{7}\}\_{k} & 0 & 0 \\
(c\_{1})\_{k} & (c\_{3})\_{k} & 0 & 0 & 0 & 0 & 0 & (c\_{5})\_{k} \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}, \\
\end{split}
$$

$$[\beta\_k] = \begin{pmatrix} -1 & -e\_k & 0 & 0 & 0 & 0 & 0 & 0\\ (a\_2)\_k & 0 & (a\_4)\_k & (a\_5)\_k & (a\_8)\_k & (a\_{10})\_k & 0 & 0\\ \left(b\_2\right)\_k & \left(b\_4\right)\_k & 0 & 0 & \left(b\_6\right)\_k & \left(b\_8\right)\_k & 0 & 0\\ (c\_2)\_k & \left(c\_4\right)\_k & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ & & & & & & 2 & $$

$$[\mathbf{C}\_{k}] = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -e\_{k} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -e\_{k} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -e\_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -e\_{k} \end{pmatrix}, 1 < k < K - 1,$$

$$[\delta\_k] = \begin{pmatrix} \delta f\_k \\ \delta u\_k \\ \delta \nu\_k \\ \delta \nu\_k \\ \delta g\_k \\ \delta s\_k \\ \delta t\_k \\ \delta t\_k \end{pmatrix}, [r\_k] = \begin{pmatrix} (r\_1)\_{k-(\frac{1}{2})} \\ (r\_2)\_{k-(\frac{1}{2})} \\ (r\_3)\_{k-(\frac{1}{2})} \\ (r\_4)\_{k-(\frac{1}{2})} \\ (r\_5)\_{k-(\frac{1}{2})} \\ (r\_6)\_{k-(\frac{1}{2})} \\ (r\_7)\_{k-(\frac{1}{2})} \\ (r\_8)\_{k-(\frac{1}{2})} \end{pmatrix}, 1 < k < K,$$

$$\mathbf{A} = \mathbf{L}\mathbf{U},\tag{29}$$

These computations are repeated until some convergence criteria are satisfied.

#### RESULTS AND DISCUSSION

Current segment is dedicated to elucidate the numerical and graphical impact of velocity parameters involved and temperature profile and coefficient of skin friction. Variation in axial velocity against Reynolds number is adorned in **Figure 3**. It is found that axial velocity at lower disk decays with increment in Re. The reason behind this fact is that Re has direct relation with inertial forces. Therefore, with increase of Re, inertial effects dominate and cause the velocity of lower disk to decelerate. **Figure 4** expresses the impact of Re on radial component of velocity. It is explored that with the increase of Re, velocity of lower disk diminishes and upper disk uplifts. It is seen from the picture that half of its portion from 0.0 to 0.6 represents velocity pattern of lower disk and from 0.6 to 1.0, it discloses radial velocity for upper plate. Impact of scaled stretching parameter on axial component of velocity at lower disk is exemplified in **Figure 5**. It is evidenced that with the increment of A1, f (ξ ) mounts at lower disk because stretching rate is decreasing continuously. **Figure 6** is portrayed to manifest the variation in radial against A<sup>1</sup> velocity for lower and upper disks. It is found that radial component of velocity increases for the lower disk as compared to upper disk. This is due to the stretching rate of the lower disk that is continuously increasing and upper disk decrements. It is exhibited in **Figure 7** that f ′ (ξ ) decays with A<sup>2</sup> for lower disk and mounts in the case of upper disk. The justification behind this impact is that stretching rate of the upper disk is more than that of the lower disk. The behavior of axial velocity with stretching scaled parameter A<sup>1</sup> of upper disk is depicted in **Figure 8**. For larger values of A2, the axial velocity of fluid decrements near the upper disk and upsurging behavior is noticed at the lower disk. By increasing A2, velocity magnitude along radial direction in the vicinity of the upper disk increases, so velocity along axial direction as an outcome depreciates. The impact of Re on tangential component of velocity g(ξ ) is disclosed in **Figure 9**. It is fond that with increase of Re, g(ξ ) suppresses. By increasing Re, inertial forces increase so more velocity is induced by the inertial forces. Behavior of tangential velocity g(ξ ) against A<sup>2</sup>

is stretched in **Figure 10**. It is found that g(ξ ) decrements and f ′ (ξ ) uplifts for A2. With increase of A2, stretching rate of the upper disk increases, and as an outcome, axial velocity increases and tangential velocity decreases. The variation in g(ξ ) with rotational parameter is revealed in **Figure 11**. Positive attribute in g(ξ ) is observed against τ . With the increase of rotational parameter τ , centrifugal force is induced, which as an outcome uplifts the tangential component of velocity. Curves investigating the aspects of β on tangential component of velocity are adorned in **Figure 12**. It is justified by the fact that momentum equation 1 β is a dimensionless parameter, so with the increase of β, momentum profile is tangential and its direction diminishes. Positive impact on thermal distribution against Re is observed in **Figure 13**. With increase of Re, viscous forces decrement and velocity of fluid particle increases. Thus, the temperature is defined as average motion of fluid molecules so thermal field molecules due to uplifts of movement of particles. The impact of θ(ξ ) against β is anticipated and shown in **Figure 14**. Increase in thermal magnitude is observed if β is increased. Since increase in β raises the rotation of disks, by increasing the rotation of disk, more rotational motion in fluid is generated and as a consequence kinetic energy of fluid molecule increases. The increase in kinetic motion raises the temperature profile. Variation in thermal profile by varying A<sup>1</sup> in the range of (0.0 ≤ A<sup>1</sup> ≤ 1.5) is revealed in **Figure 15**. It is observed that temperature of fluid boosts against the values of A1. As we increase the A1, the stretching rate increases, the fluid particles between disks exceeds, and hence temperature boosts up. The impact of Prandtl number Pr on θ(ξ ) is exhibited in **Figure 16**. It is found that thermal distribution decreases with Pr. This is due to the fact that Pr is the ratio of viscous diffusion to thermal diffusion. Thus, by increasing Pr, thermal diffusion decreases so temperature decreases. **Figure 17** is adorned to study the impact of thermal relaxation parameter on thermal distribution. Declined attribute in temperature against thermal relaxation parameter λ is depicted. It is because of the fact that with the

TABLE 1 | Influence of skin friction coefficient at wall of upper and lower disks.



TABLE 2 | Comparison of *f* ′′(0) and *g* ′ (0) with Stewartson [4], Hayat et al. [24], and Hayat et al. [25] when φ = *A*1 = *A*2 = 0 and *Re* = 1.

increase of λ, fluid particles will take more time to transfer heat to its neighboring particles, thus the temperature decreases.

**Table 1** numerically discloses the influence of porosity parameter β, stretching parameters A<sup>1</sup> and A2, and Reynolds number Re. The skin friction coefficient increases for greater value of Re and stretching parameters A<sup>1</sup> and A2, whereas it decreases for increasing values of porosity parameter β and rotating parameter τ at the upper and lower disks. **Table 2** gives assurance of present work by constructing comparison with previously published literature for skin friction coefficient along radial and tangential components. Here, τ ≥ 0 shows the rotation of both disks in the same direction, τ ≤ 0 represents the direction of rotation of both disks in opposite direction, and τ = 0 means upper disk is fixed.

#### CONCLUSIONS

Current exertion is devoted to analyze the impact of Cattaneo-Christov heat flux theory on fluid flow between the two parallel rotating disks. Equations are modeled in the form of partial differential equations and then transformed into ordinary differential expressions. These ODE (ordinary differential equations) are tackled by Keller-box scheme. The key findings are summarized as follows:

### REFERENCES


#### DATA AVAILABILITY STATEMENT

All datasets generated for this study are included in the article/supplementary material.

#### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

#### ACKNOWLEDGMENTS

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number (RGP-2019-28).


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Bilal, Tassaddiq, Majeed, Nisar, Ali and Malik. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# A Numerical Simulation for Darcy-Forchheimer Flow of Nanofluid by a Rotating Disk With Partial Slip Effects

#### Malik Zaka Ullah<sup>1</sup> \*, Stefano Serra-Capizzano2,3 and Dumitru Baleanu4,5

*<sup>1</sup> Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia, <sup>2</sup> Department of Science and High Technology, University of Insubria, Como, Italy, <sup>3</sup> Department of Information Technology, Uppsala University, Uppsala, Sweden, <sup>4</sup> Department of Mathematics, Cankaya University, Ankara, Turkey, <sup>5</sup> Institute of Space Sciences, Bucharest, Romania*

This study examines Darcy-Forchheimer 3D nanoliquid flow caused by a rotating disk with heat generation/absorption. The impacts of Brownian motion and thermophoretic are considered. Velocity, concentration, and thermal slips at the surface of the rotating disk are considered. The change from the non-linear partial differential framework to the non-linear ordinary differential framework is accomplished by utilizing appropriate variables. A shooting technique is utilized to develop a numerical solution of the resulting framework. Graphs have been sketched to examine how the concentration and temperature fields are affected by several pertinent flow parameters. Skin friction and local Sherwood and Nusselt numbers are additionally plotted and analyzed. Furthermore, the concentration and temperature fields are enhanced for larger values of the thermophoresis parameter.

Keywords: rotating disk, Darcy-Forchheimer flow, nanoparticles, heat absorption/generation, slip conditions, numerical solution

### 1. INTRODUCTION

Flow due to a rotating disk plays an indispensable role in numerous modern items encompassing rotating machinery, apparatuses, rotors, and flywheels. As of late, rotating disks have become a significant component of many pieces of machinery, for example, thermal power-creation frameworks, rotor-stator turning circle reactors, electrical controls, stopping mechanisms, pivoting sawing machines, and rotational air cleaning systems. Close investigations of laminar boundary layer flow were carried out by Von Karman [1]. Turkyilmazoglu and Senel [2] examined the linked features of heat and mass exchange arising from the revolution of a hard and permeable disk. Entropy generation in slip flow by the turning of a permeable disk with MHD and variable properties was clarified by Rashidi et al. [3]. Nanofluid flow because of the revolution of a disk was explored by Turkyilmazoglu [4]. Hatami et al. [5] investigated the impacts of the contraction, turning, and heat of disks on the movement of nanofluids. They utilized a least-square strategy for solution development. Mustafa et al. [6] deciphered the three-dimensional rotating flow of nanofluids over a stationary disk. Sheikholeslami et al. [7] created numerical models of nanofluid splashing on a slanted turning disk. Transient thermophoretic molecule deposition through the constrained convective flow of micropolar liquid over a pivoting disk was examined by Doh and Muthtamilselvan [8]. Hayat et al. [9] discussed Darcy-Forchheimer flow of carbon nanotubes in

#### Edited by:

*Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan*

#### Reviewed by:

*Nasir Shehzad, Govt. College Khayaban-E-Sir Syed, Pakistan Mustafa Turkyilmazoglu, Hacettepe University, Turkey*

> \*Correspondence: *Malik Zaka Ullah zmalek@kau.edu.sa*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *01 November 2019* Accepted: *28 November 2019* Published: *15 January 2020*

#### Citation:

*Ullah MZ, Serra-Capizzano S and Baleanu D (2020) A Numerical Simulation for Darcy-Forchheimer Flow of Nanofluid by a Rotating Disk With Partial Slip Effects. Front. Phys. 7:219. doi: 10.3389/fphy.2019.00219*

**70**

response to a turning disk. Aziz et al. [10] gave a numerical report on nanofluid flow from the pivoting of a disk, looking at the impacts of slip and heat absorption/generation. Synthetically responsive flow of third-grade nanofluid over a stretchable turning disk with heat generation was broken down by Hayat et al. [11]. The radiative flow of a suspension of nanoparticles and gyrotactic microorganisms by the variably thick surface of a turning disk was clarified by Qayyum et al. [12]. Hayat et al. [13] presented a numerical simulation of the radiative flow of carbon nanotubes due to the revolution of a disk with partial slip.

The low thermal productivity of working fluids is a guideline problem for several heat transport components in engineering applications. For this reason, some researchers are making efforts to develop an innovative course for the improvement of the thermal efficiency of working fluids. Various measures have been proposed by experts to improve the thermal efficiency of fluids. Accordingly, the incorporation of nanomaterial into the working fluid, making what is termed a nanofluid, is extremely promising. Recent assessments of nanofluids reveal that working fluid has totally different features with the addition of nanomaterial. This is because the thermal efficiency of the working liquid is lower than that of the nanomaterial. Nanofluid is suspension of fluids containing standard fluid with the particles of nano-measure. Such nanomaterials are utilized in materials, MHD control generators, oil stores, cooling of nuclear reactors, vehicle transformers, and various others [14–18]. Choi and Eastman [19] coined the term nanofluid. They proposed that nanomaterials are a groundbreaking contender for the development of heat transport via the customary fluids. Buongiorno proposed a numerical model of convective transport by nanofluid [20]. Here, thermophoresis and Brownian motion are viewed as the most important slip mechanisms. Heat transfer increase by nanofluids in a two-sided top-driven heated square hole was considered by Tiwari and Das [21]. The significance of a CuO-water nanomaterial on the outside of heat exchangers was tentatively examined by Pantzali et al. [22]. Few ongoing studies on nanofluid flow can be found in the literature [23–45].

Motivated by the above-mentioned articles, the objective here is to examine the impacts of heat absorption/generation in Darcy-Forchheimer 3D nanofluid flow caused by a rotating disk and the impacts of slip. Both Brownian diffusion and thermophoretic phenomena occur in view of the existence of nanoparticles. Velocity, concentration, and thermal slips are accounted for. The obtained framework is solved numerically by the shooting technique. Concentration, temperature, skin friction, and local Sherwood and Nusselt numbers are also analyzed through plots.

#### 2. MATHEMATICAL MODELING

Let us examine steady Darcy-Forchheimer 3D nanoliquid flow caused by a rotating disk with slip and heat absorption/generation. A disk at z = 0 rotates with constant

angular velocity (see **Figure 1**). The impacts of Brownian motion and thermophoretis are accounted for. The velocities are (u, v, w) in the directions of increase in (r, ϕ, z), respectively. The resulting boundary layer expressions are [45, 46]:

$$\frac{\partial \mu}{\partial r} + \frac{\mu}{r} + \frac{\partial \nu}{\partial z} = 0,\tag{1}$$

$$u\frac{\partial u}{\partial r} - \frac{\nu^2}{r} + w\frac{\partial u}{\partial z} = \nu \left( \frac{\partial^2 u}{\partial z^2} + \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{r^2} \right) - \frac{\nu}{k^\*} u - F u^2 \tag{2}$$

$$
\mu \frac{\partial \nu}{\partial r} + \frac{\mu \nu}{r} + \nu \frac{\partial \nu}{\partial z} = \nu \left( \frac{\partial^2 \nu}{\partial z^2} + \frac{\partial^2 \nu}{\partial r^2} + \frac{1}{r} \frac{\partial \nu}{\partial r} - \frac{\nu}{r^2} \right) - \frac{\nu}{k^\*} \nu - F \nu^2,\tag{3}
$$

$$\mu \frac{\partial \boldsymbol{w}}{\partial r} + \boldsymbol{w} \frac{\partial \boldsymbol{w}}{\partial z} = \nu \left( \frac{\partial^2 \boldsymbol{w}}{\partial z^2} + \frac{\partial^2 \boldsymbol{w}}{\partial r^2} + \frac{1}{r} \frac{\partial \boldsymbol{w}}{\partial r} \right) - \frac{\nu}{k^\*} \boldsymbol{w} - F \boldsymbol{w}^2, \tag{4}$$

$$u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = \alpha^\* \left( \frac{\partial^2 T}{\partial z^2} + \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) + \frac{Q}{(\rho c)\_f} \{T - T\_\infty\}$$

$$+ \frac{(\rho c)\_p}{(\rho c)\_f} \left( D\_\mathcal{B} \left( \frac{\partial T}{\partial r} \frac{\partial C}{\partial r} + \frac{\partial T}{\partial z} \frac{\partial C}{\partial z} \right)$$

$$+ \frac{D\_T}{T\_\infty} \left( \left( \frac{\partial T}{\partial z} \right)^2 + \left( \frac{\partial T}{\partial r} \right)^2 \right) \right), \tag{5}$$

$$\begin{split} u\frac{\partial C}{\partial r} + w\frac{\partial C}{\partial z} &= D\_B \left( \frac{\partial^2 C}{\partial z^2} + \frac{\partial^2 C}{\partial r^2} + \frac{1}{r}\frac{\partial C}{\partial r} \right) \\ &+ \frac{D\_T}{T\_\infty} \left( \frac{\partial^2 T}{\partial z^2} + \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) . \end{split} \tag{6}$$

It is subject to the boundary conditions [10]:

$$\mu = L\_1 \frac{\partial \mu}{\partial z}, \; \nu = r\Omega + L\_1 \frac{\partial \nu}{\partial z}, \; \nu = 0, \; T = T\_w + L\_2 \frac{\partial T}{\partial z},$$

$$C = C\_w + L\_3 \frac{\partial C}{\partial z} \text{ at } z = 0,\tag{7}$$

$$u \to 0, \quad \nu \to 0, \quad T \to T\_{\infty}, \quad C \to C\_{\infty} \quad \text{as } z \to \infty. \tag{8}$$

Here u, v, and w represent the velocity components in the directions r, ϕ, and z while ρ<sup>f</sup> , µ and, ν = µ/ρ<sup>f</sup> depict the fluid density, dynamic, and kinematic viscosities respectively, Cb the drag factor, L<sup>1</sup> the velocity slip factor, L<sup>2</sup> the thermal slip factor (ρc)p, the effective heat capacity of nanoparticles, T the fluid temperature, k ∗ the permeability of porous space, C the concentration (ρc)<sup>f</sup> , the heat capacity of the liquid, L<sup>3</sup> the concentration slip factor, C<sup>∞</sup> the ambient concentration, D<sup>T</sup> the thermophoretic factor, F = Cb/rk<sup>∗</sup> 1/2 the non-uniform inertia factor, k and α <sup>∗</sup> = k/(ρc)<sup>f</sup> the thermal conductivity and thermal diffusivity, respectively, D<sup>B</sup> the Brownian factor, Q the heat

generation/absorption factor and T∞ the ambient temperature. Selecting [10]:

$$\begin{array}{l} \mu = r\Omega f'(\xi), \; \nu = -(2\Omega \nu)^{1/2} f(\xi), \; \nu = r\Omega g(\xi), \\\ \phi(\xi) = \frac{C - C\_{\infty}}{C\_{\text{v}} - C\_{\infty}}, \; \xi = \left(\frac{2\Omega}{\nu}\right)^{1/2} z, \; \theta(\xi) = \frac{T - T\_{\infty}}{T\_{\text{v}} - T\_{\infty}}. \end{array} \tag{9}$$

Continuity Equation (1)istrivially verified, while Equations(2)− (8) yield

$$2f'''' + 2f''' - f'^2 + g^2 - \lambda f' - Frf'^2 = 0,\tag{10}$$

$$2\mathbf{g}^{\prime\prime} + 2\mathbf{f}\mathbf{g}^{\prime} - 2\mathbf{f}^{\prime}\mathbf{g} - \lambda\mathbf{g} - \mathbf{F}\mathbf{g}^{2} = \mathbf{0},\tag{11}$$

$$\frac{1}{\text{Pr}}\boldsymbol{\theta}^{\prime\prime} + \boldsymbol{f}\boldsymbol{\theta}^{\prime} + \text{N}\_{b}\boldsymbol{\theta}^{\prime}\boldsymbol{\phi}^{\prime} + \text{N}\_{l}\boldsymbol{\theta}^{\prime^2} + \boldsymbol{\delta}\boldsymbol{\theta} = \mathbf{0},\tag{12}$$

$$\frac{1}{\text{Sc}}\phi'' + f\phi' + \frac{1}{\text{Sc}}\frac{N\_t}{N\_b}\theta'' = 0,\tag{13}$$

$$f(0) = 0, f'(0) = \alpha f''(0), \, g(0) = 1 + \alpha g'(0),$$

$$\theta(0) = 1 + \beta \theta'(0), \, \phi(0) = 1 + \gamma \phi'(0), \tag{14}$$

$$f'(\infty) \to 0, \ g(\infty) \to 0, \; \theta(\infty) \to 0, \; \phi(\infty) \to 0. \tag{15}$$

Here, Fr stands for the Forchheimer number, α for the velocity slip parameter, λ for the porosity parameter, N<sup>t</sup> for the thermophoresis parameter, β for the thermal slip parameter, Pr for the Prandtl number, N<sup>b</sup> for Brownian motion, δ for the heat absorption/generation parameter, γ for the concentration slip

parameter, and Sc for the Schmidt number. Non-dimensional

variables are defined by

Frontiers in Physics | www.frontiersin.org

The coefficients of skin friction and the Nusselt and Sherwood numbers are

$$\begin{aligned} \text{Re}\_r^{1/2} \mathcal{C}\_f &= f''(0), \; \text{Re}\_r^{1/2} \mathcal{C}\_\emptyset = \mathfrak{g}'(0), \; \text{Re}\_r^{-1/2} \mathcal{N}u = -\theta'(0),\\ \text{Re}\_r^{-1/2} \mathcal{S}h &= -\phi'(0), \end{aligned} \tag{17}$$

where Re<sup>r</sup> = 2(r)r/ν represents the local rotational Reynolds number.

#### 3. NUMERICAL RESULTS AND DISCUSSION

This section depicts the contributions of various physical variables like thermophoresis parameter N<sup>t</sup> , Forchheimer number Fr, thermal slip parameter β, heat generation/absorption parameter δ, Brownian number N<sup>b</sup> , and concentration slip number γ on The concentration φ(ζ ) and temperature θ (ζ ) distributions. The effect of Forchheimer variable Fr on θ (ζ ) is portrayed in **Figure 2**. A larger value for Fr shows expanding behavior of θ (ζ ) and the related thermal layer. **Figure 3** shows the impact of thermal slip β on temperature θ (ζ ). Temperature is reduced by increasing thermal slip β. **Figure 4** demonstrates the effect of N<sup>t</sup> on the temperature field θ (ζ ). A larger thermophoresis parameter N<sup>t</sup> value leads to a higher temperature field and thicker dynamically warm layer. The reason for this conflict is that growth in N<sup>t</sup> yields high grounded thermophoresis control, which further allows movement of the nanoparticles in the fluid zone. A long way from the surface, a more grounded temperature scattering θ (ζ ) and continuously warm layer is thus created. The impact of N<sup>b</sup> on the temperature profile θ (ζ ) is portrayed in **Figure 5**. Physically, the irregularity of nanoparticle movement increases by enhancing Brownian motion, due to which collision of particles occurs. Thus, kinetic energy is converted into heat energy, which produces an increase in the temperature field. **Figure 6** shows how heat generation/absorption δ influences

TABLE 1 | Comparative values of *f* ′′(0) and *g* ′ (0) for value of *Fr* when λ = 0.2 and α =0.


temperature dispersion θ (ζ ). Here, δ > 0 portrays heat generation and δ < 0 for heat absorption. Both temperature θ (ζ ) and the warm layer are upgraded with increasing δ. **Figure 7** shows that concentration φ(ζ ) is higher for larger values of the Forchheimer variable Fr. **Figure 8** shows how concentration φ(ζ ) is influenced by concentration slip γ . Concentration is reduced at higher estimations of γ . **Figure 9** demonstrates how the thermophoresis parameter N<sup>t</sup> influences the concentration φ(ζ ). By improving thermophoresis parameter N<sup>t</sup> , the concentration φ(ζ ) is increased. **Figure 10** depicts the impact of Brownian motion N<sup>b</sup> on concentration φ(ζ ). It has been noted that a stronger concentration φ(ζ ) is developed by utilizing greater N<sup>b</sup> . **Figures S1**, **S2** display the impacts of Fr on <sup>C</sup><sup>f</sup> Re1/<sup>2</sup> <sup>r</sup> and <sup>C</sup>gRe1/<sup>2</sup> r , respectively. It is noted that <sup>C</sup><sup>f</sup> Re1/<sup>2</sup> r is a decaying function of Fr, while the reverse situation is observed for <sup>C</sup>gRe1/<sup>2</sup> r . The effects of N<sup>t</sup> and N<sup>b</sup> on Nu(Rer) −1/2 are highlighted in **Figures 11**, **12**, respectively. Here, Nu(Rer) −1/2 reduces for N<sup>t</sup> and N<sup>b</sup> . The effects of N<sup>t</sup> and N<sup>b</sup> on Sh(Rer) −1/2 are portrayed in **Figures 13**, **14**, respectively. Here,

#### REFERENCES


Sh(Rer) −1/2 is an increasing factor of N<sup>t</sup> , while the opposite trend is seen for N<sup>b</sup> . The figures in **Table 1** were computed to validate the present results with previously published results in a limiting sense. Here, we see that the present numerical solution is in good agreement with the previous solution by Naqvi et al. [45] in a limiting sense.

#### 4. CONCLUSIONS

In this paper, Darcy-Forchheimer 3D nanofluid flow caused by a rotating disk with heat generation/absorption is studied. Brownian motion and thermophoretic phenomena occur with the existence of nanoparticles. Velocity, concentration, and thermal slips are accounted for. A higher Forchheimer number Fr depicts similar behavior for concentration and temperature. A larger β corresponds to a lower temperature field. Higher γ depicts decreasing behavior for the concentration field. A stronger temperature field is observed for N<sup>b</sup> and N<sup>t</sup> . Concentration φ(ζ ) displays the reverse behavior for N<sup>b</sup> and N<sup>t</sup> .

### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. KEP-16-130-40. The authors, therefore, acknowledge with thanks DSR technical and financial support.

### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy. 2019.00219/full#supplementary-material

Figure S1 | Variations of *CfRe*1/<sup>2</sup> *r* for *Fr*.

Figure S2 | Variations of *CgRe*1/<sup>2</sup> *r* for *Fr*.

> variable properties. Int J Heat Mass Transf. (2014) **70**:892–917. doi: 10.1016/j.ijheatmasstransfer.2013.11.058


with internal heat generation. Int J Heat Mass Transf. (2015) **84**:903–11. doi: 10.1016/j.ijheatmasstransfer.2015.01.099


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Ullah, Serra-Capizzano and Baleanu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique

Rasool Shah1†, Umar Farooq1†, Hassan Khan1†, Dumitru Baleanu2,3†, Poom Kumam4,5 \* † and Muhammad Arif 1†

*<sup>1</sup> Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan, <sup>2</sup> Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey, <sup>3</sup> Institute of Space Sciences, Magurele, Romania, ˘ <sup>4</sup> Center of Excellence in Theoretical and Computational Science (TaCS-CoE) and KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Departments of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok, Thailand, <sup>5</sup> Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan*

#### Edited by:

*Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan*

#### Reviewed by:

*Marin I. Marin, Transilvania University of Bra ¸sov, Romania Aaqib Majeed, Bacha Khan University, Pakistan Ilyas Khan, Ton Duc Thang University, Vietnam*

#### \*Correspondence:

*Poom Kumam poom.kum@kmutt.ac.th*

*†These authors have contributed equally to this work*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *04 September 2019* Accepted: *19 December 2019* Published: *17 January 2020*

#### Citation:

*Shah R, Farooq U, Khan H, Baleanu D, Kumam P and Arif M (2020) Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique. Front. Phys. 7:244. doi: 10.3389/fphy.2019.00244* In the present article, fractional view of third order Kortewege-De Vries equations is presented by a sophisticated analytical technique called Mohand decomposition method. The Caputo fractional derivative operator is used to express fractional derivatives, containing in the targeted problems. Some numerical examples are presented to show the effectiveness of the method for both fractional and integer order problems. From the table, it is investigated that the proposed method has the same rate of convergence as compare to homotopy perturbation transform method. The solution graphs have confirmed the best agreement with the exact solutions of the problems and also revealed that if the sequence of fractional-orders is approaches to integer order, then the fractional order solutions of the problems are converge to an integer order solution. Moreover, the proposed method is straight forward and easy to implement and therefore can be used for other non-linear fractional-order partial differential equations.

Keywords: analytical solution, Mohand transform, Adomian decomposition, caputo derivatives, third order Kortewege-De Vries equations

### 1. INTRODUCTION

∂

The class of partial differential equations known as Korteweg-De Vries (KDV) equation which play a vital role in the diverse field of physics such as fluid mechanics, signal processing, hydrology, viscoelasticity and fractional kinetics [1, 2]. The KDV equation was first time derived by Korteweg and Vries in 1895. The KDV equation used to model long waves, tides, solitary waves, and wave propagating in a shallow canal. A partial differential Kortewege-De Vries equation of third order is also applied to study the non-linear model of water waves in superficial canal certain namely canal [3], in the time when wave in water was of important concentration in applications in navigational design and also for the awareness of flood and tides [4, 5]. The applications in numerous areas of physics, applied science and other scientific applications, therefore the excessive amount of investigation as a research work has been capitalized in the study of KDV equations [6–10]. We considered the third order time fractional KDV equation in the form [1]

$$\frac{\partial \mathcal{Z}^{\chi}}{\partial \chi \pi(\chi)} + \kappa \pi(\chi^{\chi}, \chi) \frac{\partial \chi}{\partial \pi(\chi^{\chi}, \chi)} + \chi \frac{\partial \chi^{\chi}}{\partial \chi^{\chi} \chi} + \psi(\chi, \chi), \quad 0 < \chi \le 1,\tag{1}$$

$$
u(\chi, 0) = \mu(\chi),$$

where, κ and λ are real numbers.

The KdV equations of fractional order can be applied to examine the influence of the higher-order wave dispersion. The KdV-Burgers equation defines the waves on lower water surfaces. The strength of fractional KdV equation is the non-local property [11–21]. For a higher order Korteweg-de Vries equation, which is a natural extension of the Korteweg-de Vries equation written in a bilinear form, a Bcklund conversion in bilinear forms is provided. For this higher-order equation the Bcklund transition is given in ordinary forms and the inverse scattering scheme [22], Korteweg-de Vries type of equations 3rd order coefficient variable [23] and Solution of the third order Korteweg-De Vries homotopy perturbation approach using elzaki transform [24].

In few decades, integral transform of various types such as Fourier transform, Laplace transform, Hankel transform, Mellin transform, Z-transform, Wavelet transform, Elzaki transform, Kamal transform, Mahgoub transform, Aboodh transform, Mohand transform, Sumudu transform, Hermite transform etc., gained a enormous importance in solving advanced model in the field mathematics, physics and engineering [25–36].

In the current article, we have applied the Mohand transform with decomposition procedure for the analytical treatment of time fractional KDV equation. The Mohand Transform is one of the new integral transform use for the analytical treatment of different physical phenomena are molded by Differential Equations (DEs) of integer order or Fractional Partial Differential Equations (FPDEs). Recently, Kumar and Viswanathan used Mohand transform and solved the mechanics and electrical circuit problems [37]. Aggarwal have Comparatively Studied Mohand and Aboodh transforms for the solution of differential equations. The numerical applications reflect that both the transforms (Mohand and Aboodh transforms) are closely related to each other [38]. Sudhanshu Aggarwal have also discussed the comparative study of Mohand and Laplace transforms, Mohand and Sumudu transforms, Mohand and Mahgoub transforms [39–41]. Sudhanshu Aggarwal have successfully discussed the Mohand transform of Bessels functions of zero, one and two orders, which is very useful for solving many equations in cylindrical or spherical coordinates such as heat equation, wave equation etc. [42]. The exact solution of second kinds of linear Volterra integral equations get by using Mohand transform. It is claimed that Mohand transform take very little time and has no large computational work [43]. Mohand transform have also used the for solution of Abel's integral equation. The obtained results show that Mohand transform is a powerful integral transform for handling Abel's integral equation [44]. The remaining section of the paper are managed as follows. In the second section, we present some related definitions of fractional calculus and basic concepts of Mohand transform. The third section presents the implementation the proposed methodology. The four section represent different models of KDV equation are examined separately and plotted. Finally, we depict our conclusions.

### PRELIMINARIES CONCEPTS

In this section, we present some basic necessary definitions and preliminaries concepts related to fractional calculus and Mohand transform.

#### DEFINITION

Mohand transform first time was define by Mohand and Mahgoub of the function u(ℑ) for ℑ ≥ 0 in the year 2017. The Mohand transform which is represented by M(.) for a function u(τ ) is define as [45]

$$M\{\boldsymbol{\mu}(\mathfrak{N})\} = R(\boldsymbol{\upsilon}) = \boldsymbol{\upsilon}^2 \int\_0^\infty \boldsymbol{\mu}(\mathfrak{N}) e^{-\boldsymbol{\upsilon}\mathfrak{N}} d\mathfrak{N}, \ k\_1 \le \boldsymbol{\upsilon} \le k\_2,\tag{2}$$

The Mohand transform of a function u(ℑ) is R(υ) then u(ℑ) is called the inverse of R(υ) which is expressed as.

$$M^{-1}\{\mathcal{R}(\upsilon)\} = \iota(\mathfrak{J}), \; M^{-1} \; is \; inverse \; Modand \; operator. \; \text{(3)}$$

#### DEFINITION

Mohand transform for nth derivatives [46]

$$M\{\boldsymbol{\mu}^{\boldsymbol{n}}(\boldsymbol{\exists})\} = \boldsymbol{\upsilon}^{\boldsymbol{n}}R(\boldsymbol{\upsilon}) - \boldsymbol{\upsilon}^{\boldsymbol{n}+1}\boldsymbol{\iota}(0) - \boldsymbol{\upsilon}^{\boldsymbol{n}}\boldsymbol{\iota}^{\boldsymbol{\prime}}(0) - \cdots - \boldsymbol{\upsilon}^{2}\boldsymbol{\iota}^{\boldsymbol{n}-1}(0),\tag{4}$$

#### DEFINITION

Mohand transform for fractional order derivatives [46]

$$R\{\boldsymbol{\omega}^{\mathcal{V}}(\mathbb{S})\} = \boldsymbol{\upsilon}^{\mathcal{V}}R(\boldsymbol{\upsilon}) - \sum\_{k=0}^{n-1} \frac{\boldsymbol{\mu}^{k}(0)}{\boldsymbol{\upsilon}^{k-(\boldsymbol{\varphi}+1)}}, \quad 0 < \boldsymbol{\chi} \le n,\tag{5}$$

#### DEFINITION

Caputo operator of fractional partial derivative [47]

$$D\_{\tau}^{\mathcal{Y}}\mathcal{g}(\boldsymbol{\chi},\boldsymbol{\upbeta}) = \begin{cases} \frac{\partial^{n}\mathcal{g}(\boldsymbol{\upchi},\boldsymbol{\upbeta})}{\partial\boldsymbol{\upbeta}^{n}}, & \boldsymbol{\upchi} = n \in N, \\\\ \frac{1}{\Gamma(n-\boldsymbol{\upchi})} \int\_{0}^{\mathbb{S}} (\boldsymbol{\upchi}-\boldsymbol{\upphi})^{n-\boldsymbol{\upchi}-1} \mathcal{g}^{n}(\boldsymbol{\upphi}) \boldsymbol{\upbeta} \boldsymbol{\upphi}, & n-1 < \boldsymbol{\upchi} < n \end{cases},\tag{6}$$

#### 2. IMPLEMENTATION OF MOHAND TRANSFORM

In this section we have considered the time fractional KDV model in the form

$$\begin{aligned} \frac{\partial \chi\_{\lambda}}{\partial \lambda^{\lambda}} + \kappa u(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial u(\chi, \mathfrak{z})} + \lambda \frac{\partial \chi^{\mathfrak{z}}}{\partial \chi^{\mathfrak{z}}} &= \psi(\chi, \mathfrak{z}), \\ 0 < \chi &\le 1, \end{aligned} \tag{7}$$

with initial source

$$
u(\chi,0) = \mathfrak{u}(\chi).$$

where, κ and λ are real numbers.

$$\begin{split} M\left\{\frac{\partial\chi\,\mu(\chi,\mathfrak{z})}{\partial\mathfrak{z}^{\chi}}\;+\;\kappa\iota(\chi,\mathfrak{z})\frac{\partial\chi}{\partial\mathfrak{z}}\bigg|\;\mathfrak{y}\,\mathfrak{z}\left(\chi,\mathfrak{z}\right)\\ =\;M\left\{\psi(\chi,\mathfrak{z})\right\},\;0\leqslant\chi\leq1,\end{split} \tag{8}$$

by using the transform property, we can simplify as

$$\begin{split} \left\| \boldsymbol{\nu}^{\boldsymbol{\upprime}} \{ \boldsymbol{R}(\boldsymbol{\upnu}) - \boldsymbol{\upnu} \boldsymbol{u}(\boldsymbol{0}) \} \right\| &+ M \left\{ \kappa \boldsymbol{u}(\boldsymbol{\upchi}, \boldsymbol{\upgamma}) \frac{\partial \boldsymbol{u}(\boldsymbol{\upchi}, \boldsymbol{\upgamma})}{\partial \boldsymbol{\upchi}} + \lambda \frac{\partial^{3} \boldsymbol{u}(\boldsymbol{x}, \boldsymbol{\uptau})}{\partial \boldsymbol{x}^{3}} \right\} \\ &= M \left\{ \boldsymbol{\uppsi}(\boldsymbol{x}, \boldsymbol{\uptau}) \right\}, \end{split} \tag{9}$$

after some evaluation, Equation (8) simplified as

$$R(\boldsymbol{\upsilon}) = \boldsymbol{\upsilon}u(0) + \frac{1}{\boldsymbol{\upsilon}^{\mathcal{Y}}}M\left\{-\kappa u(\boldsymbol{\chi},\boldsymbol{\tau})\frac{\partial u(\boldsymbol{\chi},\boldsymbol{\tau})}{\partial \boldsymbol{\chi}} - \lambda \frac{\partial^{\frac{3}{3}}u(\boldsymbol{\chi},\boldsymbol{\chi})}{\partial \boldsymbol{\chi}^{\mathcal{Y}}}\right\}$$

$$+\frac{1}{\boldsymbol{\upsilon}^{\mathcal{Y}}}M\left\{\boldsymbol{\psi}(\boldsymbol{\chi},\boldsymbol{\tilde{\chi}})\right\},\tag{10}$$

by applying inverse Mohand transform

$$u(\boldsymbol{\chi}, \boldsymbol{\upchi}) = \nu u(0) + M^{-1} \left\{ \frac{1}{\boldsymbol{\upupup}^{\boldsymbol{\upchi}}} M \left\{ -\kappa \, u(\boldsymbol{\upupup}, \boldsymbol{\upupup}) \frac{\partial u(\boldsymbol{\upupup}, \boldsymbol{\upupup})}{\partial \boldsymbol{\upupup}} \right. \right. \\ \left. -\lambda \frac{\partial^{\boldsymbol{\upupup}} u(\boldsymbol{\upupup}, \boldsymbol{\upupup})}{\partial \boldsymbol{\upupup}^{\boldsymbol{\upupup}}} \right\} + \frac{1}{\boldsymbol{\upupup}^{\boldsymbol{\upupup}}} M \left\{ \psi(\boldsymbol{\upup}, \boldsymbol{\upupup}) \right\}. \tag{11}$$

Finally we obtain the recursive general relation as

$$u\_0(\chi, \mathfrak{z}) = u(0) + M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ \psi(\chi, \mathfrak{z}) \right\} \right\} \quad m = 0$$

$$u\_{m+1}(\chi, \mathfrak{z}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ -\kappa u\_m(\chi, \mathfrak{z}) \frac{\partial u\_m(\chi, \mathfrak{z})}{\partial \chi} \right.$$

$$-\lambda \frac{\partial^3 u\_m(\chi, \mathfrak{z})}{\partial \chi^3} \right\}, \quad m \ge 0. \tag{12}$$

#### THEOREM

Let χ and Y be two Banach spaces and T : χ → Y be a contractive nonlinear operator, such that for all u; u <sup>∗</sup> ∈; χ, ||T(u)−T(u ∗ )|| ≤ K||u − u ∗ ||, 0 < K < 1 [48].

Then, in view of Banach contraction theorem, T has a unique fixed point u, such that Tu = u: Let us write the generated series (12), by the Mohand decomposition method as

$$\chi\_m = T(\chi\_{m-1}), \quad \chi\_{m-1} = \sum\_{j=1}^{m-1} \mu\_j, \quad j = 0, 1, 2, \dots$$

and supposed that χ<sup>0</sup> = u<sup>0</sup> ∈ Sp(u), where Sp(u) = {u <sup>∗</sup> ∈ χ : ||u − u ∗ || < p} then, we have

$$\begin{aligned} (B\_1) \chi\_m &\in \mathcal{S}\_\mathcal{P}(u) \\ \text{( $B\_2$ )} \lim\_{m \to \infty} \chi\_n &= u. \end{aligned}$$

#### Proof

(B1) In view of mathematical induction for m = 1, we have

$$||\chi\_1 - \mu\_1|| = ||T(\chi\_0 - T(\mu))|| \le K||\mu\_0 - \mu||.$$

Let the result is true for m − 1, then

$$||\chi\_{m-1} - u|| \le K^{m-1} ||u\_0 - u||.$$

We have

$$||\chi\_m - u|| = ||T(\chi\_{m-1} - T(u))|| \le K||\chi\_{m-1} - u|| \le K^m||u\_0 - u||.$$

Hence, using (B1), we have

$$||\chi\_m - \mathfrak{u}|| \le K^m ||\mathfrak{u}\_0 - \mathfrak{u}|| \le K^m \mathfrak{p} < \mathfrak{p}\_\* $$

which implies that χ<sup>m</sup> ∈ Sp(u).

(B2): Since ||χ<sup>m</sup> − u|| ≤ K <sup>m</sup>||u<sup>0</sup> − u|| and as a limm→∞ K <sup>m</sup> = 0.

Therefore; we have limm→∞ ||u<sup>n</sup> − u|| = 0 ⇒ limm→∞ u<sup>n</sup> = u.

#### 3. APPLICATIONS AND DISCUSSION

Here, we have implemented the Mohand transform on some time fractional KVD equations.

**Example 4.1:** Consider the third order time fractional KVD equation [49]

$$\frac{\partial \mathcal{G}^{\lambda}}{\partial \lambda^{\lambda}(\boldsymbol{\chi}, \boldsymbol{\chi})} + \epsilon \mu(\boldsymbol{\chi}, \boldsymbol{\chi}) \frac{\partial \boldsymbol{\chi}}{\partial \boldsymbol{\chi}(\boldsymbol{\chi}, \boldsymbol{\chi})} + \frac{\partial \chi^{\lambda}}{\partial \boldsymbol{\chi}^{\lambda}} = 0, \quad 0 < \boldsymbol{\chi} \le 1, \tag{13}$$

with initial source

$$u(\chi, 0) = \chi.$$

Taking Mohand transform of Equation (12), we get

$$\left\|\boldsymbol{\nu}^{\mathcal{V}}\{\boldsymbol{R}(\boldsymbol{\nu})-\boldsymbol{\nu}\boldsymbol{u}(\mathbf{0})\}\right\| = M\left\{-6\boldsymbol{u}(\boldsymbol{\chi},\boldsymbol{\mathfrak{H}})\frac{\partial\boldsymbol{u}(\boldsymbol{\chi},\boldsymbol{\mathfrak{H}})}{\partial\boldsymbol{\chi}} - \frac{\partial\boldsymbol{\chi}^{\mathcal{I}}\boldsymbol{u}(\boldsymbol{\chi},\boldsymbol{\mathfrak{H}})}{\partial\boldsymbol{\chi}^{\mathcal{I}}}\right\},\tag{14}$$

after some evaluation, Equation (13) is simplified as

$$R(\boldsymbol{\upsilon}) = \boldsymbol{\upsilon}u(0) + \frac{1}{\boldsymbol{\upsilon}^{\chi}} \left\{ M \left\{ -6\boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\varphi}) \frac{\partial \boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\varphi})}{\partial \boldsymbol{\chi}} - \frac{\partial \boldsymbol{\chi}^{\chi}}{\partial \boldsymbol{\chi}^{\chi}} \right\} \right\}, \tag{15}$$

by applying inverse Mohand transform, we get

$$u(\chi, \mathfrak{I}) = u(0) + M^{-1} \left\{ \frac{1}{\upsilon^{\chi}} M \left\{ -6u(\chi, \mathfrak{I}) \frac{g\chi}{g\mu(\chi, \mathfrak{I})} - \frac{g\chi\_{\mathfrak{I}}}{g\_{\mathfrak{I}}\mu(\chi, \mathfrak{I})} \right\} \right\},\tag{16}$$

thus, by using recursive scheme of Equation (11), we get

$$\begin{split} u\_0(\boldsymbol{\chi}, \boldsymbol{\upchi}) &= u(0) = \boldsymbol{\upchi}, \\ u\_{m+1}(\boldsymbol{\upchi}, \boldsymbol{\upchi}) &= M^{-1} \left\{ \frac{1}{\boldsymbol{\upupup}^{\boldsymbol{\upchi}}} M \left\{ -\boldsymbol{\upupup} u\_m(\boldsymbol{\upupup}, \boldsymbol{\upupup}) \frac{\partial u\_m(\boldsymbol{\upupup}, \boldsymbol{\upupup})}{\partial \boldsymbol{\upupup}} \right. \\ &\quad - \frac{\partial^3 u\_m(\boldsymbol{\upupup}, \boldsymbol{\upupup})}{\partial \boldsymbol{\upupup}^3} \right\}, \quad m = 0, 1, \cdots \end{split} \tag{17}$$

From the recursive formula (17),

for m = 0

$$u\_1(\chi, \mathfrak{J}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{V}}} M \left\{ -6\mu\_0(\chi, \mathfrak{J}) \frac{\partial \chi}{\partial \chi\_0} - \frac{\partial \chi^3}{\partial \chi^3} \right\} \right\},\tag{19}$$
 
$$u\_1(\chi, \mathfrak{J}) = -6\chi \frac{\chi\_{\mathcal{V}}}{\chi\_{\mathcal{V}}},$$

for m = 1

$$u\_2(\chi, \mathfrak{z}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ -\mathfrak{e} u\_0(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi} \right\} \right\},$$

$$-\mathfrak{e} u\_1(\chi, \mathfrak{z}) \frac{\partial u\_0(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial^3 u\_1(\chi, \mathfrak{z})}{\partial \chi^3} \right\},$$

$$u\_2(\chi, \mathfrak{z}) = 72 \chi \frac{\mathfrak{z}^{2\mathcal{I}}}{(2\chi)!},\tag{20}$$

for m = 2

$$\begin{split} u\_3(\chi, \mathfrak{z}) &= M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{Y}}} M \left\{ -6u\_0(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial \chi} \right\}, \\ &- 6u\_1(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi^3} \right\}, \\ u\_3(\chi, \mathfrak{z}) &= -864\chi \frac{\mathfrak{z}^{3\mathcal{Y}}}{(3\mathcal{Y})!} - 216\chi(2\mathcal{Y})! \frac{\mathfrak{z}^{3\mathcal{Y}}}{(3\mathcal{Y})!\mathcal{Y}!\mathcal{Y}!}, \\ u\_3(\chi, \mathfrak{z}) &= -864\chi \frac{\mathfrak{z}^{3\mathcal{Y}}}{(3\mathcal{Y})!} - 216\chi(2\mathcal{Y})! \frac{\mathfrak{z}^{3\mathcal{Y}}}{(3\mathcal{Y})!\mathcal{Y}!\mathcal{Y}!}, \end{split} \tag{21}$$

Similarly for m = 3, we can get

$$\mu\_4(\chi, \mathfrak{I}) = 10368 \chi \frac{\mathfrak{I}^{4\chi}}{(4\chi)!} + 2592 \chi (2\chi)! \frac{\mathfrak{I}^{4\chi}}{(4\chi)!\chi!}$$

$$+5184 \chi (3\chi)! \frac{\mathfrak{I}^{4\chi}}{(4\chi)!(2\chi)!\chi!},\tag{22}$$

$$\vdots$$

The Mohand transform solution for example 4.1 is

$$\begin{aligned} \mu(\chi,\mathfrak{z}) &= \chi - 6\pi \frac{\chi}{\chi!} + 72\chi \frac{\chi}{(2\chi)!} - 864\chi \frac{\chi^{3\chi}}{(3\chi)!} \\ \mu(\chi,\mathfrak{z}) &= \chi - 6\pi \frac{\chi^{3}}{\chi!} + 72\chi \frac{\chi^{2\chi}}{(2\chi)!} - 864\chi \frac{\chi^{3\chi}}{(3\chi)!} \\ &- 216\chi(2\chi)! \frac{\chi^{3\chi}}{(3\chi)!\chi!\chi!} + 10368\chi \frac{\chi^{4\chi}}{(4\chi)!} \\ &+ 2592\chi(2\chi)! \frac{\chi^{4\chi}}{(4\chi)!\chi!} + 5184\chi(3\chi)! \frac{\chi^{4\chi}}{(4\chi)!(2\chi)!\chi!} + \cdots \end{aligned} \tag{23}$$

For particular case γ = 1, the Mohand transform solution become as

$$u(\mathbf{x}, \mathbf{r}) = \chi (1 - 6\mathfrak{H} + 36\mathfrak{J}^2 - 216\mathfrak{J}^3 + 1296\mathfrak{J}^4 + \cdots). \tag{25}$$

The calculated result provide the exact solution in the close form

$$
\mu(\chi, \chi) = \frac{\chi}{1 + \mathfrak{G}\mathfrak{J}}.\tag{26}
$$

**Example 4.2:** Consider the third order time fractional KVD equation [50]

$$\frac{\partial \mathcal{G}^{\lambda}}{\partial \boldsymbol{\chi}(\boldsymbol{\chi}, \boldsymbol{\chi})} + \boldsymbol{\eta}(\boldsymbol{\chi}, \boldsymbol{\chi}) \frac{\partial \boldsymbol{\chi}}{\partial \boldsymbol{\eta}(\boldsymbol{\chi}, \boldsymbol{\chi})} + \frac{\partial \boldsymbol{\chi}^{\lambda}}{\partial \boldsymbol{\chi}^{\lambda}} = \boldsymbol{0}, \quad \boldsymbol{0} < \boldsymbol{\chi} \le 1,\tag{27}$$

with initial source

$$u(\chi,0) = 1 - \chi.$$

Taking Mohand transform of Equation (26)

$$\{\boldsymbol{\nu}^{\mathcal{I}}(\boldsymbol{R}(\boldsymbol{\nu}) - \boldsymbol{\nu}\boldsymbol{u}(0))\} = \boldsymbol{M} \left\{ -\boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\mathfrak{I}}) \frac{\partial \boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\mathfrak{I}})}{\partial \boldsymbol{\chi}} - \frac{\partial \boldsymbol{\chi}^{\mathcal{I}}(\boldsymbol{\chi}, \boldsymbol{\mathfrak{I}})}{\partial \boldsymbol{\chi}^{\mathcal{I}}} \right\},\tag{28}$$

after some evaluation, Equation (27) is simplified as

$$R(\nu) = \nu u(0) + \frac{1}{\upsilon^{\mathcal{I}}} \left\{ M \left\{ -u(\chi, \mathfrak{z}) \frac{\partial u(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial^{\frac{3}{2}} u(\chi, \mathfrak{z})}{\partial \chi^{\mathcal{I}}} \right\} \right\},\tag{29}$$

taking inverse Mohand transform of Equation (28)

$$u(\chi, \mathfrak{z}) = u(0) + M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ -u(\chi, \mathfrak{z}) \frac{\partial u(\chi, \mathfrak{z})}{\partial \chi} - \frac{\varrho \chi\_{\mathfrak{z}}}{\varrho \chi\_{\mathfrak{z}}} \right\} \right\}, \tag{30}$$

by using the recursive scheme Equation (11), we get

$$u\_0(\chi, \mathfrak{Z}) = u(0) = 1 - \chi,\tag{31}$$

$$u\_{m+1}(\chi, \mathfrak{Z}) = M^{-1} \left\{ \frac{1}{\upsilon \chi} M \left\{ -\mu\_m(\chi, \mathfrak{Z}) \frac{\partial u\_m(\chi, \mathfrak{Z})}{\partial \chi} \right\} \right.$$

$$-\frac{\partial^3 u\_m(\chi, \mathfrak{Z})}{\partial \chi^3} \right\},\tag{32}$$

From the recursive formula (31),

for m = 0

$$u\_1(\chi, \mathfrak{J}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{V}}} M \left\{ -u\_0(\chi, \mathfrak{J}) \frac{\partial u\_0(\chi, \mathfrak{J})}{\partial \chi} - \frac{\partial \chi^3}{\partial \chi^3} \right\} \right\},\tag{33}$$
 
$$u\_1(\chi, \mathfrak{J}) = (1 - \chi) \frac{\mathfrak{J}^{\mathcal{V}}}{\nu^{\mathcal{I}}},\tag{34}$$

for m = 1

$$\begin{aligned} u\_2(\chi, \mathfrak{z}) &= M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ -u\_0(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi} \right\} \right\}, \\ &- u\_1(\chi, \mathfrak{z}) \frac{\partial u\_0(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial \chi^1(\chi, \mathfrak{z})}{\partial \chi} \bigg\}, \end{aligned}$$

$$
\mu\_2(\chi, \mathfrak{J}) = \mathfrak{L}(1 - \chi) \frac{\mathfrak{J}^{2\nu}}{(2\nu)!},\tag{34}
$$

for m = 2

$$u\_3(\chi, \mathfrak{z}) = M^{-1} \left\{ \frac{1}{\upsilon \mathcal{Y}} M \left\{ -u\_0(\chi, \mathfrak{z}) \frac{\partial u\_2(\chi, \mathfrak{z})}{\partial \chi} \right\} \right.$$

$$\begin{split} -u\_1(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi} - u\_2(\chi, \mathfrak{z}) \frac{\partial u\_2(\chi, \mathfrak{z})}{\partial \chi} \\ -\frac{\partial^3 u\_2(\chi, \mathfrak{z})}{\partial \chi^3} \Big|\_{\mathfrak{z}} \Big|\_{\mathfrak{z}} \\ \mu\_3(\chi, \mathfrak{z}) = 6(1 - \chi) \frac{\mathfrak{z}^3 \mathsf{z}}{(3\lambda)!} .\end{split}$$

The Mohand transform solution for example 3.2 is

$$u(\chi, \mathfrak{z}) = 1 - \chi + (1 - \chi)\frac{\mathfrak{z}\chi}{\chi!} + 2(1 - \chi)\frac{\mathfrak{z}^{2\chi}}{(2\chi)!}$$

$$+6(1 - \chi)\frac{\mathfrak{z}^{3\chi}}{(3\chi)!} + \dotsb.\tag{37}$$

For particular case γ = 1, the Mohand transform solution become as

$$u(\chi, \mathfrak{z}) = 1 - \chi(1 + \mathfrak{z} + \mathfrak{z}^2 + \mathfrak{z}^3 + \cdots). \tag{38}$$

The calculated result provide the exact solution in the close form

$$
\mu(\chi, \mathfrak{z}) = \frac{1 - \chi}{1 - \mathfrak{z}}.\tag{39}
$$

**Example 4.3** Consider the third order time fractional KVD equation [6]

$$\frac{\partial \mathfrak{H}^{\lambda}}{\partial \boldsymbol{\lambda}^{\lambda}(\boldsymbol{\chi}, \boldsymbol{\chi})} - \mathfrak{e}\boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\chi}) \frac{\partial \boldsymbol{\chi}}{\partial \boldsymbol{\mu}(\boldsymbol{\chi}, \boldsymbol{\chi})} + \frac{\partial \boldsymbol{\chi}^{\lambda}}{\partial \boldsymbol{\chi}^{\lambda}} = \boldsymbol{0}, \quad \boldsymbol{0} < \boldsymbol{\chi} \le 1, \tag{40}$$

with initial source

$$u(\chi,0) = 6\chi.$$

Taking Mohand transform of Equation (39)

$$\left\|\boldsymbol{\nu}^{\mathcal{V}}\{\boldsymbol{R}(\boldsymbol{\nu})-\boldsymbol{\nu}\boldsymbol{u}(\boldsymbol{0})\}\right\| = M\left\{6\boldsymbol{u}(\boldsymbol{\chi},\boldsymbol{\chi})\frac{\partial\boldsymbol{u}(\boldsymbol{\chi},\boldsymbol{\chi})}{\partial\boldsymbol{\chi}} - \frac{\partial\boldsymbol{\chi}^{\mathcal{I}}}{\partial\boldsymbol{\chi}^{\mathcal{I}}}\right\},\tag{41}$$

after some evaluation, Equation (40) is simplified as

$$R(\boldsymbol{\upsilon}) = \boldsymbol{\upsilon}u(0) + \frac{1}{\boldsymbol{\upsilon}^{\boldsymbol{\chi}}} \left\{ M \left\{ 6u(\boldsymbol{\chi}, \boldsymbol{\chi}) \frac{\partial u(\boldsymbol{\chi}, \boldsymbol{\chi})}{\partial \boldsymbol{\chi}} - \frac{\partial \boldsymbol{\chi}^{\boldsymbol{\chi}}}{\partial \boldsymbol{\chi}^{\boldsymbol{\chi}}} \right\} \right\},\tag{42}$$

by applying inverse Mohand transform, we get

$$u(\chi, \mathfrak{z}) = u(0) + M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ 6u(\chi, \mathfrak{z}) \frac{\partial u(\chi, \mathfrak{z})}{\partial \chi} - \frac{\mathfrak{z}\chi\_{\mathfrak{z}}}{\mathfrak{z}^{\mathcal{J}}\chi} \right\} \right\},\tag{43}$$

thus, by using recursive scheme of Equation (11), we get

$$\begin{split} u\_0(\chi, \mathfrak{Z}) &= \, u(0) = 6\chi \\ u\_{m+1}(\chi, \mathfrak{Z}) &= M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{Y}}} M \left\{ 6\mu\_m(\chi, \mathfrak{Z}) \frac{\partial u\_m(\chi, \mathfrak{Z})}{\partial \chi} \right. \right. \\ &\left. - \frac{\partial^3 u\_m(\chi, \mathfrak{Z})}{\partial \chi^3} \right\}, \quad m = 0, 1, \cdots \; . \end{split} \tag{45}$$

From the recursive formula (44), for m = 0

$$u\_1(\chi, \mathfrak{J}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{Y}}} M \left\{ \mathfrak{G} u\_0(\chi, \mathfrak{J}) \frac{\partial \mathfrak{J}}{\partial \chi} - \frac{\partial \chi^{\mathcal{Y}}}{\partial \chi^{\mathcal{Y}}} \right\} \right\},$$

$$u\_1(\chi, \mathfrak{J}) = 216 \chi \frac{\mathfrak{J}^{\mathcal{Y}}}{\nu^{\mathcal{I}}},\tag{46}$$

for m = 1

$$\begin{split} u\_2(\chi, \mathfrak{z}) &= M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{V}}} M \left\{ \mathfrak{e} u\_0(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi} \right\} \right. \\ &\left. + \mathfrak{e} u\_1(\chi, \mathfrak{z}) \frac{\partial u\_0(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial^3 u\_1(\chi, \mathfrak{z})}{\partial \chi^3} \right\}, \\ u\_2(\chi, \mathfrak{z}) &= 15552 \chi \frac{\mathfrak{z}^{2\mathcal{V}}}{(2\gamma)!}, \end{split} \tag{47}$$

for m = 2

$$\begin{split} u\_{3}(\chi,\mathbb{S}) &= M^{-1} \left\{ \frac{1}{\upsilon \lambda} M \left\{ 6u\_{0}(\chi,\mathbb{S}) \frac{\chi}{\partial \lambda} - \frac{\chi}{\beta \lambda} \right\} \right\}, \\ &+ 6u\_{2}(\chi,\mathbb{S}) \frac{\mathfrak{g}\chi}{(3\gamma)!} - \frac{\mathfrak{g}\chi^{3}}{9\lambda^{2}(\chi,\mathbb{S})} \right\}, \\ u\_{3}(\chi,\mathbb{S}) &= 1119744 \chi \frac{\mathfrak{g}^{3\chi}}{(3\gamma)!} + 279936 \chi(2\chi)! \frac{\mathfrak{g}\chi}{(3\gamma)!\gamma!\gamma!\gamma!}, \\ &\vdots \end{split} \tag{48}$$

The Mohand transform solution for example 4.3 is

$$\begin{split} \mu(\chi,\chi) &= 6\chi + 216\chi \frac{\chi}{\chi!} + 15552\chi \frac{\chi}{(2\chi)!} + \\ \mu(\chi,\chi) &= \eta \chi + 216\chi \frac{\chi^{3\chi}}{\chi!} + 15552\chi \frac{\chi^{2\chi}}{(2\chi)!} + \\ \mu(\chi,\chi) &= \eta\sigma(\chi,\chi) + \eta\sigma(\chi,\chi)! \frac{\chi^{3\chi}}{(3\chi)!\chi!\chi!} + \dotsb \end{split} \tag{49}$$

For particular case γ = 1, the Mohand transform solution become as

$$\mu(\chi,\mathfrak{z}) = 6\chi(1 + 36\mathfrak{z} + 1296\mathfrak{z}^2 + 46656\mathfrak{z}^3 + \cdots). \tag{51}$$

The calculated result provide the exact solution in the close form

$$
\mu(\chi, \chi) = \frac{\chi\_{\mathfrak{G}}}{1 - 36\mathfrak{G}}.\tag{52}
$$

**Example 4.4** Consider the third order time fractional KVD equation [6]

$$\frac{\partial \chi\_{\lambda} u(\chi, \mathfrak{z})}{\partial \lambda^{\lambda}} - \mathfrak{e}u(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial u(\chi, \mathfrak{z})} + \frac{\partial \chi\_{\lambda}}{\partial \chi^{\lambda}} = 0, \quad 0 < \chi \le 1, \tag{53}$$

with initial source

$$u(\chi, 0) = \frac{6}{\chi^2}.$$

Taking Mohand transform of Equation (52)

$$
\omega^{\mathbb{V}} \{ R(\nu) - \upsilon \mu(0) \} = M \left\{ \Theta \mu(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial \chi} - \frac{\partial \chi^{\mathbb{V}}}{\partial \chi^{\mathbb{V}}} \right\},
$$

after some evaluation, Equation (53) is simplified as

$$R(\nu) = \nu u(0) + \frac{1}{\upsilon \nu} \left\{ M \left\{ \mathfrak{H} \{ \mathfrak{u}(\chi, \mathfrak{z}) \frac{\partial \mathfrak{u}(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial \chi^3}{\partial \chi^3} \right\} \right\},\tag{55}$$

by applying inverse Mohand transform, we get

$$u(\chi, \mathfrak{z}) = u(0) + M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{I}}} M \left\{ 6u(\chi, \mathfrak{z}) \frac{\partial u(\chi, \mathfrak{z})}{\partial \chi} - \frac{\varrho \chi\_{\mathfrak{z}}}{\varrho \chi\_{\mathfrak{z}}} \right\} \right\}.\tag{56}$$

Thus, by using recursive scheme of Equation (11), we get

$$u\_0(\chi, \mathfrak{z}) = u(0) = \frac{6}{\chi^2},\tag{57}$$

$$u\_{m+1}(\chi, \mathfrak{z}) = M^{-1} \left\{ \frac{1}{\upsilon^{\chi}} M \left\{ 6u\_m(\chi, \mathfrak{z}) \frac{\partial u\_m(\chi, \mathfrak{z})}{\partial \chi} \right. \right.$$

$$-\frac{\partial^3 u\_m(\chi, \mathfrak{z})}{\partial \chi^3} \right\}, \quad m = 0, 1, \dots, \tag{58}$$

From the recursive formula (44),

for m = 0

$$u\_1(\chi, \mathfrak{J}) = M^{-1} \left\{ \frac{1}{\upsilon^{\mathcal{V}}} M \left\{ \mathfrak{A} u\_0(\chi, \mathfrak{J}) \frac{\partial u\_0(\chi, \mathfrak{J})}{\partial \chi} - \frac{\partial^3 u\_0(\chi, \mathfrak{J})}{\partial \chi^3} \right\} \right\},$$

$$u\_1(\chi, \mathfrak{J}) = \frac{-288}{\chi^5} \frac{\mathfrak{J}^{\mathcal{V}}}{\mathcal{V}^{\mathcal{I}}},\tag{59}$$

for m = 1

$$u\_2(\chi, \mathfrak{z}) = M^{-1} \left\{ \frac{1}{\upsilon \varkappa} M \left\{ \mathfrak{e} u\_0(\chi, \mathfrak{z}) \frac{\partial u\_1(\chi, \mathfrak{z})}{\partial \chi} \right\} \right\},$$

$$+ 6u\_1(\chi, \mathfrak{z}) \frac{\partial u\_0(\chi, \mathfrak{z})}{\partial \chi} - \frac{\partial \chi^1(\chi, \mathfrak{z})}{\partial \chi^2} \right\},$$

$$u\_2(\chi, \mathfrak{z}) = \frac{12096}{\chi^8} \frac{\mathfrak{z}^{2\chi}}{(2\chi)!},\tag{60}$$

for m = 2

$$\begin{aligned} \mu\_3(\chi, \mathfrak{z}) &= M^{-1} \left\{ \frac{1}{\upsilon^{\chi}} M \left\{ \mathfrak{e} u\_0(\chi, \mathfrak{z}) \frac{\partial u\_2(\chi, \mathfrak{z})}{\partial \chi} \right\} \right\}, \\ &\quad + \mathfrak{e} u\_1(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial \mathfrak{z}} + \mathfrak{e} u\_2(\chi, \mathfrak{z}) \frac{\partial \chi}{\partial \mathfrak{w}(\chi, \mathfrak{z})} \,. \end{aligned}$$

$$u^{\mathfrak{J}}(\boldsymbol{\chi}, \boldsymbol{\upchi}) = \frac{4354560}{\chi^{11}} \frac{\mathfrak{J}^{3\boldsymbol{\upchi}}}{(\mathfrak{J}\boldsymbol{\upchi})!} - \frac{2488320}{\chi^{11}} (2\boldsymbol{\upupchi})! \frac{\mathfrak{J}^{3\boldsymbol{\upupup}}}{(\mathfrak{J}\boldsymbol{\upupup})!\boldsymbol{\upupup}!\boldsymbol{\upupup}!}, \tag{61}$$

The Mohand transform solution for example 4.3 is

$$u(\chi,\chi) = \frac{\mathfrak{u}(\chi,\mathfrak{z}) + \mathfrak{u}\_1(\chi,\mathfrak{z}) + \mathfrak{u}\_2(\chi,\mathfrak{z}) + \mathfrak{u}\_3(\chi,\mathfrak{z}) + \dots}{\mathfrak{x}^2 - \frac{288}{\chi^5} \frac{\mathfrak{z}\chi}{\mathcal{V}!} + \frac{12096}{\mathfrak{z}^8} \frac{\mathfrak{z}^{2\mathcal{V}}}{(2\mathcal{V})!} + \frac{4354560}{\chi^{11}} \frac{\mathfrak{z}^{3\mathcal{V}}}{(3\mathcal{V})!}}{(3\mathcal{V})!}$$

$$-\frac{2488320}{\chi^{11}}(2\mathcal{V})! \frac{\mathfrak{z}^{3\mathcal{V}}}{(3\mathcal{V})! \mathcal{V}! \mathcal{V}!} . \tag{63}$$

For particular case γ = 1, the Mohand transform solution become as

$$u(\chi, \mathfrak{z}) = \frac{6}{\chi^2} - \frac{288}{\chi^5} \mathfrak{z} + \frac{6048}{\chi^8} \mathfrak{z}^2 - \frac{103680}{\chi^{11}} \mathfrak{z}^3 + \dotsb \,. \tag{64}$$

The calculated result converge to the exact solution in the close form

$$
\pi(\chi, \chi) = \frac{6\chi(\chi^3 - 24\mathfrak{J})}{(\chi^3 + 12\mathfrak{J})^2}.\tag{65}
$$

### 4. RESULTS AND DISCUSSION

In **Figure 1**, the exact and analytical solutions of example 4.1 are presented. The solution-graph have confirmed that the obtained results are in good contact with the exact solutions of example 4.1. In **Figure 2**, the fractional-order solutions are calculated at fractional-order γ = 1, 0.9, 0.7, and 0.5. The solutions graphs are expressed in both two and three dimensions. The convergence phenomena can be observed from **Figure 2**. The similar implementation and results can be seen in **Figures 3**–**7** for example 4.3 and 4.4 also. In **Table 1**, the results of MDM are compared with the results of HPTM which provide identical results. It is observed that the proposed method has the sufficient accuracy and rate of convergence to the exact solutions of the problems. It is also investigated that the proposed method provided the simple and straightforward implementation for all examples 1, 2, 3, and 4. These investigations of results have confirmed that the present method can be extended to other fractional-order problems arising in science and engineering.

## 5. CONCLUSION

The proposed method is considered to be one of the pre-eminent and new analytical technique, to solve fractional order partial differential equation. In current research article, the proposed method is applied to solve fractional-order kortewege-De Vries equations. The current method is constructed by using Mohand transformation along with Adomian decomposition method. The new hybrid method is very useful to handle the analytical

TABLE 1 | Comparison of MDM and HPTM [49] of example 1 at ℑ = 0.5.


solutions of fractional-order partial differential equations. To verify, the validity of the suggested method some numerical examples of time fractional third order KdV equations are considered to solve it analytically. The solution graphs have confirmed the validity and reliability of the suggested method toward the solutions of other fractional-order non-linear partial differential equations.

## DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

## AUTHOR CONTRIBUTIONS

HK, RS, and UF has the primary contribution to produce this manuscript. PK has provided the financial support to publish this article. DB and MA have provided their expert opinion and writing draft of the paper.

### FUNDING

The research grant for this manuscript was provided by PK, Center of Excellence in Theoretical and Computational Science (TaCS-CoE) and KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.

### REFERENCES


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Shah, Farooq, Khan, Baleanu, Kumam and Arif. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Marangoni Driven Boundary Layer Flow of Carbon Nanotubes Toward a Riga Plate

Anum Shafiq<sup>1</sup> , Islam Zari <sup>2</sup> , Ilyas Khan<sup>3</sup> \*, Tahir Saeed Khan<sup>2</sup> , Asiful H. Seikh<sup>4</sup> and El-Sayed M. Sherif 4,5

*<sup>1</sup> School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China, <sup>2</sup> Department of Mathematics, University of Peshawar, Peshawar, Pakistan, <sup>3</sup> Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>4</sup> Centre of Excellence for Research in Engineering Materials, King Saud University, Riyadh, Saudi Arabia, <sup>5</sup> Electrochemistry and Corrosion Laboratory, Department of Physical Chemistry, National Research Centre, Cairo, Egypt*

The objective of this article is to explore radiative Marangoni boundary layer flow of carbon nanotubes along a surface that is an electromagnetic actuator, such as a Riga surface. A comparative study is conducted to investigate the behavior of Lorentz forces on the basis of nanoparticle temperature fluxes with two different types of carbon nanotubes, namely single-wall carbon nanotube and multi-wall carbon nanotubes saturated into water as the base fluid. The proposed schemes of governing equations are then converted into ordinary differential equations by similarity transformation. One of best analytical methods, the homotopy analytical method, is utilized for the solution of the governing equations and the convergence of the control parameters. Embedded dimensionless parameters of the flow fields are examined via graphical illustrations. It is observed that an increase in the modified Hartmann number increases the velocity field but reduces the temperature distribution.

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Arshad Riaz, University of Education Lahore, Pakistan Anwar Shahid, International Islamic University, Islamabad, Pakistan*

> \*Correspondence: *Ilyas Khan ilyaskhan@tdtu.edu.vn*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *13 September 2019* Accepted: *26 November 2019* Published: *27 January 2020*

#### Citation:

*Shafiq A, Zari I, Khan I, Khan TS, Seikh AH and Sherif E-SM (2020) Marangoni Driven Boundary Layer Flow of Carbon Nanotubes Toward a Riga Plate. Front. Phys. 7:215. doi: 10.3389/fphy.2019.00215* Keywords: Marangoni boundary layer flow, carbon nanotubes, Riga plate, thermal radiation, series solutions

## 1. INTRODUCTION

Marangoni boundary layer flow phenomena are characterized by gradients in surface tension due to variations in surfactant concentration, concentration of solute, and variations in temperature along the interface. In light of the enhanced significance of surface forces and greater interface extensions, Marangoni boundary layer flows become pertinent in microgravity and in earth gravity. On the other hand, for a duly defined sufficient large Reynolds number, Marangoni boundary layers are edge dissipative flows and form thin dissipative films near unrestricted surfaces [1]. These types of flows have widespread application in diverse fields of engineering and practical projects such as for stabilizing soap films, drying silicon wafers after wrap processing steps, growing crystals, spreading thin films, nucleating vapor bubbles, processing semiconductors, and welding and for use in packed distillation columns, falling film spectator, artificial rain, and materials science. In view of their importance, many researchers have studied and reported results for these types of flows. The first contribution to this research area was by Napolitano [2] during his survey of steady dissipative layers. Lin and Zheng [3] theoretically investigated the problem of Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk. They concluded that the Marangoni parameter has a destabilizing effect on all the other parameters such as temperature, shear stress, velocity, and boundary layer velocity. Moreover, to achieve an analytical solution of

**89**

the said model, they employed the Homotopy Analysis Method (HAM). Recently, Tiwari et al. [4] presented a mathematical model for electrically conducting Marangoni MHD flow saturated with carbon nanotubes (CNTs) as the nanoparticles in a base fluid over a porous medium. An analytical method was adopted to achieve a solution for this project. A similarity solution of Marangoni convection boundary layer flow, taking into account the impacts of gravitational and external pressure, has been studied by Zhang and Zheng [5]. They showed that flow and heat transfer phenomena were substantially affected by the Marangoni convection parameter and the Prandtl number. A numerical method was adopted by Mehdi et al. to investigate the influence of different nanoparticles on Marangoni convection boundary layer flow [6]. The study showed that some nanoparticles with low thermal conductivity have a greater amplification effect on heat transfer phenomena than other recommended particles. Sheikholeslami and Ganji [7] considered Marangoni boundary layer flow to investigate the effect of the magnetic field on various nanofluids. The results illustrated that a Lorentz force increase causes the velocity of nanofluid to decrease. Remeli et al. [8] investigated suction and injection in a nanofluid via Marangoni-driven boundary layer flow. The effect of the injection parameter is to decrease the velocity profile, while the suction parameter increases the velocity profile and delays the separation of the boundary layer. The effects of particle shape on Marangoni convection boundary layer flow of a nanofluid were addressed by Ellahi et al. [9]. They considered different types of nanoparticles, such as needle-shaped, disc-shaped, and sphereshaped. They discussed the said flow model in the context of nanoparticles and found that with an increase in the volume fraction and size of the particles, the surface temperature gradient fluctuated correspondingly. The maximum heat transfer rate at the surface was found in the case of sphere-shaped particles. Further, numerous studies on Marangoni boundary layer flow of several types can be found in the literature [10–12].

The characteristic of nanofluid of tremendously intensifying heat transfer and thermal convection has led to its broad application in innumerable fields, such as in biomedical devices and in highly advanced technical contexts such as the cooling of microchips, nanodrug delivery, nuclear reactions and radiators, etc. To reconcile the issues associated with hightemperature mixtures and to improve thermal conductivity in practice, nanoparticles are soaked into a base fluid. Many contributions have been made to the literature on nanoparticles that disseminate in the base fluid to attain excellent thermal properties [13–15]. According to Tiwari et al. [16], the adding of nanoparticles within a base fluid alone is not enough. CNTs had a six-times improved thermal conductivity compared to other nanomaterials [17]. Through the enhancement of various models of CNTs, these tubes have a wide span of properties such as thermal and electronic [18]. Similarly, solid nanoparticles have higher conductivity than do liquids. Therefore, CNTs are a topic of interest for advanced technology due to their electrical and isolated structure. In recent years, different applications of CNTs [19–21] have been investigated to develop ideal materials ranging from ultra-strong fibers to field emission. These tubes have extensive uses and applications in various fields such as providing increased energy density for capacitors, modeling the structures of catalysts, detecting proteins that indicate the existence of oral cancer, for gas storage, for water purification devices, for detecting bacteria in drinking water, for minimizing the weight of coaxial cable in aerospace applications, for improving battery lifetime, as an extra powerful fiber, etc. In this regard, Hayat et al. [22] utilized carbon nanotubes in water flow under homogenous-heterogeneous reactions and with melting heat transfer effects. Two different types of CNTs, i.e., SWCNT and MWCNT, were incorporated in water for the flow model. It was found that, in comparison with other nanofluids, the minimum thermal resistance and maximum heat transfer was achieved when MWCNT was disseminated in the base fluid. Moreover, the surface thickness of carbon nanotubes with heat transfers in stagnation point flow was examined by Hayat et al. [23].

From the last few decades, many researchers have turned their attention toward the study of flow fields with different configurations. One of the new geometries devised by Gailitis and Lielausis [24] for weakly conducting fluids is the so-called Riga plate. The novelty of this plate is that it incorporates and imposes magnetic and electric fields, which properly instigates Lorentz forces parallel to the wall to constrain the flow of weakly conducting fluid. Avoiding boundary layer separation, it may be utilized as an efficient agent for submarine pressure drag, skin friction, and radiation. The related theory has important features and is employed in many areas such as engineering, geophysics, astrophysics, industrial procedures, and MHD generators. Pantokratoras et al. [25] addressed boundary layer flow based on a weakly conducting fluid passing through a Riga plate. Their analysis demonstrated that by keeping quality, suitable size of nanoparticles, and adjusting the magnitude of flow, aiding and opposing Lorentz force due to the Riga plate in order to control the skin friction. Magyari and Pantokratoras [26] carried out an investigation to extend the idea of opposing and aiding mixed convection flows through a Riga plate. Further, Pantokratoras [27] addressed Blasius and Sakiadis type flows over a Riga plate. Hayat et al. [28] explored the flow of nanofluid through a convectively heated Riga plate with variable thickness. The results demonstrated that for larger values of the modified Hartman number, the velocity distribution exhibited decreasing behavior. Shafiq et al. [29] analyzed the impact of radiation in stagnation point flow of Walters' B fluid through a Riga plate. Their observations indicated that due to enhancement of the strength of Newtonian heating, the temperature and surface heat transfer significantly increased. Theoretical and numerical discussion by Bhatti et al. has shown the effects of thermal radiation with EMD through a Riga plate [30]. Further, a Cattaneo-Christov model for third-grade nanofluid flow toward a Riga plate has been developed by Naseem et al. [31] by using a semi-analytical method, i.e., the optimal homotopy analysis method (OHAM). The proposed theory was adopted together with the newly esteemed zero nanoparticles mass flux condition to investigate mass and thermal diffusions. Thermal radiation and heat transfer phenomena play a central role in advanced technological systems through boundary layer flow. The important applications of this flow in the aforementioned fields can be seen in the literature [32–34]. Henceforth, in

various flow fields, the importance of thermal radiation cannot be overlooked. Non-linear radiation and Joule heating in Marangoni mixed convection flow were demonstrated by Hayat et al. [35]. The impact of exponential temperature on radiation effects and particle shape was examined by Lin et al. by utilizing heat transfer of copper water-based nanofluid and Marangoni boundary layer flow [36]. Hayat et al. [37] reported the effect of Joule heating and thermal radiation in the flow of third-grade fluid over a radiative surface. They studied whether the presence of an electric field and the radiation parameter caused the temperature and velocity to increase. A revised model of second-grade nanofluid magnetohydrodynamic Falkner Skan flow was examined by Hayat et al. [38]. A few other interesting investigations are given in Ellahi et al. [39–44], Bhatti et al. [30, 45, 46], Ellahi and Riaz [47], and Waqas et al. [48].

The main intention of the present study is to interpret radiative Marangoni-driven boundary layer flow utilizing different types of CNTs (SWCNTs and MWCNTs) over a Riga plate. To the best of our knowledge, such a study does not yet exist in the literature. Suitable transformations are utilized to establish a non-linear system of equations. The homotopy analysis method (HAM) is utilized for convergent series solutions. The impacts of several influential parameters on the physical quantities of interest are analyzed through tables and graphs. The upcoming sections illustrate the mathematical model and explore the effects of the different physical parameters on the velocity and temperature profiles, respectively.

#### 2. MATHEMATICAL SCHEME OF THE PROBLEM

We consider Marangoni boundary layer flow of carbonnanoliquids (SWCNTs and MWCNTs) toward a Riga surface along with radiation phenomenon. The Riga plate comprises a spanwise connected array of permanent magnets and irregular electrodes attached to a horizontal surface. Lorentz forces generated by the Riga plate and directed along the free stream are responsible for optimally controlling the proposed flow field. The base fluid, water, is packed with SWCNTs and MWCNTs. Further, the governing equations for the flow form may be expressed as (1–5):

$$
\frac{\partial \check{u}}{\partial \check{\mathbf{x}}} + \frac{\partial \check{\mathbf{v}}}{\partial \check{\mathbf{y}}} = \mathbf{0},
\tag{1}
$$

$$\check{u}\frac{\partial\check{u}}{\partial\check{\chi}} + \check{\nu}\frac{\partial\check{u}}{\partial\check{\jmath}} = \check{u}\_{\epsilon}\frac{d\check{u}\_{\epsilon}}{d\check{\chi}} + \frac{\mu\_{\eta f}}{\rho\_{\eta f}}\frac{\partial^{2}\check{u}}{\partial\check{\jmath}^{2}} + \frac{\pi\,M\_{0}\,f\_{0}\,\operatorname{Exp}[-\frac{\pi}{b}\check{\chi}]}{8\rho\_{\eta f}}$$

$$-g\,\check{\beta}(T - T\_{w}),\tag{2}$$

$$\begin{split} \check{u}\frac{\partial T}{\partial \check{\boldsymbol{x}}} + \check{\nu}\frac{\partial T}{\partial \check{\boldsymbol{y}}} &= \alpha\_{\boldsymbol{n}\boldsymbol{f}} \, \frac{\partial^{2}T}{\partial \check{\boldsymbol{y}}^{2}} - \frac{1}{(\rho c\_{\boldsymbol{p}})\_{\boldsymbol{n}\boldsymbol{f}}} \frac{\partial q\_{\boldsymbol{r}}}{\partial \check{\boldsymbol{y}}} \\ &+ \frac{\mu}{(\rho c\_{\boldsymbol{p}})\_{\boldsymbol{n}\boldsymbol{f}}} \left( \frac{\partial \check{\boldsymbol{u}}}{\partial \check{\boldsymbol{y}}} \right)^{2}, \end{split} \tag{3}$$

and the boundary conditions are set as

$$\check{\nu} = 0, \qquad T = T\_0, \qquad \frac{\mu\_{nf}}{\mu\_f} \frac{\partial \check{u}}{\partial \check{\jmath}} = \frac{\partial T}{\partial \check{\varkappa}} \qquad \text{at } \check{\wp} = 0,$$

$$\check{u} = \check{u}\_{\varepsilon}, \qquad T = T\_{\varepsilon} \qquad \text{at } \check{\wp} \to \infty. \qquad \text{(4)}$$

The velocity components in the x˘ and y˘ directions mentioned in the expressions are u˘ and v˘ , the fluid density is denoted by ρ, the velocity of external flow is u˘ <sup>e</sup>(x), ρnf indicates the nanofluid density, µnf is the nanofluid dynamic viscosity, j<sup>0</sup> stands for the applied current density within the electrodes, M<sup>0</sup> is the magnetization of the permanent magnets, b indicates the width of the magnets and electrode, the constant temperature of the Riga plate is denoted by T<sup>w</sup> where T<sup>w</sup> > 0, K is the thermal conductivity, c<sup>p</sup> represents the specific heat, µ is the dynamic viscosity, and T is the nanofluid temperature. The boundary temperature distribution is T0(x), and (ρcp)nf is the nanofluid heat capacity. The nanofluid effective density is αnf . The radiative heat flux q<sup>r</sup> is defined as

$$q\_r = -\frac{4\sigma^\*}{3k\_1} \frac{\partial T^4}{\partial \check{\mathbf{y}}},\tag{5}$$

where σ ∗ is the Stefan-Boltzmann constant and k<sup>1</sup> is the mean absorption coefficient. Through Taylor's series, we have T <sup>4</sup> ∼= 4T 3 <sup>e</sup> T − 3T 4 e , where T<sup>e</sup> is the ambient temperature, and then energy equation now reduces to the following expression:

$$(\rho c\_{\mathcal{V}})\_{\rm nf} \left( \check{u} \frac{\partial T}{\partial \check{\boldsymbol{\chi}}} + \check{\boldsymbol{\nu}} \frac{\partial T}{\partial \check{\boldsymbol{\chi}}} \right) = \left( \frac{16 \boldsymbol{\sigma}^{\*} T\_{\infty}^{3}}{3k\_{1}} + K \right) \frac{\partial^{2} T}{\partial \check{\boldsymbol{\chi}}^{2}} + \mu \left( \frac{\partial \check{\boldsymbol{u}}}{\partial \check{\boldsymbol{\chi}}} \right)^{2} . \tag{6}$$

Moreover, the mathematical properties of CNTs are demonstrated by the following Equation (19–22)

$$\alpha\_{\eta f} = \frac{k\_{\eta f}}{(\rho c\_p)\_{\eta f}}, \ \mu\_{\eta f} = \frac{\mu\_f}{(1-\phi)^{2.5}}, \ \check{\nu}\_{\eta f} = \frac{\mu\_{\eta f}}{\rho\_{\eta f}},$$

$$\frac{k\_{\eta f}}{k\_f} = \frac{2\phi \frac{\check{k}\_{\text{CNT}}}{k\_{\text{CNT}} - k\_f} \ln \frac{\check{k}\_{\text{CNT}} + \check{k}\_f}{2\check{k}\_f} + (1-\phi)}{2\phi \frac{\check{k}\_f}{k\_{\text{CNT}} - k\_f} \ln \frac{\check{k}\_{\text{CNT}} + \check{k}\_f}{2\check{k}\_f} + (1-\phi)},$$

$$\rho\_{\eta f} = \rho\_f (1-\phi) + \rho\_i \langle c\_p \rangle\_{\text{CNT}} \, \phi \, , \tag{7}$$

where ˘k<sup>f</sup> is the fluid thermal conductivity, knf is the nanofluid thermal conductivity, the nanofluid solid volume fraction is φ, and µ<sup>f</sup> is the fluid dynamic viscosity. <sup>µ</sup><sup>ˆ</sup> nf µˆf ∂u ∂y y=0 = ∂T ∂x y=0 denotes Marangoni condition at the interface. The linear relation of surface tension σ is given as:

$$
\sigma = \sigma\_0 \left[ 1 - \chi\_1 \left( T - T\_e \right) \right], \tag{8}
$$

where γ<sup>1</sup> = − 1 σ0 ∂σ <sup>∂</sup><sup>T</sup> > 0 represents the surface tension temperature coefficient, and σ<sup>0</sup> represents surface tension. The directions of the driving forces depend on the orientation of the temperature gradients in nanoliquids ∇T.

The similarity transformation is introduced:

$$\begin{aligned} u\left(\mathbf{x},\boldsymbol{\uprho}\right) &= u\_0 \mathbf{x}^{(2r-1)/3} f'\left(\boldsymbol{\upeta}\right), \\ \nu\left(\mathbf{x},\boldsymbol{\uprho}\right) &= \frac{1}{3} u\_0 l\_0 \mathbf{x}^{(r-2)/3} \left[ (2-r)\,\boldsymbol{\uprho} f'\left(\boldsymbol{\upeta}\right) - (1+r)f\left(\boldsymbol{\upeta}\right) \right], \\ T\left(\mathbf{x},\boldsymbol{\uprho}\right) &= T\_\varepsilon - h\_0 \mathbf{x}^r \boldsymbol{\uptheta}\left(\boldsymbol{\upeta}\right) \,, \quad \boldsymbol{\upeta} = \mathbf{x}^{(r-2)/3} \frac{\boldsymbol{\uprho}}{l\_0}, \end{aligned} \tag{9}$$

where h0, u0, and l<sup>0</sup> represent constants. The values of u<sup>0</sup> and l<sup>0</sup> take the following form when h<sup>0</sup> = 1:

$$u\_0 = \left(\frac{3}{1+r}\right)^{1/3} r^{2/3}, \quad l\_0 = \left(\frac{3}{1+r}\right)^{1/3} r^{-1/3}, \quad \text{(10)}$$

after the above-mentioned transformations, Equations (1)–(6) take the following form:

$$\frac{1}{(1-\phi)^{2.5}(1-\phi+\frac{\rho\_{\rm{CNT}}}{\rho\_{\rm{f}}}\phi)}f^{\prime\prime} + f^{\prime\prime} - (\frac{2r-1}{1+r})[(f^{\prime})^2 - 1]$$

$$+\frac{3}{1+r}\lambda\theta(\eta) + \frac{3}{1+r}Q\exp[-c\eta] = 0,\tag{11}$$

$$\begin{split} \mathbb{P}\left(1+\frac{4}{3}R\right) \frac{\frac{k\_{nf}}{k\_f}}{[(1-\phi)+\frac{(\rho c\_p)\_{\text{CNT}}}{(\rho c\_p)\_f}\phi]} \theta'' \\ -\frac{3}{1+r}\Pr\left[rf'\theta-\frac{1+r}{3}f\,\theta'\right]-\Pr\left[E\,\left(f\right)^2\right] &= 0,\end{split} \tag{12}$$

in which

$$\mathbf{Q} = \frac{\pi M\_0 l\_0 \check{\mathbf{x}}}{8 \rho\_{\eta f} \check{\mathbf{u}}\_{\varepsilon}^2}, \quad \mathbf{R} = \frac{4 \sigma^3 T\_{\infty}^3}{3 k\_{\eta f} k^\*}, \quad \mathbf{Pr} = \frac{c\_p \mu}{k}, \quad \mathbf{Ec} = \frac{\check{\mathbf{u}}\_{\varepsilon}^2 \check{\mathbf{x}}^{\frac{1}{3}(r-2)}}{\rho\_{\varepsilon} c\_p^2 h\_0},\tag{13}$$

where Q denotes the modified Hartmann number, R represents the radiation parameter, Pr indicates the Prandtl number, and Ec symbolizes the Eckert number.

#### 3. SOLUTION METHODOLOGY

To find the series solution of the underlying problem, the Homotopy Analysis Method is adopted. Therefore, the auxiliary linear operators (I<sup>f</sup> , <sup>I</sup><sup>θ</sup> ) and the initial guess ( ˘ f0, θ˘ 0 ) may be defined as:

$$\check{f}\_0(\eta) = \eta + (1 - \phi)^{2.5} (1 - e^{-\eta}), \quad \check{\theta}\_0(\eta) = e^{-\eta}, \tag{14}$$

$$\mathcal{I}\_{\theta}(\check{\theta}) = \frac{d^2 \check{\theta}}{d\eta^2} - \check{\theta}, \quad \mathcal{I}\_f(\check{f}) = \frac{d^3 \check{f}}{d\eta^3} - \frac{d\check{f}}{d\eta}, \tag{15}$$

$$\begin{aligned} \mathcal{T}\_{\emptyset} \left[ K\_1 + K\_2 \text{Exp}(\eta) + K\_3 \text{Exp}(-\eta) \right] &= 0, \\ \mathcal{T}\_{\emptyset} \left[ K\_4 \text{Exp}(\eta) + K\_5 \text{Exp}(-\eta) \right] &= 0, \end{aligned} \tag{16}$$

where K<sup>h</sup> (h = 1 − 5) are arbitrary constants.

The zeroth-order problem design is

$$
\hat{\rho}(1-\check{p})\mathbb{Z}\_f[\check{f}(\eta,\check{p})-\check{f}\_0(\eta)]=\check{p}h\_f\mathcal{M}\_f\left[\check{f}(\eta,\check{p})\right],\tag{17}
$$

$$\left. \frac{\partial \check{f}(\eta \,; \check{p})}{\partial \eta} \right|\_{\eta=0} = 0, \quad \frac{1}{(1-\phi)^{2.5}} \frac{\partial^2 \check{f}(\eta \,; \check{p})}{\partial \eta^2} \bigg|\_{\eta=0} = -1,$$

$$\left. \frac{\partial \check{f}(\eta \,; \check{p})}{\partial \eta} \right|\_{\eta \to \infty} = 1,\tag{18}$$

$$(1 - \not p) \mathbb{Z}\_{\theta} \left[ \not\! \not p (\eta \,, \not p) - \not\! \not p (\eta \,, \not p) \right] = \not\! \not p \hbar\_{\theta} \mathcal{M}\_{\theta} \left[ \not\!\not (\not p \,, \not p) \not\! \not p \right] \,, \tag{19}$$
 
$$\not\!\not (\eta \,, \not p) \Big|\_{\eta = 0} = 1, \quad \not\!\not (\eta \, \not p) \Big|\_{\eta \to \infty} = 0,\tag{20}$$

The non-linear operators are

M<sup>f</sup> h ˘ f(η ; p˘) i = 1 (1 − φ) 2.5[1 − φ + (ρ˜cp) CNT (ρ˜cp)<sup>f</sup> φ] ∂ 3 ˘ f(η , p˘) ∂η3 + ˘ f(η , p˘) ∂ 2 ˘ f(η , p˘) ∂η2 − 2r − 1 1 + r ∂ ˘ f(η , p˘) ∂η !<sup>2</sup> + 2r − 1 1 + r + 3 1 + r λθ(η) + 3 1 + r Q Exp[−cη ], (21) M<sup>θ</sup> h ˘ f(η ; p˘), θ˘(η ; p˘) i = 1 + 4 3 Rd knf kf [1 − φ + (ρ˜cp) CNT (ρ˜cp)<sup>f</sup> φ] ∂ 2 θ˘(η , p˘) ∂η2 − 3 1 + r Pr " r ∂ ˘ f(η , p˘) ∂η θ˘(η , p˘) # − 3 1 + r Pr " − 1 + r 3 ˘ f(η , p˘) ∂θ˜(η , p˘) ∂η # <sup>−</sup> Pr Ec ∂ 2 ˘ f(η , p˘) !2 , (22)

where, 0 ≤ p˘ ≤ 1 and h¯<sup>f</sup> and h¯<sup>θ</sup> designate zero free auxiliary parameters.

∂η2

The mth-order deformation problem is

$$\mathcal{L}\_f \left[ \check{f}\_m(\eta) - \check{\chi}\_m \check{f}\_-(\eta) \right] = h\_f \mathcal{P}\_m^{\check{\mathfrak{f}}}(\eta) \,, \tag{23}$$

$$\left. \frac{\partial^2 \check{f}\_m(\eta \,\, \check{p})}{\partial \eta^2} \right|\_{\eta = 0} = 0, \quad \left. \frac{\partial \check{f}\_m(\eta \,\, \check{p})}{\partial \eta} \right|\_{\eta \to \infty} = 0,$$

$$\check{f}\_m(\eta \,\, \check{\mathfrak{g}}) \Big|\_{\eta = \infty} = 0 \,\tag{24}$$

$$\left.\bar{f}\_{m}(\eta \,;\check{p})\right|\_{\eta=0} = 0,\tag{24}$$

$$\mathcal{T}\_{\theta}\left[\check{\theta}\_{m}(n) - \check{\chi}\_{m}\check{\theta}\_{m-1}(n)\right] = \hbar\mathcal{P}^{\tilde{\theta}}(n).\tag{25}$$

$$\mathcal{I}\_{\theta} \left[ \check{\theta}\_{m} \left( \eta \right) - \check{\chi}\_{m} \check{\theta}\_{m-1} \left( \eta \right) \right] = h\_{\theta} \mathcal{P}\_{m}^{\tilde{\theta}} \left( \eta \right) \,, \tag{25}$$
 
$$\left. \check{\rho}\_{m} (\eta \cdot \mathfrak{H}) \right| \qquad - \text{o} \quad \check{\rho}\_{m} (\eta \cdot \mathfrak{H}) \qquad - \text{o} \tag{26}$$

$$\left. \delta\_m(\eta : \not p) \right|\_{\eta=0} = 0, \left. \not \delta\_m(\eta : \not p) \right|\_{\eta \to \infty} = 0,\tag{26}$$

Frontiers in Physics | www.frontiersin.org

$$\begin{split} \mathcal{P}\_{m}^{\tilde{f}}(\eta) &= \frac{1}{(1-\phi)^{2.5} [1-\phi + \frac{\rho\_{CNT}}{\rho\_{f}}\phi]} f\_{m-1}^{\prime\prime\prime}(\eta) \\ &- \sum\_{k=0}^{m-1} f\_{m-1-k} f\_{k}^{\prime\prime} - \frac{2r-1}{1+r} \left[ \sum\_{k=0}^{m-1} f\_{k} f\_{m-1-k}^{\prime} - (1-\chi\_{m}) \right] \\ &+ \frac{3}{1+r} \lambda \theta\_{m-1}(\eta) + \frac{3}{1+r} Q \operatorname{Exp}[-c\eta], \\ \mathcal{P}\_{m}^{\tilde{\delta}}(\eta) &= \left( 1 + \frac{4}{3} R\_{d} \right) \frac{\bar{\dot{k}}\_{\tilde{\delta}}}{\operatorname{Im} \left[ \tanh \frac{\bar{\dot{k}}\_{\tilde{\delta}}}{\tanh \eta} \theta\_{m-1}^{\prime\prime}(\eta) \right] \end{split} \tag{27}$$

$$\begin{split} \boldsymbol{\zeta}(\boldsymbol{\eta}) &= \left(1 + \frac{1}{3}\boldsymbol{R}\_{d}\right) \frac{\boldsymbol{r}}{[1 - \phi + \frac{(\rho c\_{p})\_{\text{CNT}}}{(\rho c\_{p})\_{f}}\phi]} \boldsymbol{\theta}\_{m-1}^{\prime\prime}(\boldsymbol{\eta}) \\ &- \frac{3}{1+r} \Pr \sum\_{k=0}^{m-1} \left[r f\_{k}^{\prime} \boldsymbol{\theta}\_{m-1-k} - \frac{1+r}{3} f\_{m-1-k} \boldsymbol{\theta}\_{k}^{\prime}\right] \\ &- \Pr \boldsymbol{E} \sum\_{k=0}^{m-1} f\_{m-1-k}^{\prime} \boldsymbol{f}\_{k}^{\prime}, \end{split} \tag{28}$$

where

$$\check{\chi}\_m = \begin{cases} 0, m \le 1 \\ 1, m > 1 \end{cases} \tag{29}$$

For p˘ = 0 and p˘ = 1, we have

$$
\check{f}(\eta,0) = \check{f}\_0(\eta), \quad \check{f}(\eta,1) = \check{f}(\eta), \tag{30}
$$

$$
\check{\theta}(\eta,0) = \check{\theta}\_0(\eta), \qquad \check{\theta}(\eta,1) = \check{\theta}(\eta). \tag{31}
$$

The solutions ˘ f(η; p˘) and θ˘(η; p˘) vary from the primary solutions ˘ f0(η) and θ˘ <sup>0</sup>(η) to the final solutions ˘ f(η) and θ˘(η), respectively, where p˘ differs from 0 to 1. The Taylor series expansion follows:

$$\check{f}(\boldsymbol{\eta}, \check{\boldsymbol{p}}) = \check{f}\_0(\boldsymbol{\eta}) + \sum\_{m=1}^{\infty} \check{f}\_m(\boldsymbol{\eta}) \check{\boldsymbol{p}}^m, \\ \check{f}\_m(\boldsymbol{\eta}) = \frac{1}{m!} \left. \frac{\partial^m \check{f}\_m(\boldsymbol{\eta}, \boldsymbol{\tilde{p}})}{\partial \check{\boldsymbol{p}}^m} \right|\_{\check{\boldsymbol{p}} = 0}, \tag{32}$$
 
$$\check{\theta}(\boldsymbol{\eta}, \boldsymbol{\tilde{p}}) = \check{\theta}\_0(\boldsymbol{\eta}) + \sum\_{m=1}^{\infty} \theta\_m(\boldsymbol{\eta}) \check{\boldsymbol{p}}^m, \\ \check{\theta}\_m(\boldsymbol{\eta}) = \frac{1}{m!} \left. \frac{\partial^m \check{\theta}(\boldsymbol{\eta}, \boldsymbol{\tilde{p}})}{\partial \check{\boldsymbol{p}}^m} \right|\_{\check{\boldsymbol{p}} = 0}.$$

The above series solutions converge if the auxiliary parameters are properly nominated. Therefore,

$$\check{f}(\eta) = \check{f}\_0(\eta) + \sum\_{m=1}^{\infty} \check{f}\_m(\eta),\tag{34}$$

$$\check{\theta}(\eta) = \check{\theta}\_0(\eta) + \sum\_{m=1}^{\infty} \check{\theta}\_m(\eta), \tag{35}$$

The general solutions ˘ fm, θ˘<sup>m</sup> via special solutions f × <sup>m</sup> , θ × m are

$$\check{f}\_m(\eta) = f\_m^\times(\eta) + K\_1 + K\_2 \text{Exp}(\eta) + K\_3 \text{Exp}(-\eta), \tag{36}$$

$$\check{\theta}\_m(\eta) = \theta\_m^\times(\eta) + K\_4 \text{Exp}(\eta) + K\_5 \text{Exp}(-\eta), \tag{37}$$

where K<sup>h</sup> (h = 1 − 5) are the elaborated constants.

#### 4. CONVERGENCE OF SERIES SOLUTIONS

The convergence phenomenon of HAM solution is dependent on auxiliary parameters ℏ<sup>f</sup> and ℏ<sup>θ</sup> , which control and adjust the convergence of the derived series solution. Therefore, ℏ− curves are portrayed in **Figures 2A,B** for different values of the physical parameters in terms of SWCNT and MWCNT. The suitable values of these parameters ℏ<sup>f</sup> and ℏ<sup>θ</sup> are −0.78 ≤ ℏ<sup>f</sup> < −0.19, −0.19 ≤ ℏ<sup>θ</sup> < −0.03 for SWCNT and −0.66 ≤ ℏ<sup>f</sup> < −0.1, −0.22 ≤ ℏ<sup>θ</sup> < −0.01 for MWCNT.

#### 5. DISCUSSION

The major contribution of this section is to explore the physical influence of different dimensionless parameters on the velocity and temperature profiles. A physical sketch of the problem is given in **Figure 1**. The graphs in **Figures 3**–**11** depict the results of the comprehensive analysis. We divide this section into three subsections for simplicity and clarity. In the first subsection, exploration is made of the physical impact of various parameters on the velocity profile. A discussion of the effects of the same parameters along with the radiation parameter on the temperature profile is given in the next subsection. The third subsection is based on the performance of the Nusselt number under the influence of different parameters.

The impact of fluid parameter r on the velocity profile is illustrated in **Figure 3** for both SWCNT and MWCNT. The velocity distribution is noted to decrease with the intensification of parameter r. Further, the velocity profile is seen to be higher in the case of MWCNT with a base fluid of water. **Figure 4** is plotted to indicate the effect of the nanofluid solid volume fraction φ for both SWCNT and MWCNT. It is shown that the velocity distribution increases with the enhancement of the nanofluid volume fraction. In addition, a stronger response is seen with MWCNT than with SWCNT. This is due to the low density of MWCNT. **Figures 5**, **6** display the effects of variation in the convection parameter λ on the velocity profile in both opposing (λ < 0) and assisting (λ > 0) flows. Under these circumstances, the velocity profile and the momentum boundary layer thickness exhibit increasing behavior with the buoyancy

parameter for both SWCNT and MWCNT. Basically, the mixed convection may be defined as the ratio of buoyancy forces to inertial forces. The reason behind enhancement in the velocity of the fluid is the buoyancy force, which influences the inertial force, increasing the value of the mixed convection parameter. Moreover, MWCNT shows an increasing trend throughout the field in comparison to SWCNT. The effect of the Hartmann number Q on the velocity profile is shown in **Figure 7** for both SWCNT and MWCNT. Physically, an increase in the modified Hartmann number increases the velocity field. The structure of the Hartmann number is the ratio of electromagnetic force to viscous force. Since the increasing phenomenon of the velocity profile is dependent on an increase in Q, this indicates development in the Lorentz force, which is generated by the presence of a magnetic field in the flow field and acts against the flow if the magnetic field is applied in the normal direction. The significance of the modified Hartmann number Q is exhibited in **Figure 8** for both SWCNT and MWCNT. An increment in the Hartmann number correlates with a reduction in the temperature distribution.

The effect of variation of radiative parameter R on temperature profile θ(η) is plotted in **Figure 9** for both MWCNT and SWCNT. The behavior inferred from this figure is that there is an enhancement in temperature distribution and in the related boundary layer thickness due to the increment in R. Hence, the temperature profile is an increasing function of the radiative parameter. Hence, the enhancement in the temperature profile due to an increase in the radiative parameter causes a reduction in the absorption coefficient. Further, SWCNT shows a stronger response compared with MWCNT.

The physical effect of fluid parameter r on the velocity field is depicted in **Figure 10** for both SWCNT and MWCNT. The velocity profile is noted to increase with escalation in parameter

TABLE 1 | Thermophysical characteristics of base fluid and nanoparticles (SWCNT and MWCNT).


TABLE 2 | Convergence of homotopy solutions when α = 0.01, β = 0.1, γ = 0.1, δ = 0.3, *M* = 0.1, *Re* = 2, Pr = 6.2, *Ec* = 0.1, and ℏ*<sup>f</sup>* = ℏ<sup>θ</sup> = −0.7.


TABLE 3 | Numerical values of the Nusselt number for both SWCNT and MWCNT at different values of other parameters.


r. **Figure 11** depicts the influence of the nanofluid solid volume fraction φ for both SWCNT and MWCNT. It is observed that with the augmentation of the nanofluid volume fraction, the velocity profile shows a reduction.

**Table 1** presents the thermophysical properties (density, specific heat, and thermal conductivity) of the base fluid (water) and carbon nanotubes (SWCNT and MWCNT). **Table 2** shows that the series solutions are convergent up to four decimal places for the velocity profile at the 10th order of approximation for MWCNT and at the 20th order of approximation for SWCNT. Similarly, for the case of the temperature field, the 20th order of approximation for SWCNT and 15th order of approximation for MWCNT were observed for convergence. Further, **Table 3** displays the behavior of the local Nusselt number for different values of physical parameters such as fluid parameter r, convection parameter λ, Hartman number Q, volume fraction φ, radiative parameter R , and parameter γ . The desired results were observed for both SWCNT and MWCNT. It is concluded that the Nusselt number shows decreasing behavior for larger values of r, λ, and φ in the cases of both SWCNT and MWCNT. On the other hand, the Nusselt number shows stronger behavior for larger values of Q, R , and γ for both SWCNT and MWCNT.

### 6. FINAL OBSERVATION

The key points are as follows:


### DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

## AUTHOR CONTRIBUTIONS

AS formulated the problem. IZ solved the problem. TK computed the results. IK discussed the results with the conclusion. All authors contributed to writing the manuscript.

### ACKNOWLEDGMENTS

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through Researchers Supporting Project number (RSP-2019/33), King Saud University, Riyadh, Saudi Arabia. AS was supported by the Talented Young Scientist Program of Ministry of Science and Technology of China (Pakistan-19-007).

## REFERENCES


the presence of a motile gyrotactic microorganism: Buongiorno's nanofluid model. Phys Script. (2019) **94**:1–15. doi: 10.1088/1402-4896/ ab2ddc

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Shafiq, Zari, Khan, Khan, Seikh and Sherif. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# Entropy Generation in C6H9NAO<sup>7</sup> Fluid Over an Accelerated Heated Plate

Tarek Nabil Ahmed1,2 \* and Ilyas Khan1,3 \*

*<sup>1</sup> Basic Engineering Sciences Department, College of Engineering Majmaah University, Al-Majmaah, Saudi Arabia, <sup>2</sup> Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt, <sup>3</sup> Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, Saudi Arabia*

This study considers sodium-alginate (C6H9NaO7) fluid over an accelerated vertical plate. The plate is heated from the bottom. A non-Newtonian model of C6H9NaO<sup>7</sup> is considered. The convection term in the momentum equations is also considered. The dimensionless form of the problem is constructed based on dimensionless variables. The integral transformation of Laplace is used to develop the exact solution to the problem. Explicit expressions are obtained for the velocity field and temperature distribution. The corresponding skin-friction and Nusselt number results are computed based on this. Equations for entropy generation (EG) and Bejan number (BN) are developed. The results are plotted and discussed for embedded parameters. Most significantly, the results for EG and BN are computed and discussed.

#### Edited by:

*Sara I. Abdelsalam, National Autonomous University of Mexico, Mexico*

#### Reviewed by:

*Kh S. Mekheimer, Al-Azhar University, Egypt Abdullah Zaher, Benha University, Egypt Hina Sadaf, National University of Sciences and Technology, Pakistan*

#### \*Correspondence:

*Tarek Nabil Ahmed t.ahmed@mu.edu.sa Ilyas Khan i.said@mu.edu.sa*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *20 October 2019* Accepted: *23 December 2019* Published: *07 February 2020*

#### Citation:

*Ahmed TN and Khan I (2020) Entropy Generation in C6H9NAO<sup>7</sup> Fluid Over an Accelerated Heated Plate. Front. Phys. 7:250. doi: 10.3389/fphy.2019.00250* Keywords: heat transfer, entropy generation, Casson fluid, exact solutions, integral transform

### INTRODUCTION

Entropy generation (EG) is a tool that helps to assess improved results, enhance achievements, and reduce the loss of energy in thermal engineering systems (TES) [1]. Recently, this technique has been applied to TES operating with nanofluids [2]. The EG method is used to develop performance standards for thermal engineering equipment. In the literature, Bejan is considered to be the first to point out the various factors behind EG [3, 4] in TES. Bejan [5] introduced the EG number, referred to as the Bejan number, which is the ratio of EG due to heat transfer to the total EG of the system. Moreover, he indicated the conditions of the second law of thermodynamics related to the convection problems of nanofluids. Selimefendigil et al. [6] demonstrated the magneticresistive convection flow of nanofluids (CuO-water and Al2O3-water) in a restricted trapezoidal cavity. Quing et al. [7] investigated EG in radiative flow of Casson nanofluids over permeable stretchable sheets. A detailed review of EG in nanofluid flow was presented by Mahian et al. [8], who collected and critically discussed recent studies with a wide range of applications. The study organized different aspects of heat-transfer problems and EG in the current state of the art, making suggestions for useful future directions.

Darbari et al. used the response surface method (RSM) to conduct a numerical sensitivity analysis of the effect of nanoparticles (Al2O3) in water-based nanofluids on EG [9]. The results indicated that the total EG comprised EG due to friction and due to heat conduction. The sensitivity analysis of EG highlighted the influence of the Reynolds number, particle size, and solid-volume fraction. Ellahi et al. [10] mathematically analyzed EG in natural-convection boundary-layer flow of nanofluids near an inverted cone. It was found that EG was produced because of nanoparticles. Sheikholislami et al. [11] revealed that the EG and heat-transfer rate were enhanced by volume friction and Rayleigh number during the flow of various types of nanofluids in a cavity containing

**100**

square-shell rectangular heated objects. Saqib et al. [12] developed a Caputo-type fractional model for the mixedconvection flow of different types of nanofluids. The exact analytical results for velocity, temperature, EG, and Bejan number were obtained via the Laplace-transform technique and presented in figures and tables with physical explanations. Khan et al. [13] described the EG in unsteady magnetic fluid dynamics (MHD) flow through porous media, combining the effects of mass and heat transfer. The effects of several factors on EG, Bejan number, and velocity distribution were reported in numerous figures. Bhatti et al. [14] analyzed the EG of Eyeling-Powell nanofluids through a permeable stretchable surface. The effects of magnetohydrodynamics (MHD) and non-linear thermal radiation were also considered. Li et al. [15] considered EG in forced-convection flow of Al2O3-water nanofluids. They reported the impact of Reynolds number (Re), height ratio, and pitch ratio on EG.

Khan et al. [16] obtained an exact solution for the problem of convection-MHD flow of sodium-alginate-based Cassontype nanofluids with the effects of MHD and Newtonian heating. Haq et al. [17] used an exact analysis and developed an exact solution for the free-convection problem of viscous fluid, which depends strongly on time and the slippage condition. Khan et al. [18] generated exact solutions for a rotating viscous fluid such that the fluid exhibits eccentricconcentric rotation. Ahmed and Khan [19] examined the mixed-convection flow of SA-NaAlg nanofluids such that the base fluid is taken as MoS2. Khater et al. [20, 21] studied two different problems using the magnetohydrodynamics effect with a Hall current. In this problem, the analysis of entropy generation is considered for Casson fluid over an accelerated plate. The problem in dimensionless form is solved by using the Laplace transform technique, and the results are plotted and discussed.

#### DESCRIPTION OF THE PROBLEM

Consider the unsteady, incompressible mixed-convection flow of a Casson fluid near an infinite vertical plate. It is assumed that, at τ ≤ 0, the system is at rest at a temperature of θ∞. At τ = 0 +, the plate starts moving with a variable velocity of v (0, τ ) = Aτ , and the temperature of the plate increases from θ (η, 0) = θ<sup>∞</sup> to θ (0, τ ) = θw. At this stage, mixed convection occurs owing to the change in temperature and the motion of the plate. The initial fluid motion is in the vertical direction and is governed by the following partial differential equations (momentum and energy equations) [16, 19].

$$\rho \frac{\partial \nu \left(\eta, \tau\right)}{\partial \tau} = \mu \left(1 + \frac{1}{\beta}\right) \frac{\partial^2 \nu}{\partial \eta^2} + \rho g \beta \theta \left(\theta(\eta, \tau) - \theta\_{\infty}\right) \text{,} \text{(1)}$$

$$
\rho c\_p \frac{\partial \theta \{\eta, \tau\}}{\partial \tau} = k \frac{\partial^2 \theta \{\eta, \tau\}}{\partial \eta^2}, \tag{2}
$$

These are associated with the following physical initial and boundary conditions.

$$\begin{cases} V\left(\eta,0\right) = 0, & \theta\left(\eta,0\right) = \theta\_{\infty} \\ \nu\left(0,\tau\right) = A\tau, & \nu\left(\infty,\tau\right) = 0 \\ \theta\left(0,\tau\right) = \theta\_{\text{w}}, & \theta\left(\infty,\tau\right) = \theta\_{\infty} \end{cases} \right\},\tag{3}$$

where ρ is the density, v (η, τ ) the x-component of the velocity vector, µ the dynamic viscosity, g the gravitational acceleration, β<sup>θ</sup> the volumetric thermal expansion, θ(η, τ ) the x-component of the temperature vector, c<sup>p</sup> the heat capacitance, and k the thermal conductivity of the fluid. To remove the units, the following dimensionless variables are introduced into Equations (1)–(3).

$$\nu^\* = \frac{\nu}{(\nu A)^{\frac{1}{3}}}, \ \eta^\* = \frac{\eta A^{\frac{1}{3}}}{\nu^{\frac{2}{3}}}, \ \text{\textbullet}^\* = \frac{\tau A^{\frac{2}{3}}}{\nu^{\frac{1}{3}}}, \ \theta^\*(\eta, \tau) = \frac{\theta - \theta\_{\infty}}{\theta\_{\text{w}} - \theta\_{\infty}},$$

This yields the following form.

$$\frac{\partial \nu}{\partial x} = \left(1 + \frac{1}{\beta}\right) \frac{\partial^2 \nu}{\partial \eta^2} + Gr \theta,\tag{4}$$

$$\text{Pr}\frac{\partial\theta}{\partial x} = \frac{\partial^2\theta}{\partial\eta^2},\tag{5}$$

$$\begin{cases} V\left(\eta,0\right) = 0, \; \nu\left(0,\tau\right) = \tau, \quad \nu\left(\infty,\tau\right) = 0\\ \theta\left(0,\tau\right) = 1, \; \theta\left(\infty,\tau\right) = 0, \quad \theta\left(\eta,0\right) = 0 \end{cases} \right\},\tag{6}$$

where Gr = gβθ 1θ A ,Pr = µcp k .

#### Entropy Generation

The following entropy-generation relation is developed to optimize the heat transfer and minimize the energy loss in the system defined in Equations (4)–(6) [3–5, 12, 13].

$$s\_{\rm gen} = \frac{k}{\theta\_{\infty}^2} \left(\frac{\partial \theta}{\partial \eta}\right)^2 + \frac{\mu}{\theta\_{\infty}} \left(1 + \frac{1}{\beta}\right) \left(\frac{\partial \nu}{\partial \eta}\right)^2. \tag{7}$$

Using the non-similarity variable, ∂θ/∂η = 1θA 1 3 ν − 2 <sup>3</sup> ∂θ <sup>∗</sup> /∂η<sup>∗</sup> and ∂v/∂η = A 2 3 ν − 1 <sup>3</sup> ∂v ∗ /∂η<sup>∗</sup> are derived and incorporated into Equation (7), which yields

$$N\_s = \left(\frac{\partial\theta}{\partial\eta}\right)^2 + \frac{Br}{\Omega}\left(1 + \frac{1}{\beta}\right)\left(\frac{\partial\nu}{\partial\eta}\right)^2,\tag{8}$$

where

$$N\_{\theta} = \frac{s\_{\text{gen}} \nu^{\frac{4}{3}} \theta^{\frac{2}{3}} \infty}{k A^{2/3} (\Delta \theta)^{2}}, \text{ } Br = \frac{\mu A^{\frac{2}{3}} \nu^{\frac{2}{3}}}{k \Delta \theta}, \text{ } \Omega = \frac{\Delta \theta}{\theta\_{\infty}} = \frac{\theta\_{\text{w}} - \theta\_{\infty}}{\theta\_{\infty}}.$$

#### Bejan Number

Bejan is generally considered in the literature to be the first person to point out various factors for optimizing the performance of thermal systems. He developed Bejan's number, which is the ratio of heat-transfer entropy production to total entropy production, and proposed aspects of the second law of thermodynamics that consider various problems associated with mixed convection. The Bejan number is given by

$$Be = \frac{\frac{k}{\theta\_{\infty}^{2}} \left(\frac{\partial \theta}{\partial \eta}\right)^{2}}{\frac{k}{\theta\_{\infty}^{2}} \left(\frac{\partial \theta}{\partial \eta}\right)^{2} + \frac{\mu}{\theta\_{\infty}} \left(1 + \frac{1}{\beta}\right) \left(\frac{\partial \nu}{\partial \eta}\right)^{2}} \tag{9}$$

and

$$Be = \frac{\left(\frac{\partial\theta}{\partial\eta}\right)^2}{\left(\frac{\partial\theta}{\partial\eta}\right)^2 + \frac{Br}{\Omega}\left(1 + \frac{1}{\beta}\right)\left(\frac{\partial\nu}{\partial\eta}\right)^2}.\tag{10}$$

#### EXACT SOLUTIONS

In the literature, mixed-convection problems are handled using numerical or approximate methods, and exact solutions are limited. Here, the exact solutions are obtained using the Laplace transform method. Applying the Laplace transform to Equations (4)–(6) gives

$$q\overline{\nu}(\eta,q) = \left(1 + \frac{1}{\beta}\right) \frac{\partial^2 \overline{\nu}(\eta,q)}{\partial \eta^2} + Gr\overline{\theta}(\eta,q) \tag{11}$$

$$\overline{\nu}\left(0,q\right) = \frac{1}{q^2}, \quad \overline{\nu}\left(\infty,q\right) = 0\tag{12}$$

$$\Pr q \overline{\theta} \left( \eta, q \right) = \frac{\partial^2 \overline{\theta} (\eta, q)}{\partial \eta^2} \tag{13}$$

$$\overline{\partial}\left(0,q\right) = \frac{1}{q}, \quad \overline{\partial}\left(\infty,q\right) = 0\tag{14}$$

The second-order partial differential Equation (13) is solved using the transform boundary conditions (14) as follows.

$$\overline{\theta}\left(\eta, q\right) = \frac{e^{-\eta\sqrt{\text{Pr}\,q}}}{q} \tag{15}$$

Inverting the Laplace transform yields

$$\theta\left(\eta,\tau\right) = \text{erfc}\left(\frac{\eta\sqrt{\text{Pr}}}{2\sqrt{\tau}}\right) \tag{16}$$

Similarly, the solution of Equation (11) using Equations (12) and (15) is given by

$$\overline{\nu}\left(\eta,q\right) = \frac{a\_1}{q^2}e^{-\eta\sqrt{\mathcal{V}q}} + \frac{a\_0}{q^2}e^{-\eta\sqrt{\mathbb{P}\mathbf{r}\cdot q}}\tag{17}$$

where

$$\left(1 + \frac{1}{\beta}\right) = \frac{1}{\gamma}, \quad a\_0 = \frac{Gr\gamma}{\gamma - \text{Pr}}, \quad a\_1 = 1 - a\_0.$$

With the inverse Laplace transform,

$$\begin{split} \nu(\eta,\tau) &= a\_1 \left[ \left( \frac{1}{2} \eta^2 \mathcal{V} + \tau \right) \operatorname{erfc} \left( \frac{\eta \sqrt{\mathcal{V}}}{2 \sqrt{\tau}} \right) - \eta \sqrt{\frac{\tau \mathcal{V}}{\pi}} e^{\frac{\eta^2 \mathcal{V}}{4 \tau}} \right] \\ &+ a\_0 \left[ \left( \frac{1}{2} \eta^2 \operatorname{Pr} + \tau \right) \operatorname{erfc} \left( \frac{\eta \sqrt{\operatorname{Pr}}}{2 \sqrt{\tau}} \right) - \eta \sqrt{\frac{\tau \operatorname{Pr}}{\pi}} e^{\frac{\eta^2 \operatorname{Pr}}{4 \tau}} \right] . \end{split} \tag{18}$$

**Special Case:** Note that Equation (18) is reduced to the following form for Newtonian fluid ( <sup>1</sup> <sup>β</sup> →0):

$$\begin{split} \nu(\eta,\tau) &= (1-\frac{Gr}{1-\text{Pr}}) \left[ \left(\frac{1}{2}\eta^{2}+\tau\right) \text{erfc}\left(\frac{\eta}{2\sqrt{\pi}}\right) - \eta\sqrt{\frac{\tau}{\pi}}\varepsilon\frac{\eta^{2}}{4\tau} \right] \\ &+ \frac{Gr}{1-\text{Pr}} \left[ \left(\frac{1}{2}\eta^{2}\text{Pr}+\tau\right) \text{erfc}\left(\frac{\eta\sqrt{\text{Pr}}}{2\sqrt{\pi}}\right) - \eta\sqrt{\frac{\tau}{\pi}\frac{\text{Pr}}{\pi}}\varepsilon\frac{\eta^{2}\text{Pr}}{4\tau} \right]. \end{split} \tag{19}$$

#### Skin Friction

In the dimensionless form, skin friction is defined as

$$c\_f = \left(1 + \frac{1}{\beta}\right) \left. \frac{\partial \nu(\eta, \tau)}{\partial \eta} \right|\_{\eta = 0} \tag{20}$$

#### Nusselt Number

The heat-transfer rate in the dimensionless form is given by

$$Nu = \left. \frac{\partial \theta(\eta, \mathbf{r})}{\partial \eta} \right|\_{\eta = 0} \tag{21}$$

### RESULTS AND DISCUSSION

In this paper, we conducted an entropy generation (EG) analysis for accelerated flow of non-Newtonian fluid. EG, also known as the second law of thermodynamics, is quite useful in heat transfer problems such as in analyzing heat exchangers. This section highlights the influence of different parameters on velocity, temperature, entropy generation, and Bejan number. Since, in this work, sodium-alginate is taken as a counter-example of a Casson fluid, the Prandtl number (Pr) value is taken as 13.09

FIGURE 4 | Temperature plot for τ.

in all of these figures. This value of Pr is computed from Pr = µcp/k, µ = 0.002; k = 0.6376; c<sup>p</sup> = 4175.

**Figure 1** shows the effects of time τ on velocity. It is found that an increase in time results in an increase in the velocity profile. Physically, the fluid is considered to be unsteady, and thus velocity increases with time. **Figure 2** highlights the effect of Gr: the velocity profile increases with increasing Gr Value. The increase in Gr enhances the buoyancy force, causing the velocity to increase. The physical interpretation indicates that positive values of Gr show heating of the fluid or cooling of the boundary surface. The effect of the Casson parameter, β, is highlighted in **Figure 3**; a dual effect is generated. Initially, near the plate, the velocity is found to increase, and then away from the plate, it decreases for large values of β. This is because an increase in β reduces the boundary-layer thickness. **Figure 4** shows the influence of time τ on the temperature profile, where the maximum values of time τ lead to an increase in temperature.

The impact of EG (Ns) for dissimilar values of τ is highlighted in **Figure 5**. An increase in time τ leads to an increase in EG. **Figure 6** presents the EG values for different values of . is defined as the temperature difference, and an increase in temperature difference decreases entropy generation. **Figure 7** presents the influence of unlike values of Gr on EG. The buoyancy forces increase with increasing Gr values, which results in an

increase in entropy generation. In addition, an increase in Gr could save energy in the system.

The effect of β is shown in **Figure 8**; it is significant to note that the thickness of the velocity boundary layer decreases with increasing Casson parameter value, and hence EG increases. Furthermore, at high values of β, i.e., β → ∞, Newtonian fluid behavior is observed. The decrease in the Casson parameter leads to an increase in fluid plasticity. The influence of Brickman's number, Br, is investigated in **Figure 9**. A large value of Brickman's number produces a high amount of heat via viscous dissipation and vice versa. Therefore, high values of Brickman's number increase entropy generation.

The influence of time parameter τ on Bejan number variation is highlighted in **Figure 10**. The influence of time τ leads to a decrease in Bejan number. **Figure 11** shows the effect of the temperature difference, , on the Bejan number; the maximum value of corresponds with an increase in the Bejan number. **Figure 12** highlights the change in Bejan number with respect to Gr. It is detected that a greater Gr value decreases the Bejan number. This is because heat-transfer reunification becomes dominant in the region near the plate with increasing Gr value. In **Figure 13**, the Bejan number can be seen to decrease with increasing Casson parameter β. The Bejan number variation for different Br values is reported in **Figure 14**. Larger values of Br are associated with decreasing Bejan number.


TABLE 2 | Effect of variation of different variables on Nu.


**Table 1** examines the effect of different factors on skin friction. It is observed that the skin friction increases with increasing τ , β, and Gr values. **Table 2** highlights the effect of the variation in τ and Pr on Nusselt number. The Nusselt number decreases up to the maximum value of τ and increases for the maximum value of Pr.

#### REFERENCES


#### CONCLUDING REMARKS

An exact analysis of entropy generation in sodium-alginate fluid over an accelerated heated plate is conducted via Laplacetransform methods. The Bejan number Be and local entropy generation Ns are discussed for various parameters. The effects are displayed for different embedded parameters. The main conclusions are:


### AUTHOR CONTRIBUTIONS

TA formulated and solved the problem. IK plotted and discussed the results and revised the manuscript. TA and IK wrote the manuscript.

#### ACKNOWLEDGMENTS

The authors acknowledge with thanks the Deanship of Scientific Research (DSR) at Majmaah University, Majmaah, Saudi Arabia, for technical and financial support through vote number 38/107 for this research project.


Hall current. Eur Phys J B Condens Matter Complex Syst. (2006) **50**:17. doi: 10.1140/epjb/e2006-00138-5

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Ahmed and Khan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


## GREEK SYMBOLS


]


# Swimming of Motile Gyrotactic Microorganisms and Nanoparticles in Blood Flow Through Anisotropically Tapered Arteries

Muhammad M. Bhatti 1,2 \*, Marin Marin<sup>3</sup> , Ahmed Zeeshan<sup>4</sup> , R. Ellahi 4,5,6 and Sara I. Abdelsalam<sup>7</sup>

*<sup>1</sup> College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China, <sup>2</sup> Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China, <sup>3</sup> Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, Romania, <sup>4</sup> Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences (FBAS), International Islamic University (IIU), Islamabad, Pakistan, <sup>5</sup> Department of Mechanical Engineering, University of California, Riverside, Riverside, CA, United States, <sup>6</sup> Center for Modeling and Computer Simulation, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, <sup>7</sup> Basic Science, Faculty of Engineering, The British University in Egypt, Cairo, Egypt*

#### Edited by:

*Dumitru Baleanu, University of Craiova, Romania*

#### Reviewed by:

*Amin Jajarmi, University of Bojnord, Iran Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria*

#### \*Correspondence:

*Muhammad M. Bhatti muhammad09@shu.edu.cn; mmbhatti@sdust.edu.cn*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *15 November 2019* Accepted: *13 March 2020* Published: *08 April 2020*

#### Citation:

*Bhatti MM, Marin M, Zeeshan A, Ellahi R and Abdelsalam SI (2020) Swimming of Motile Gyrotactic Microorganisms and Nanoparticles in Blood Flow Through Anisotropically Tapered Arteries. Front. Phys. 8:95. doi: 10.3389/fphy.2020.00095* In the present article, we have presented a theoretical study on the swimming of migratory gyrotactic microorganisms in a non-Newtonian blood-based nanofluid via an anisotropically narrowing artery. Sutterby fluid model is used in order to understand the rheology of the blood as a non-Newtonian fluid model. This fluid pattern has the ability to show Newtonian and non-Newtonian features. The mathematical formulation is performed via continuity, temperature, motile microorganism, momentum, and concentration equation. The series solutions are obtained using the perturbation scheme up to the third-order approximation. The resulting solutions are discussed with the help of graphs for all the leading parameters. The graphical results are also presented for non-tapered, diverging, and converging artery. We further discuss the velocity, temperature, swimming microorganism and temperature distribution. Moreover, the variation of impedance and the impact of wall shear stress are discussed and presented through the graphs.

Keywords: Sutterby fluid, wall shear stress, motile microorganism, anisotropically tapered artery, nanoparticles, perturbation solutions

### INTRODUCTION

Throughout the previous decade, nanofluids have gained essential importance due to their extensive fields of applications especially in the biomedical sciences. Different theoretical and experimental studies have been presented based on the formulation of nanofluids [1–4]. Nanofluids are beneficial in improving the thermo-physical features i.e., thermal diffusivity, convection, and conductivity of the governing fluid. In biomedical science, nanofluids are helpful for the bacteriostatic activity, nano-drug delivery, labeling of cancerous tissues, magnetic resonance imaging (MRI), localized therapy, cancer therapeutics, production of ferrofluids and magnetic resonance imaging, etc. Further, they are also beneficial in nano-cryosurgery. Ferrofluids can be utilized as contrast agents for MRI and are helpful in cancer detection. In this case, the ferrofluids are made up of iron oxide nanoparticles and are recognized as superparamagnetic iron oxide nanoparticles (SPIONs). Recently, the localized delivery of cancer medicine to the cancer patient at the affected part. With the help of the heat transfer process, it can also be used for detergency. Because of these significant applications, different authors examined the behavior of nanofluids in different situations. Bég and Tripathi [5] presented a Mathematica simulation of the bioengineering model with the help of peristaltic configuration and nanofluids. Tripathi and Bég [6] analyzed the drug delivery systems using a peristaltic flow of nanofluids and presented the exact mathematical solutions. Kothandapani and Prakash [7] explored the behavior of a heat source on an MHD non-Newtonian hyperbolic tangent nanofluid model in an asymmetric tapered conduit. El-Dabe et al. [8] discussed the influence of slip in mild stenosis tapered artery using peristaltic simulation. Akbar [9] addressed the blood flow with thermal conductivity in a nontapered stenosis artery filled with blood. She further discussed the shape properties of the nanoparticles. Abbas et al. [10] presented a blood flow model using nanofluids and explained the applications of drug delivery and magnetic field phenomena. Akbar [11] studied the metal-based nanomaterials suspended in the blood propagating via a tapered stenotic artery and explained the applications of Nanomedicines. Bhatti et al. [12] discussed the heat transfer properties and the applications of the blood clot model with variable viscosity. They considered the twophase model with peristalsis. Bhatti et al. [13] also discussed the behavior of titanium magneto-nanoparticles suspended in Sisko fluid. Some more essential studies on the blood flow and nanofluids can be found from Shit et al. [14], Riaz et al. [15], Ijaz and Nadeem [16], and Abdelsalam and Bhatti [17] and in the references therein.

The macroscopic movement of the fluid as a result of the spatial variation of density over an area causes additive mobility in the swimming microorganisms known as bioconvection. The self-driven motile microorganisms tend to improve the base fluid in a particular direction producing a bio-convective stream. The moving microorganisms are divided into various types i.e., chemotaxis or oxytactic, gyrotactic microorganisms, and negative gravitaxis. The nanoparticles are not self-driven as compared with motile microorganisms, and their motion is due to the impact of the Brownian motion and the thermophoresis effect. Bioconvection in the nanofluids is anticipated to be feasible if the concentration of nanoparticles is small and as a result it won't be able to produce an essential enhancement in the base fluid viscosity. Bioconvection in the presence of nanoparticles was initially considered in Kuznetsov and Avramenko [18, 19]. Later, Kuznetsov [20] presented the suspension of nanoparticles with gyrotactic microorganisms using the Buongiorno's theory. Bég et al. [21] investigated the bioconvection flow with nanofluids through a porous medium numerically. Akbar [22] considered the bioconvection flow through a symmetric channel filled with nanoparticles and presented a bio nano-engineering model. Bhatti et al. [23] also inspected the behavior of a varying magnetic field and clot blood model using Jeffrey fluid model with nanoparticles and microorganisms. Ahmed et al. [24] considered the magnetized laminar flow of nanofluid and gyrotactic microorganisms through a non-Darcy porous medium. Chakraborty et al. [25] researched the extrinsic magnetic influence and bioconvection flow with nanoparticles with convective boundary conditions. Few important studies on the motile gyrotactic microorganisms and nanofluids can be found in Shahid et al. [26], Waqas et al. [27], Waqas et al. [28], and Sohail et al. [29].

From the above survey, it is observed that blood flow in the presence of nanoparticles has been discussed, but no attention has been devoted to discussing the simulation of motile gyrotactic microorganisms and nanoparticles suspended in the blood propagating through an anisotropically tapered artery. In most of the aforementioned studies, work has been done with nanoparticles propagating through tapered artery, however, no one considered the presence of gyrotactic microorganism in blood. Mathematical modeling has been performed on the basis of temperature, momentum, concentration and motile microorganism equations followed by an approximation in wavelength being long with and inertia-free flow. The Homotopy perturbation scheme is employed to obtain the series results. The governing equations are nonlinear and coupled and the exact solutions are not possible, whereas some other numerical/analytical methods [30–32] are beneficial to solve these kinds of problems. All the outcomes are presented graphically and plotted against the leading parameters. The behavior of temperature, velocity, concentration, and motile microorganism profile have been considered. Furthermore, wall shear and variation of impedance are also investigated and presented graphically. According to the results, it is found that the flow behavior through converging, diverging and non-tapered arteries are uniform throughout the whole channel.

#### MATHEMATICAL MODELING

We consider a tube having finite length "L" filled with nanofluids and motile gyrotactic microorganisms. We present here the theoretical model of the swimming of nanoparticles with motile gyrotactic microorganisms in non-Newtonian blood flow propagating in an anisotropically tapered artery. A Sutterby fluid model is used to represent the rheology of the blood. The governing fluid is incompressible and having constant density. Let (r, θ, z) be the cylinderical polar coordinates while z lies along the axis, whereas r, θ are considered along the radial and circumferential direction (see **Figure 1**). We consider the temperature and concentration at the wall of the tube as T<sup>1</sup> and C1, respectively. The anisotropically tapered stenosed artery with time-variant stenosis is geometrically defined as

$$\frac{R(z)}{R\_0} = \begin{cases} \tau\left(t\right) \left[\xi z + R\_0 - \frac{\delta \cos\Psi}{L\_0}\left(11 - \frac{94}{3L\_0}(z-d)\right) \right. \\\quad + \frac{32}{L\_0^2}(z-d)^2 - \frac{32}{L\_0^3}(z-d)^3 \right]; & d \le z \le \frac{3}{2}L\_0, \text{(1)};\\\tau\left(t\right)\left(1 + \xi z\right); & \text{otherwise} \end{cases}$$

where R(z) denotes the tapered arterial segment and the artery radius with composite stenosis, t the time, L<sup>0</sup> the stenosis length,

δ is the stenosis height, R<sup>0</sup> is the normal artery radius in the nonstenotic zone, 9 is the tapering angle, and ξ = tan 9 shows the slope of the tapered vessel i.e.,

$$\Psi = \begin{cases} < 0, \text{ converging array} \\ = 0, \text{ non-tapered array} \\ > 0, \text{ diverging array} \end{cases} \tag{2}$$

The time-variant, τ (t), is defined as

$$
\pi\_\*(t) = 1 + \frac{\alpha \left(1 - \cos \omega t\right)}{e^{\alpha \alpha t}},\tag{3}
$$

where α is constant and ω is the radial frequency of the forced oscillation.

The equations governing the flow model can then be written as

$$\begin{aligned} \nabla \cdot \tilde{\mathbf{V}} &= 0, \end{aligned} \tag{4}$$

$$\begin{aligned} \rho\_f \left( \frac{\partial \tilde{\mathbf{V}}}{\partial t} + \tilde{\mathbf{V}} \cdot \nabla \tilde{\mathbf{V}} \right) &= -\nabla \cdot \boldsymbol{\rho} + \nabla^2 \cdot \tilde{\mathbf{V}} \\ &+ \left[ \rho\_f T\_\varepsilon \left( 1 - C\_1 \right) \left( T - T\_1 \right) \right. \\ &- \left( \rho\_\mathbb{P} - \rho\_f \right) T\_\varepsilon \left( C - C\_1 \right) \\ &- \left( n - n\_1 \right) \Theta \left( \rho\_m - \rho\_f \right) \Big] \mathbf{g}, \end{aligned} \tag{5}$$

where **V**˜ = [U,V] are the components of velocity, T is the nanofluid temperature, T<sup>1</sup> is the reference temperature, p is the pressure, 2 is the average volume of a microorganism, n is the concentration of microorganisms, ρ<sup>f</sup> is the density basefluid at the reference temperature, ρ<sup>p</sup> is the nanoparticles' density, ρ<sup>m</sup> is the density of microorganisms, **g** is the gravity vector, T<sup>e</sup> is the base fluid volumetric coefficient of thermal expansion, and µ the viscosity of the suspension (the suspension contains the nanoparticles, microorganisms, and base fluid).

The temperature equation reads as

$$\begin{split} (\rho c)\_f \left( \frac{\partial T}{\partial t} + \mathbf{\tilde{V}} \cdot \nabla T \right) &= \nabla \cdot \left( k\_f \nabla T \right) \\ &+ (\rho c)\_p \left[ D\_B \nabla C \cdot \nabla T + \frac{D\_T}{T\_1} \nabla T \cdot \nabla T \right], \end{split} \tag{6}$$

where D<sup>T</sup> and DB, k<sup>f</sup> , (ρc)<sup>f</sup> and (ρc)<sup>p</sup> are the thermophoretic diffusion and Brownian coefficient, thermal conductivity, volumetric heat capacities for the nanofluid and nanoparticles, respectively.

The concentration equation with no chemical reaction reads as

$$\left(\frac{\partial \mathbf{C}}{\partial t} + \tilde{\mathbf{V}} \cdot \nabla \mathbf{C}\right) = D\_B \nabla^2 \mathbf{C} + \frac{D\_T}{T\_1} \nabla T \cdot \nabla T,\tag{7}$$

The conservation of microorganisms' reads as

$$\left(\frac{\partial n}{\partial t} + \tilde{\mathbf{V}} \cdot \nabla n\right) + \frac{bW\_c}{\mathbf{C}\_0 - \mathbf{C}\_1} \nabla \left(n \cdot \nabla \mathbf{C}\right) = -D\_{\text{mo}} \nabla^2 n,\tag{8}$$

where b is the chemotaxis constant, W<sup>c</sup> is the maximum cell swimming speed, and Dmo is the diffusivity of microorganisms.

The stress tensor for Sutterby fluid reads as

$$\mathbf{S} = \frac{\mu}{2} \left[ \frac{\sinh^{-1} B \boldsymbol{\zeta}}{B \boldsymbol{\zeta}} \right]^m \mathbf{A}\_1. \tag{9}$$

where A, B are material constants and

$$
\zeta = \sqrt{\frac{\text{trace } \mathbf{A}\_1^2}{2}},
\tag{10}
$$

**A**1=grad **V** + grad **V** T . The boundary conditions are given by

$$\begin{cases} \frac{\partial u}{\partial r} = \frac{\partial T}{\partial r} = \frac{\partial C}{\partial r} = \frac{\partial n}{\partial r} = 0, \text{ at } r = 0\\ u = 0, T = T\_1, C = C\_1, n = n\_1, \text{ at } r = R(z) \end{cases} \tag{11}$$

The non-dimensional quantities are defined as

$$\begin{split} \hat{r} &= \frac{r}{R\_0}, \hat{z} = \frac{z}{R\_0}, \hat{\nu} = \frac{L\_0}{U\_d \delta} \boldsymbol{\nu}, \hat{\mathsf{R}} = \frac{R}{R\_0}, \hat{\mathsf{p}} = \frac{R\_0^2}{U\_d L\_0 \mu} \boldsymbol{\rho}, T \\ &= (1 - \theta) \, T\_1 + T\_0, \\ \mathrm{C} &= (1 - \phi) \, \mathrm{C}\_1 + \mathrm{C}\_0, n = (1 - \chi) \, n\_1 + n\_0, \hat{\delta} = \frac{\delta}{R\_0}, \hat{\mathrm{L}} \\ &= \frac{L}{L\_0}, \hat{\xi} = \frac{L\_0 \xi}{R\_0}. \end{split} \tag{12}$$

where U<sup>a</sup> is the averaged velocity over a section of the whole tube.

Substituting with Equation (12) into the governing mathematical model assuming the case of mild stenosis and creeping flow yields (after dropping the hat)

$$\frac{\partial p}{\partial r} = 0,\tag{13}$$

$$\frac{\partial p}{\partial z} = \frac{1}{r} \frac{\partial}{\partial r} \left[ r \mathbf{S}\_{rz} \right] + T\_{\xi} \theta + \mathbf{N}\_{\xi} \phi - \mathbf{R}\_{b} \chi,\tag{14}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \theta}{\partial r}r\right) + \frac{\partial \theta}{\partial r}\left[T\_b\frac{\partial \Phi}{\partial r} + T\_t\left(\frac{\partial \theta}{\partial r}\right)\right] = 0,\tag{15}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \phi}{\partial r}r\right) + \frac{T\_t}{T\_b}\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \theta}{\partial r}r\right) = 0,\tag{16}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \chi}{\partial r}r\right) = P\_l \left[\frac{\partial \chi}{\partial r}\frac{\partial \phi}{\partial r} + \left(\bar{\Theta} + \chi\right)\frac{\partial^2 \phi}{\partial r^2}\right],\tag{17}$$

and

$$\mathbf{S}\_{rz} = \left[1 - \beta \left(\frac{\partial \boldsymbol{u}}{\partial r}\right)^2\right] \left(\frac{\partial \boldsymbol{u}}{\partial r}\right),\tag{18}$$

whereas for Newtonian fluid the results can be achieved by taking β = 0.

The parameters used above are defined as

$$\beta = \frac{mB^2 U\_a^2}{6R\_0^2}, T\_b = \frac{D\_B \left(C\_0 - C\_1\right) \left(\rho c\right)\_p}{k\_f \left(\rho c\right)\_f},$$

$$\begin{split} T\_l &= \frac{D\_B \left(T\_0 - T\_1\right) \left(\rho c\right)\_p}{k\_f T\_1 \left(\rho c\right)\_f}, \\ N\_\xi &= -\frac{T\_\xi \lg\left(\rho\_p - \rho\right) R\_0^2 \left(T\_0 - T\_1\right)}{\mu U\_a}, \end{split}$$

$$\begin{split} P\_l &= \frac{bW\_c}{D\_{\text{mo}}}, \Theta = \frac{n\_1}{n\_0 - n\_1}, \\ T\_\xi &= \frac{T\_\xi \varrho \rho\_f R\_0^2 \left(1 - C\_1\right) \left(T\_0 - T\_1\right)}{\mu U\_a}, \\ R\_b &= \frac{\left(n - n\_1\right) \Theta \left(\rho\_m - \rho\_f\right) g R\_0^2}{\mu U\_a}. \end{split} \tag{19}$$

In the above equation, T<sup>g</sup> is the local temperature Grashof number, N<sup>g</sup> is the local particle Grashof number, R<sup>b</sup> is the bioconvection Rayleigh number, T<sup>b</sup> is the Brownian motion parameter, T<sup>t</sup> is the thermophoresis parameter, P<sup>l</sup> is the Peclet number, 2¯ is a constant, and β is the fluid parameter.

The boundary conditions read.

$$\begin{aligned} \mu' &= 0, \; \theta' = 0, \phi' = 0, \chi' = 0, \; \text{at} \; r = 0, \\\mu &= 0, \; \theta = 0, \phi = 0, \; \chi = 0, \; \text{at} \; r = R. \end{aligned} \tag{20}$$

#### SERIES SOLUTIONS

Ps

The solutions of Equations (13) to (17) can be obtained using a Homotopy perturbation method. And thus, the Homotopy P<sup>s</sup> for Equations (13) to (17) are defined as

$$\begin{split} P\_s(\bar{\boldsymbol{u}}, \boldsymbol{\zeta}) &= (1 - \boldsymbol{\zeta}) \left[ \ell \left( \bar{\boldsymbol{u}} \right) - \ell \left( \bar{\boldsymbol{u}}\_0 \right) \right] \\ &+ \boldsymbol{\zeta} \left[ \ell \left( \bar{\boldsymbol{u}} \right) - 3 \beta \frac{\partial^2 \bar{\boldsymbol{u}}}{\partial r^2} \left( \frac{\partial \bar{\boldsymbol{u}}}{\partial r} \right)^2 - \frac{\beta}{r} \left( \frac{\partial \bar{\boldsymbol{u}}}{\partial r} \right)^3 \right. \\ &+ \left. T\_{\mathcal{S}} \bar{\boldsymbol{\theta}} + N\_{\mathcal{S}} \bar{\boldsymbol{\phi}} - R\_b \bar{\boldsymbol{\chi}} - \frac{\partial p}{\partial z} \right], \end{split} \tag{21}$$

$$\begin{aligned} \left(\bar{\theta}, \xi\right) &= (1 - \xi) \left[\ell\left(\bar{\theta}\right) - \ell\left(\bar{\theta}\_0\right)\right] \\ &+ \xi \left[\ell\left(\bar{\theta}\right) + T\_b \frac{\partial \bar{\phi}}{\partial r} \frac{\partial \bar{\theta}}{\partial r} + T\_l \left(\frac{\partial \bar{\theta}}{\partial r}\right)^2\right], \end{aligned} \tag{22}$$

$$\begin{split} P\_s\left(\bar{\phi},\xi\right) &= \left(1-\xi\right)\left[\ell\left(\bar{\phi}\right)-\ell\left(\bar{\phi}\_0\right)\right] \\ &+ \xi \left[\ell\left(\bar{\phi}\right)+\frac{T\_t}{T\_b}\frac{1}{r}\frac{\partial}{\partial r}\left(\frac{\partial \bar{\theta}}{\partial r}r\right)\right], \\ P\_{\bar{\phi},\bar{\phi}}(\bar{\tau},\eta) &= \left(\bar{\kappa}-1\right)L^{\bar{\phi},\bar{\zeta}}(\bar{\tau})-\ell\left(\bar{\kappa}\right). \end{split} \tag{23}$$

$$\begin{split} P\_s\left(\bar{\boldsymbol{\chi}},\boldsymbol{\xi}\right) &= \left(\boldsymbol{\zeta}-1\right) \left[\ell\left(\bar{\boldsymbol{\chi}}\_0\right) - \ell\left(\bar{\boldsymbol{\chi}}\right)\right] \\ &+ \left.\boldsymbol{\zeta}\right[\ell\left(\bar{\boldsymbol{\chi}}\right) - P\_l\frac{\partial}{\partial r}\left(\left(\bar{\boldsymbol{\chi}}+\bar{\boldsymbol{\Theta}}\right)\frac{\partial\bar{\boldsymbol{\phi}}}{\partial r}\right)],\end{split} \tag{24}$$

where ζ ∈ [0, 1]the embedding parameter.

The linear operator reads as

$$\ell = \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r},\tag{25}$$

and the initial guesses read as

$$
\bar{\psi}\_0 = \bar{\theta}\_0 = \bar{\phi}\_0 = \bar{\chi}\_0 = \frac{r^2 - R^2}{c^2},\tag{26}
$$

where c(6= 0) is a constant.

The above initial guess is chosen in such a way that the following initial guess satisfied the linear operator as given in Equation (25) as well as satisfy all the governing boundary conditions as given in Equation (20).

Defining the following expansions

$$
\bar{u} = \bar{u}\_0 + \zeta \,\bar{u}\_1 + \zeta^2 \bar{u}\_2 + \dots,\tag{27}
$$


Using the series expansions in Equations (27–30) in the Homotopy equations [see Equations (21) to (24)], we get the set of linear differential equations, after comparing the powers of ζ . By applying the property of Homotopy perturbation method, i.e., ζ → 1, we get

$$
\mu = \bar{u} = \bar{u}\_0 + \bar{u}\_1 + \bar{u}\_2 + \dots,\tag{31}
$$

$$
\theta = \bar{\theta} = \bar{\theta}\_0 + \bar{\theta}\_1 + \bar{\theta}\_2 + \dots , \tag{32}
$$

$$
\phi = \bar{\phi} = \bar{\phi}\_0 + \bar{\phi}\_1 + \bar{\phi}\_2 + \dots , \tag{33}
$$

$$\chi = \bar{\chi} = \bar{\chi}^0 + \bar{\chi}^1 + \bar{\chi}^2 + \dots,\tag{34}$$

The final results for all the governing equations are obtained as

$$u(r) = u\_0 + r^2 u\_1 + r^4 u\_2 + r^5 u\_3 + r^6 u\_4 + \cdots,\qquad \text{(35)}$$

$$\theta \; (r) = \theta\_0 + r^3 \theta\_1 + r^4 \theta\_2 + r^5 \theta\_3 + r^6 \theta\_4 + \dotsb,\tag{36}$$

$$
\phi\left(r\right) = \phi\_0 + r^3 \phi\_1 + r^4 \phi\_2 + \cdots \ , \tag{37}
$$

$$\chi\_{\;}(r) = \chi\_0 + r^2 \chi\_1 + r^4 \chi\_2 + r^6 \chi\_4 + \cdots \, . \tag{38}$$

where un, θn, φn, χn, with n = 1, 2, 3 . . . are the constants which can be found using the calculations through a computational software Mathematica 10.3v.

The flux Q can be determined as

$$Q = \int\_0^R 2ru \, (r, z) \, \text{d}r.\tag{39}$$

$$Q = \frac{\wp}{f\left(z\right)},\tag{40}$$

where ℘ = − dp dz .

The impedance can be determined as

$$
\lambda = \frac{1}{Q} \int\_0 \wp \, dz,\tag{41}
$$

The wall shear stress is calculated as.

$$S\_{rz} = \left. \frac{1}{2} \wp R \right|\_{r=R}. \tag{42}$$

#### DISCUSSION

We have discussed the graphical behavior of all the leading parameters for the temperature, velocity, motile microorganism and concentration profiles. The effects of wall shear stress and the variation of impedance are also investigated to see the behavior of blood during the swimming of microorganisms and the movement of nanoparticles. With the aid of said perturbation scheme, we obtained the third order approximation against each profile. All the numerical computations have been performed using computational software Mathematica. **Figures 2**–**7** are plotted for different profiles with all the emerging parameters i.e., Peclet number P<sup>l</sup> , height of stenosis δ, angular frequency ω, fluid parameter β, local temperature Grashof number T<sup>g</sup> , local particle Grashof number N<sup>g</sup> , bioconvection Rayleigh number R<sup>b</sup> , thermophoresis parameter T<sup>t</sup> , and Brownian motion T<sup>b</sup> . All three cases i.e., diverging, converging, and non-tapered artery, have been plotted with the help of Equation (2).

**Figure 2** presents the behavior of the velocity profile against the fluid parameter β, thermophoresis parameter T<sup>t</sup> , and

FIGURE 7 | Impedance profile for various values of (A) β; Black line: β = 0, Red line: β = 1. (B) *R<sup>y</sup>* ; Black line: *R<sup>y</sup>* = 0, Red line: *R<sup>y</sup>* = 12. (C) *T<sup>t</sup>* ; Black line: *T<sup>t</sup>* = 0.2, Red line: *T<sup>t</sup>* = 2. (D) *Tb*; Black line: *T<sup>b</sup>* = 0.1, Red line: *T<sup>b</sup>* = 1.

Brownian motion parameter T<sup>b</sup> . In order to understand the behavior of hemodynamics in a specific artery or lesion, it is necessary to have a knowledge of blood velocity within the flow pattern. The hemodynamic velocity in the artery is not the same at all the points [33]. We can see from **Figure 2A** that the distribution of velocity at the center of the channel is maximum while it attains a minimum value when it gets close to the wall. Further, we can notice that in the case of non-Newtonian fluid, β = 4, the velocity of the blood diminishes. However, we can see a turning point between r ∈ (0.6, 0.8) the artery where the velocity turns opposite as compared with the core of the channel and decreases as it gets closer to the wall of the artery. The significant change in the velocity gradient among different points in the artery exists because of the friction forces that play an essential role among the fluid at the walls and the flowing fluid. The friction forces occur because of the viscosity features. The viscosity represents the resistance to the flow, and it attains a minimum value if the trivial force on the fluid layer generates a velocity higher than that layer associated with the adjoining layer, and the converse is true [34, 35]. **Figure 2B** shows the behavior of the thermophoresis parameter T<sup>t</sup> on the velocity profile. It is noticed from this figure that by enhancing the thermophoresis parameter, the nanoparticles start moving quickly and tends to repel from the hotter to a colder area. But it doesn't affect the velocity of the fluid. However, it causes resistance in the velocity of the fluid. Brownian motion plays a simultaneous role with thermophoresis. However, both parameters similarly affect the velocity profile (see **Figure 2C**). Brownian motion occurs due to the collision of suspended particles in random direction in the working fluid. Higher values of Brownian motion reveals that the particles collide very quickly which causes the resistance in the motion of the base fluid.

**Figures 3**, **4** are plotted for temperature and concentration distributions for multiple values of T<sup>t</sup> and T<sup>b</sup> . In **Figure 3**, we can see that the temperature profile rises with the increment in T<sup>t</sup> and T<sup>b</sup> . The enhancement of both parameters tends to repel the particles quickly. Therefore, the particles start moving from one region to the other area (i.e., hotter to colder part). Both parameters produce a force i.e., thermophoretic force and random movement of suspended particles which resist the fluid motion and as result the temperature profile increases. **Figure 4** shows that the concentration profile that is shown to be inversely proportional to the temperature profile. By increasing both parameters, the concentration profile tends to diminish remarkably. **Figure 5** is plotted to judge the variation of motile microorganisms with P<sup>l</sup> , Tt , and T<sup>b</sup> . It can be noticed from **Figure 5A** that the motile microorganisms 'distribution rises due to the strong influence of the Brownian motion parameter. However, a converse behavior has been observed for the thermophoresis parameter. In **Figure 5B**, we can see that the Peclet number produces resistance in the motile microorganism profile. By increasing Peclet number, it is noticed that advection propagation transport in more dominant as compared with diffusion propagation rate, which suppress the motile microorganism profile.

**Figure 6** shows that the behavior of wall shear stress, that has been plotted using Equation (42), with δ, ω, T<sup>g</sup> , N<sup>g</sup> , R<sup>b</sup> , T<sup>t</sup> , and Tb . The wall shear stress is an essential part of the blood flow, and it can be described as the fluid flowing over the surface of the conduit artery. From Equation (42) we can see that wall shear stress is directly proportional to the velocity gradient close to the wall of the artery. That shows how quickly the velocity of the fluid is when propagating from one point on the artery wall to another point adjacent to the point in the perpendicular direction of the wall. However, low wall shear stress belongs to low velocities, accordingly, the higher residence time of the fluid closer to the wall. And as a result, this velocity gradient close to the wall is known as the wall shear rate. We can see from **Figure 6A** that wall shear stress is reduced due to the strong influence of the thermophoresis parameter, however, an inverse behavior has been noticed with a variation of T<sup>b</sup> . In **Figure 6B**, we can see that local temperature Grashof number and local particle Grashof number suppress the wall shear stress remarkably. However, we noticed that the height of the clot enhances the wall shear stress, whereas the angular frequency tends to diminish the wall shear stress as shown in **Figure 6C**. In **Figure 6D**, we found that bioconvection Rayleigh number doesn't affect the wall shear stress significantly and the effect is minimal.

**Figure 7** is schemed to judge the variation of impedance distribution for multiple values of β, R<sup>b</sup> , **Tt**, and T<sup>b</sup> . **Figure 7A** is sketched for impedance vs. height of the clot for various values of the fluid parameter. We can see from this figure that impedance profile rises with an increase in the height of the clot, whereas it decreases simultaneously due to an enhancement in the fluid parameter. In **Figure 7B**, the effect of bioconvection Rayleigh number is shown incrementally decreasing. Further, it is noticed that an increase in the angular frequency ω implies to a decrease in the impedance profile. Also, it is seen in the whole domain that the thermophoresis parameter tends to suppress the impedance profile, as shown in **Figure 7C**. In **Figure 7D**, the impedance profile rises due to the strong influence of the Brownian motion parameter. However, it is seen that the impedance profile tends to reduce with an increment in time.

### CONCLUDING REMARKS

A theoretical study on the swimming of nanoparticles with motile gyrotactic microorganisms in non-Newtonian

### REFERENCES


blood flow propagating in an anisotropically tapered artery has been presented. Sutterby fluid model is presented to understand the rheology of the blood. The mathematical modeling is formulated using continuity, temperature, motile microorganism, momentum and concentration equation. The Homotopy perturbation method is applied to obtain the series solutions. All the graphical results are presented for diverging, converging, and non-tapered artery. The main results from the present study has been summarized below:


### AUTHOR CONTRIBUTIONS

MB and MM performed mathematical formulation, AZ and RE made the analysis and wrote the paper. SA made the geometry of problem and arranged the setting of the paper.

non-uniform channel. RSC Adv. (2018) **8**:7904–15. doi: 10.1039/C7RA 13188G


containing gyrotactic microorganism. Microvasc Res. (2017) **110**:32–42. doi: 10.1016/j.mvr.2016.11.007


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling editor declared a past co-authorship with the author MM.

Copyright © 2020 Bhatti, Marin, Zeeshan, Ellahi and Abdelsalam. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# A New Computational Technique Design for EMHD Nanofluid Flow Over a Variable Thickness Surface With Variable Liquid Characteristics

#### Muhammad Irfan, Muhammad Asif Farooq\* and Tousif Iqra

*Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad, Pakistan*

The objective of this paper comprises two key aspects: to establish descriptive mathematical models for constant and variable fluid flows over a variable thickness sheet by inducting applied electric and magnetic fields, porosity, radiative heat transfer, and heat generation/absorption, and to seek their solution by constructing a novel numerical method, the Simplified Finite Difference Method (SFDM). We resort to similarity transformations to implicate partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). Optimal results for a pair of ODEs obtained from SFDM are assessed by drawing a comparison with *bvp4c* and existing literature values. SFDM has been implemented in MATLAB for both constant and variable fluid properties. Tabulated numerical values of the skin friction coefficient and local Nusselt and Sherwood numbers are measured and analyzed against different parameters. The influence of distinct parameters on velocity, temperature, and nanoparticle volume fraction are explained in great detail via diagrams. The skin friction coefficient for variable fluid properties is greater than for constant fluid properties. However, the local Nusselt number is lower for variable fluid properties than with constant fluid properties. Surprisingly, high-precision computational results are achieved from the SFDM.

Keywords: electrical magnetohydrodynamics (EMHD), variable thicked surface, nanofluid, simplified finite difference method (SFDM), variable fluid properties

### 1. INTRODUCTION

Fluid mechanics has many applications in contexts from the human biological system to the manufacturing industry. For example, the study of breathing in biological systems uses biofluid dynamics. Cooling is another such phenomenon, which is important in electronics and the automobile industry. Investigating stretching sheet flows is relevant to many significant applications. All of this plays a vital role in technological advances such as those of polymer manufacturing and cooling processes in glass and paper production Hayat et al. [1]. Having variable thickness becomes useful in minimizing the weight of architectural elements Hayat et al. [1].

Hayat et al. [1, 2] analyzed the consequences of Cattaneo-Christov heat flux and a temperaturedependent fluid thermal conductivity on fluid flow over a variable thickness sheet and showed that variable conductivity enhances the temperature distribution. They also maintained that the temperature profile decreases with the thermal relaxation parameter. Mabood et al. [3] discussed the non-Darcian MHD convective flow and claimed that temperature rise depends on the Eckert

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Rahmat Ellahi, University of California, Riverside, United States Muhammad Ibrahim, University of Science and Technology Beijing, China*

#### \*Correspondence:

*Muhammad Asif Farooq asiffarooq.2007@gmail.com*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *02 December 2019* Accepted: *28 February 2020* Published: *08 April 2020*

#### Citation:

*Irfan M, Farooq MA and Iqra T (2020) A New Computational Technique Design for EMHD Nanofluid Flow Over a Variable Thickness Surface With Variable Liquid Characteristics. Front. Phys. 8:66. doi: 10.3389/fphy.2020.00066*

**118**

number. In the context of a stretching sheet for variable thickness, Fang et al. [4] has tackled the boundary layer flow and analyzed multiple solutions. Khader and Ahmed [5] computed a numerical solution for variable sheet thickness with slip velocity and pointed out that the skin friction coefficient increases with the wall thickness parameter. Daniel et al. [6] discussed the thermal stratification effects on MHD radiative flow of nanofluid for a variable thickness sheet. They submitted that the thermal stratification effect reduces temperature. Reddy et al. [7] investigated the MHD flow and heat transfer of Williamson nanofluid over a variable thickness sheet with variable thermal conductivity and identified that the velocity profile decreases with the wall thickness parameter when m < 1. Daniel et al. [8] examined the effect of thermal radiation on electrical MHD flow of nanofluid over a stretching sheet with variable thickness and concluded that the thermal radiation did impact the nanofluid temperature.

Magnetohydrodynamics is the study of the flow of electrically conducting fluids in an electro-magnetic-field. The study of MHD flow is of considerable interest in modern metallurgical and metal-working processes. Noreen et al. [9] examined the numerical solutions of magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid toward a stretching sheet. They showed that the skin friction coefficient increases with an increase in M. Mukhopadhyay et al. [10] conducted a study to assess the effects of fluid flow with constant and changeable viscosity on a heated surface. They noticed that a decrease in viscosity causes the velocity to decrease with increasing distance along the stretching sheet. Nadeem et al. [11] examined MHD three-dimensional Casson fluid flow through a porous linear stretching plate and concluded that the stretching parameter resulted in decreasing behavior of the velocity profile. Mabood et al. [12] investigated MHD boundary layer flow and heat transfer of nanofluid over a non-linear stretching sheet. They note that the boundary layer thickness grows with Brownian motion. Zhang et al. [13] discussed the MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. They examined three types of nanoparticles. Popley et al. [14] addressed the overall impact of varying liquid characteristics upon hydro-magnetic motion and heat transfer across a non-linear stretching surface. They demonstrated that the free stream velocity induces a reduction in the boundary layer thickness. Mohsen et al. [15] discussed nanofluid flow with convective heat transfer considering Lorentz forces and showed that heat transfer rises with the Hartmann number. Patel [16] studied the effects of heat generation, thermal radiation, and Hall current on MHD Casson fluid flow past an osculating plate in a porous medium. They stated that the Hall current boosts mobility in both directions. Farooq et al. [17] discussed the MHD flow of Maxwell fluid with nanomaterials due to an exponentially stretching surface. The influence of the thermophoresis parameter on the temperature distribution is negligible. Magnetohydrodynamic (MHD) boundary layer flow past a wedge with heat transfer and viscous effects of nanofluid embedded in porous media was investigated by Ibrahim and Tulu [18]. They discover that the pressure gradient influences the boundary layer thickness. The impact of 3D Maxwell nanofluid flow over an exponentially stretching surface in terms of heat and mass transfer was explored by Ali et al. [19]. They showed that the skin friction coefficient decreases with the Deborah number.

Nanofluids solid-liquid suspensions consisting of solid nanoparticles of size 1-100 nm and liquid Mabood et al. [3]. Due to reports of their having significantly enhanced thermal properties, nanofluids have drawn great interest recently Mabood et al. [3]. The term nanofluid was proposed by Choi and Eastman [20], who demonstrated that the introduction of a small number of nanoparticles (< 1 percent by volume fraction) to traditional liquids increased the fluid thermal conductivity by approximately two times Nadeem et al. [11]. The numerical simulation of nanofluid flow with convective boundary conditions was studied by Das et al. [21], who demonstrated that the surface convection parameter enhances the heat transfer rate. Mabood and Das [22], in their analysis, communicated melting heat transfer of hydromagnetic nanofluid flow with a second-order slip condition. Cao et al. [23] discussed the MHD flow and heat transfer of fractional Maxwell viscoelastic nanofluid over a moving plate by using a finite difference method and found that the average Nusselt number is higher with a rise in the fractional derivative parameter. Das et al. [24] studied the effects of a magnetic field on an unsteady mixed convection flow of nanofluids containing spherical and cylindrical nanoparticles. Narayana et al. [25] discussed the effects of thermal radiation and a heat source on an MHD nanofluid past a vertical plate in a rotating system with a porous medium. They used three different nanoparticles and showed that they enhance the heat transfer rate, a result that can be used in heat exchanger technology. The influences on stagnation-point flow toward a stretching/shrinking sheet in a nanofluid were discussed by Mansur et al. [26] using the Buongiorno model. They proved that the thermophoresis parameter reduces the heat transfer rate. Makinde [27] studied viscous dissipation and Newtonian heating over a flat plate in a nanofluid. The heat transfer rate rises with the nanoparticle volume fraction and the Biot number. Ali et al. [28] discussed a numerical study of unsteady MHD Couette flow and heat transfer of nanofluids in a rotating system with convective cooling and indicated that the rotation has a significant effect on velocity and heat transfer. Ashwinkumar and Sulochana [29] investigated the effect of radiation absorption and buoyancy force on the MHD mixed convection flow of Casson nanofluid. They noticed that the volume fraction of nanoparticles governs the temperature distribution. Under temperature control, Andersson and Aarseth [30] revisited the fluid properties of a liquid. The effect of variable fluid properties on the hydromagnetic flow and heat transfer over a non-linearly stretching sheet was discussed by Prasad et al. [31]. Hayat et al. [32] discussed mixed convection flow across a porous sheet and reported that the thermal boundary layer thickness is lowered with Pr. Reddy et al. [33] probed the effect of variable thermal conductivity on MHD flow of nanofluid over a stretching sheet. They considered convective boundary conditions. Zaka et al. [34] applied numerical simulation for Darcy-Forchheimer flow of nanofluid by considering a rotating disk. They reported that the temperature distribution is enhanced with the thermophoresis parameter. Shah et al. [35] discussed the nanofluid flow for

different shape factors. They managed to show that the shape factor causes stronger convection. Zeeshan et al. [36] reported the effect of radiative nanofluid flow under a pressure gradient due to entropy generation and observed an increase in entropy with an increase in the pressure gradient. Ellahi et al. [37] investigated flow of a power-law nanofluid with entropy generation and noted that the skin friction coefficient increases at the heated wall. Yousif et al. [38] analyzed the momentum and heat transfer of MHD Carreau nanofluid over an exponentially stretching surface and used the shooting method to compute the solution. Sarafraz et al. [39] discussed the pool boiling heat transfer characteristics of an iron oxide nano-suspension considering a constant magnetic field and found that bubble formation is intensified due to the magnetic field. Fujimoto [40] described multi-scale simulation on adaptive meshes.

This paper is arranged in the following way. A mathematical formalism of the physical model is explained in section Problem Formulation. Section Fluid Properties Analysis addresses constant as well as varying liquid characteristics. Section Physical Quantities provides physical quantities, and an overview of the numerical process has been given in section Numerical Procedure. Results and discussion are presented in section Result and Discussion. In section Conclusions, the conclusion is drawn.

#### 2. PROBLEM FORMULATION

We assume an electrical magnetohydrodynamic (EMHD), twodimensional, steady, laminar flow of nanofluid over a non-linear stretching sheet with variable thickness. A variable magnetic field B(x) = Bo(x + b) n−1 <sup>2</sup> (n 6= 1) and variable electrical field E(x) = Eo(x + b) n−1 <sup>2</sup> (n 6= 1) are applied normal to the direction of flow. The sheet is stretching with non-linear velocity U<sup>w</sup> = Uo(x + b) n (n 6= 1), where b is the dimensional constant and U<sup>o</sup> is the reference velocity. Therefore, the surface is considered not to be flat, and its thickness varies as y = A(x + b) 1−n <sup>2</sup> (n 6= 1), where A is a very small constant to hold the sheet thin enough. We also observe that for n = 1, the current problem reduces to a flat sheet. The geometry of the problem is shown in **Figure 1**, where the x-axis has been taken along the sheet and y-axis is normal to it.

The induced magnetic field has been neglected under the assumption of a small magnetic Reynolds number. The boundary layer equations governing this flow are Daniel et al. [6, 8] and Irfan et al. [41]

$$
\frac{\partial \mu}{\partial x} + \frac{\partial \nu}{\partial y} = 0,
$$

$$\begin{split} \mu \frac{\partial u}{\partial \mathbf{x}} + \nu \frac{\partial u}{\partial \mathbf{y}} &= \frac{1}{\rho} \frac{\partial}{\partial \mathbf{y}} (\frac{\mu \partial u}{\partial \mathbf{y}}) + \frac{\sigma}{\rho} (E(\mathbf{x})B(\mathbf{x}) - B^2(\mathbf{x})u) - \frac{\mu}{\rho K(\mathbf{x})} u(2)) \\ \mu \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} &= \frac{1}{\rho C\_p} \frac{\partial}{\partial \mathbf{y}} (\frac{k \partial T}{\partial \mathbf{y}}) + \tau (D\_B \frac{\partial T}{\partial \mathbf{y}} \frac{\partial C}{\partial \mathbf{y}} + \frac{D\_T}{T\_{\infty}} (\frac{\partial T}{\partial \mathbf{y}})^2) \\ \mu \frac{\partial T}{\partial \mathbf{x}} &= 0 \end{split}$$

$$-\frac{1}{\rho C\_{\mathcal{P}}} \frac{\partial q\_{\mathcal{P}}}{\partial \boldsymbol{y}} + \frac{\sigma}{\rho C\_{\mathcal{P}}} (\mu B(\mathbf{x}) - E(\mathbf{x}))^2 + \frac{Q(\mathbf{x})}{\rho C\_{\mathcal{P}}} (T - T\_{\infty}),\tag{3}$$

$$\rho C\_{\mathcal{P}} \quad \text{e} \quad \text{e} \quad \text{e} \quad \text{e} \quad \text{e} \quad \text{e} \quad \text{e} \quad \text{e}$$

$$
\mu \frac{\partial C}{\partial x} + \nu \frac{\partial C}{\partial y} = D\_B \frac{\partial^2 C}{\partial y^2} + \frac{D\_T}{T\_\infty} \frac{\partial^2 T}{\partial y^2},\tag{4}
$$

Here, u and v are the velocity components parallel to the x− and y− axis, respectively. Further, µ is the viscosity, ρ is the density, ν is the kinematic viscosity, C<sup>p</sup> is the specific heat capacity, B is the magnetic field. T and C are the fluid temperature and nanoparticle fraction, respectively. The temperature of the fluid at the wall and ambient temperature are denoted by T<sup>w</sup> and T∞, respectively. D<sup>B</sup> and D<sup>T</sup> are the Brownian diffusion coefficient and thermophoretic diffusion coefficient, respectively. τ = (ρc)p (ρc)<sup>f</sup> is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the fluid, q<sup>r</sup> is the radiative heat flux, Q(x) = Q0(x + b) n−1 <sup>2</sup> is the volumetric rate of heat generation, and K(x) = K0(x + b) n−1 is a variable permeability.

The above system is completed with the following appropriate boundary conditions, taking to view of [32] and [33]:

$$\begin{aligned} \mu &= U\_{\text{w}}(\mathbf{x}) = U\_{o}(\mathbf{x} + b)^{n}, \; \nu = 0, \; -k \frac{\partial T}{\partial \mathbf{y}} = h\_{i}(T\_{f} - T), \\ D\_{B} \frac{\partial C}{\partial \mathbf{y}} + D\_{T} \frac{\partial T}{\partial \mathbf{y}} &= 0 \; \text{at } \; \mathbf{y} = A(\mathbf{x} + b)^{\frac{1 - n}{2}} \\ \mu &\longrightarrow 0, \; \quad T \longrightarrow T\_{\infty}, \; \quad C \longrightarrow C\_{\infty} \qquad \text{as} \qquad \mathbf{y} \longrightarrow \infty \ (5) \end{aligned}$$

To the above equations, (1)-(4), the following relevant transformations will be utilized:

$$\begin{aligned} \psi &= \sqrt{\frac{2}{n+1}} v \, U\_o(\mathbf{x} + b)^{n+1} f(\eta), \\ \xi &= \sqrt{(\frac{n+1}{2}) \frac{U\_o(\mathbf{x} + b)^{n-1}}{\upsilon}}, \alpha = A \left( \frac{(n+1)U\_0}{2\upsilon} \right)^{\frac{1}{2}} \\ \eta &= \xi - \alpha = \wp \sqrt{(\frac{n+1}{2}) \frac{U\_o(\mathbf{x} + b)^{n-1}}{\upsilon}} - \alpha \\ \theta &= \frac{T - T\_{\infty}}{T\_w - T\_{\infty}}, \quad \phi = \frac{C - C\_{\infty}}{C\_w - C\_{\infty}}, \qquad u = U\_o(\mathbf{x} + b)^n f'(\eta), \\ \nu &= -\sqrt{\frac{2}{n+1}} v \, U\_o(\mathbf{x} + b)^{n-1} \langle f(\eta) + \eta \frac{n-1}{n+1} f'(\eta) \rangle. \end{aligned}$$

Equation (1) is identically satisfied. In addition, when the above similarity variables are applied to Eqations (2), (3), and (4), it yields:

$$\left(\frac{\mu}{\mu\_o}f''\right)' - \frac{2n}{n+1}f'^2 + f'' + M(E\_1 - f') - Kp\frac{\mu}{\mu\_o}f' = 0,\,\text{(7)}$$

$$\left(1 + \frac{4}{3}Rd\big(\frac{k}{k\_o}\boldsymbol{\theta}'\right)' + Pr\_o(f\boldsymbol{\theta}' + Nb\boldsymbol{\theta}'\boldsymbol{\phi}' + Nt(\boldsymbol{\theta}')^2$$

$$+ MEc(\boldsymbol{\xi}' - E\_1)^2 + \frac{2}{n+1}s\boldsymbol{\theta}\boldsymbol{\theta} = 0,\tag{8}$$

$$
\boldsymbol{\phi}'' + \frac{\text{Nt}}{\text{N}b} \boldsymbol{\theta}'' + \text{LePr}\_0 \boldsymbol{f} \boldsymbol{\phi}' = \boldsymbol{0}.\tag{9}
$$

The equivalent boundary conditions in terms of similarity variables are specified as:

$$f(0) = \alpha(\frac{1-n}{1+n}), f'(0) = 1, \; f'(\infty) = 0, \; \boldsymbol{\theta}'(0) = -B\_i(1-\boldsymbol{\theta}(0))$$

$$\boldsymbol{\theta}(\infty) = 0, \; \text{Nb}\boldsymbol{\phi}'(0) + \text{N}\boldsymbol{\theta}'(0) = 0, \quad \boldsymbol{\phi}(\infty) = 0,\tag{10}$$

where M = 2σB 2 o <sup>ρ</sup>Uo(n+1) is a magnetic parameter, α is the wall thickness parameter, E<sup>1</sup> = Eo BoUo(x+b) <sup>n</sup> is the electric field, and

Kp = 2ν <sup>K</sup>oUo(n+1) is the permeability constant. Pr<sup>o</sup> <sup>=</sup> µoCp ko is the Prandtl number, Nb = τDB(Cw−C∞) ν is the Brownian motion parameter, Nt = τDT(Tw−T∞) T∞ν is the thermophoresis parameter, Ec = U 2 w Cp(Tw−T∞) is the local Eckert number, Rd = 4σ ∗T 3 ∞ kok ∗ denotes the radiation parameter, s = Qo(x+b) ρuwCp is the heat source parameter, B<sup>i</sup> is the Biot number, and Le = ν DB is Lewis number Irfan et al. [41].

#### 3. FLUID PROPERTIES ANALYSIS

We illustrate the main theme of this work through the following two subsections.

#### 3.1. Case A: Constant Fluid Characteristics

For this case, we rewrite Equations (7), (8), and (9) into the following set of equations Irfan et al. [41]:

$$\left(f'' - \frac{2n}{n+1}f'^2 + f'' + M(E\_1 - f') - Kpf' = 0\right) \tag{11}$$

$$(1 + \frac{4}{3}Rd)\boldsymbol{\theta}'' + Pr\_o(\boldsymbol{f}\boldsymbol{\theta}' + \text{Nb}\boldsymbol{\theta}'\boldsymbol{\phi}' + \text{Nt}(\boldsymbol{\theta}')^2 + ME\boldsymbol{c}(\boldsymbol{f}' - \boldsymbol{E}\_1)^2)$$

$$(1 + \frac{2}{n+1}s\theta) = 0\tag{12}$$

$$\left(\boldsymbol{\phi}\right)' + \frac{\text{Nt}}{\text{Nb}}\boldsymbol{\theta}\,\,^{\prime} + \text{Pr}\_{o}\text{L}\boldsymbol{f}\boldsymbol{\phi}\,\,^{\prime} = \text{0} \tag{13}$$

#### 3.2. Case B: Variable Fluid Properties

In this case, we express viscosity and thermal conductivity as a function of temperature Andersson and Aarseth [30]

$$
\mu(T) = \frac{\mu\_{\rm ref}}{1 + \chi(T - T\_{\rm ref})} \tag{14}
$$

In (14), above, γ is a property of a fluid. Assuming T<sup>o</sup> ≈ Tref , we get

$$\mu = \frac{\mu\_o}{1 - \frac{T - T\_o}{\theta\_r (T\_w - T\_o)}} = \frac{\mu\_o}{1 - \frac{\theta \{ \eta \}}{\theta\_r}} \tag{15}$$

Here, θ<sup>r</sup> = −1 γ (Tw−To) . Inserting Equation (15) into Equation (7), we get

$$\frac{\theta\_r}{\{\theta r - \theta\}}{\{\theta r - \theta\}}{\{\theta r - \theta\}}^{\prime\prime} + \frac{f^{\prime\prime}\theta^{\prime}\theta\_r}{\{\theta\_r - \theta\}^2} - \frac{2n}{n+1}f^{\prime\prime} + f^{\prime\prime} + M(E\_1 - f^{\prime}) - \text{Kp}\frac{\theta\_r}{\theta\_r - \theta}f^{\prime} = 0 \tag{16}$$

Following Prasad et al. [31], the changeable thermal conductivity is expressed as

$$k(T) = k\_o(1 + \epsilon \theta) \tag{17}$$

Using Equation (17) in Equation (8), we get.

$$(1 + \frac{4}{3}Rd)((1 + \epsilon\theta)\boldsymbol{\theta}^{\prime} + \epsilon(\boldsymbol{\theta}^{\prime})^2) + Pr\_o(\boldsymbol{f}\boldsymbol{\theta}^{\prime} + Nb\boldsymbol{\theta}^{\prime}\boldsymbol{\phi}^{\prime}$$

$$+ Nt(\boldsymbol{\theta}^{\prime})^2 + MEc(\boldsymbol{\theta}^{\prime} - E\_1)^2 + \frac{2}{n+1}s\boldsymbol{\theta}) = 0\tag{18}$$

#### 4. PHYSICAL QUANTITIES

The important physical parameters are defined as follows.

#### 4.1. Skin Friction Coefficient

The wall friction coefficients for case A and case B are defined as

$$\mathbf{C}\_{\circ} = \frac{\mathbf{r}\_{\mathrm{w}}}{\rho \,\mu\_{\mathrm{w}}^{2}} = \sqrt{\frac{1+n}{2\mathrm{Re}\_{\mathrm{x}}}} f'(0) \text{ (CASE A)}$$

$$C\_f = \frac{\mathfrak{r}\_w}{\rho u\_w^2} = \frac{\theta\_r}{\theta\_r - \theta(0)} \sqrt{\frac{1+n}{2Re\_\chi}} f'(0) \text{ (CASE B)}$$

#### 4.2. Local Nusselt Number

The local Nusselt numbers for Cases A and B are the same and can be written as

$$Nu\_{\mathfrak{x}} = -\frac{(\mathfrak{x} + b)q\_{\mathfrak{w}}}{k\_{\mathfrak{o}}(T\_{\mathfrak{w}} - T\_{\infty})} = -(1 + \frac{4}{3}Rd)\sqrt{\frac{(1+n)Re\_{\mathfrak{x}}}{2}}\theta'(0)$$

#### 4.3. Local Sherwood Number

The local Sherwood number for both Case A and Case B is

$$Sh\_{\chi} = -\frac{(\varkappa + b)j\_{\w}}{C\_{\w} - C\_{\infty}} = -\sqrt{\frac{(1+n)Re\_{\chi}}{2}}\phi'(0) \tag{19}$$

#### 5. NUMERICAL PROCEDURE

The system of ODEs for Case A and Case B, along with the boundary conditions, are first transformed into a system of first-order ODEs. We use two numerical methods to find the solution of these ODEs. The first method is the SFDM [42], and the second is implemented through MATLAB's built-in solver bvp4c. The details of the methods and the implications are described below.

#### 5.1. Simplified Finite Difference Method (SFDM)

The algorithm and necessary details for the simplified FDM are as follows:


An explanation of SFDM has been illustrated in the flowchart. Generally, we find the results when N = 1, 000 grid points in the η direction. The domain to achieve steady state varies due to the effects of different parameters, but the domain η = 7 seems sufficient for our results. To initiate, we assume f ′ = F in (11), and then we get

$$\frac{d^2F}{d\eta^2} = \frac{2n}{n+1}F^2 - f\frac{dF}{d\eta} - M(E\_1 - F) + KpF \tag{20}$$

We can write this expression for the function f as

$$\chi\_1(\eta, F, F') = \frac{2n}{n+1} F^2 - f \frac{dF}{d\eta} - M(E\_1 - F) + K\rho F \tag{21}$$

Let us approximate dF dη in the above equation by forward difference approximation

$$\chi\_1(\eta, F, F') = \frac{2n}{n+1} F\_i^2 - f\_i(\frac{F\_{i+1} - F\_i}{h}) - M(E\_1 - F\_i) + KpF\_i \tag{22}$$

The coefficients of second-order ODE read as

$$A\_n = -\frac{\partial \chi\_1}{\partial F} = -(-f) = f = f\_i \tag{23}$$

$$B\_n = -\frac{\partial \chi\_1}{\partial F} = -(\frac{4n}{n+1}F + M + Kp) = -(\frac{4n}{n+1}F\_i + M + Kp) \tag{24}$$

$$D\_n = \chi\_1(\eta, F, \boldsymbol{F}') + B\_n F\_i + A\_n \frac{F\_{i+1} - F\_i}{h} \tag{25}$$

After some manipulation, (25) becomes

$$a\_i F\_{i-1} + b\_i F\_i + c\_i F\_{i+1} = r\_i, \qquad i = 1, 2, 3..., N \tag{26}$$

where

$$a\_i = 2 - hA\_n, \quad b\_i = 2h^2 B\_n - 4, \quad c\_i = 2 + hA\_n, \quad r\_i = 2h^2 D\_n \tag{27}$$

In matrix-vector form, it is written compactly as

$$AF = s\tag{28}$$

where

$$A = \begin{bmatrix} b\_1 & c\_1 \\ a\_2 & b\_2 & c\_2 \\ & & \cdots \\ & & & a\_{N-2} & b\_{N-2} & c\_{N-2} \\ & & & & a\_{N-1} & b\_{N-1} \end{bmatrix} \tag{29}$$
 
$$\begin{bmatrix} F\_1 \\ F\_2 \end{bmatrix} \qquad \qquad \begin{bmatrix} s\_1 \\ s\_2 \end{bmatrix}$$

$$F = \left\lfloor \begin{array}{c} F\_2 \\ \cdot \\ \cdot \\ F\_{N-1} \end{array} \right\rfloor \qquad s = \left\lfloor \begin{array}{c} s\_2 \\ \cdot \\ \cdot \\ \cdot \\ s\_{N-1} \end{array} \right\rfloor \tag{30}$$

The matrix A is a tridiagonal matrix and is written in LU-Factorization as [43]

$$A = LU\tag{31}$$

where

$$L = \begin{bmatrix} \beta\_1 \\ a\_2 \ \beta\_2 \\ \vdots \\ \beta\_{N-2} \ \beta\_{N-2} \\ \vdots \\ \beta\_{N-1} \ \beta\_{N-1} \end{bmatrix} \tag{32}$$

and

$$U = \begin{bmatrix} 1 & \wp\_1 \\ & 1 & \wp\_2 \\ & & \cdots \\ & & & 1 & \wp\_{N-2} \\ & & & & 1 \end{bmatrix} \tag{33}$$

where L and U are the lower and upper triangular matrices, respectively. Here the unknowns (β<sup>i</sup> , γi), i = 1, 2, ..., N − 1 are to be related as [43]

$$\beta\_1 = -1 - \frac{\lambda}{h}, \quad \gamma\_1 = \frac{\lambda}{\beta\_1 h} \tag{34}$$

$$
\beta\_i = b\_i - a\_i \wp\_{i-1}, \quad i = 2, 3, \ldots, N - 1 \tag{35}
$$

$$
\beta\_i \nu\_i = c\_i, \quad i = 2, 3, \dots, N - 2 \tag{36}
$$

After defining these relations, (31) becomes

$$LUF = s, \qquad UF = z, \qquad \text{and} \quad Lz = s \tag{37}$$

and we have

$$
\begin{bmatrix}
\beta\_1 \\ a\_2 & \beta\_2 \\ & \cdots \\ & & a\_{N-2} & \beta\_{N-2} \\ & & & a\_{N-1} \\ & & & & a\_{N-1}
\end{bmatrix}
\begin{bmatrix}
z\_1 \\ z\_2 \\ z\_3 \\ \cdot \\ \cdot \\ \cdot \\ z\_{N-2} \\ z\_{N-1}
\end{bmatrix} = 
\begin{bmatrix}
s\_1 \\ s\_2 \\ s\_3 \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ s\_{N-2}
\end{bmatrix} \tag{38}
$$

The unknown elements of z can be found by

$$z\_1 = s\_1 / \beta\_1, z\_i = \frac{s\_i - a\_i z\_{i-1}}{\beta\_i}, i = 2, 3, \dots, N - 1 \tag{39}$$

and

$$
\begin{bmatrix} 1 & \mathbb{M} & & & \\ & 1 & \mathbb{M}\_2 & & \\ & & \cdots & & \\ & & & 1 & \mathbb{M}\_{N-2} \\ & & & & 1 \end{bmatrix} \begin{bmatrix} F\_1 \\ F\_2 \\ \cdot \\ \cdot \\ \cdot \\ F\_{N-2} \\ F\_{N-1} \end{bmatrix} = \begin{bmatrix} z\_1 \\ z\_2 \\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ z\_{N-2} \\ z\_{N-1} \end{bmatrix} \tag{40}
$$

We then get

$$F\_{i-1} = z\_{i-1}, \quad F\_i = z\_i - \gamma\_i F\_{i+1}, \quad i = N-2, N-3, \dots, 3, 2, 1$$

which is a solution of (20). We can easily find f from f ′ = F, which in discretization form

$$\frac{f\_{i+1} - f\_i}{h} = F\_i \tag{42}$$

gives a required solution of (11). A similar procedure can also adopted for solutions θ and φ. For the sake of brevity, we only present coefficients for these ODEs and leave out the details that follow on the same line as presented above. For example, the energy equation (12) is

$$\begin{split} \frac{d^2\theta}{d\eta^2} &= -(\frac{Pr\_o}{\left(1 + \frac{4}{3}Rd\right)} \langle f \frac{d\theta}{d\eta} + Nb \frac{d\theta}{d\eta} \frac{d\phi}{d\eta} + Nt(\frac{d\theta}{d\eta})^2 \\ &+ MEc(\frac{df}{d\eta} - E\_1)^2 + \frac{2}{n+1}s\theta) \rangle \\ \text{s.t.} &(\text{\$\eta \gg 0\$}, \alpha^{\circ}\$) = -e^{\alpha \star \alpha^{\circ}} \frac{Pr\_o}{\left(\text{\$\xi \ll \ell\$}\right)} \theta^{\natural} - \theta i\_{\text{\$\epsilon \ll \ell\$}} \end{split} \tag{43}$$

$$\chi\_2(\eta, \theta, \theta') = -(\frac{1 \cdot \eta}{(1 + \frac{4}{3}Rd)})f\_i(\frac{c\_l - c\_{i-1}}{h})$$

$$+ Nb(\frac{\theta\_i - \theta\_{i-1}}{h})(\frac{\phi\_i - \phi\_{i-1}}{h})\tag{44}$$

$$-Nt(\frac{\theta\_i - \theta\_{i-1}}{h})^2 + MEc(F\_i - E\_1)^2 + \frac{2}{n+1}s\theta\_i) \text{\(\}}$$

$$A\_{\text{nn}} = -\frac{\partial \chi}{\partial \theta'} = -(-\frac{Pr\_o}{(1 + \frac{4}{3}Rd)}(f + Nb\phi' + (2Nt\theta')) \tag{45}$$

$$A\_{nn} = \frac{Pr\_o}{(1 + \frac{4}{3}Rd)} (f\_i + Nb(\frac{\phi\_i - \phi\_{i-1}}{h}) + 2N\_t(\frac{\theta\_i - \theta\_{i-1}}{h})) \tag{46}$$

$$B\_{nn} = \frac{2Pr\_o}{(n+1)(1+4/3Rd)}s\tag{47}$$

$$\frac{d^2\phi}{d\eta^2} = \frac{-Nt}{Nb} \frac{d^2\theta}{d\eta^2} - L\varepsilon Pr\_f \phi^{'}\tag{48}$$

$$\chi\_3(\eta, \phi, \phi') = \frac{-\text{Nt}}{\text{Nb}} \frac{\theta\_{i-1} - 2\theta\_i + \theta\_{i+1}}{h^2} - L e Pr\_o(f\_l \frac{\phi\_i - \phi\_{i-1}}{h}) \text{(49)}$$

Similarly, the coefficients for (13) are written as

$$A\_{mm} = Pr\_o Lef\_i, \qquad B\_{mm} = 0 \tag{50}$$

Boundary conditions can easily be discretized by following the above procedure.

#### 5.2. bvp4c

To solve the system of ODEs for Case A and Case B, we first transformed the system into first-order ODEs to compute the solution using bvp4c. For Case A it gives,

(a) Case A:

$$\begin{aligned} f &= \nu\_1, f' = \nu\_2, f'' = \nu\_3, f''' = \nu\_3' = \frac{2n}{n+1}\nu\_2^2 - \nu\_1\nu\_3, \\ &- M(E\_1 - \nu\_2) + Kp\nu\_2, \end{aligned}$$


TABLE 2 | Resemblance of the values of −*f* ′′ (0) for different values of parameters *<sup>M</sup>*, *<sup>n</sup>*, <sup>α</sup>, *<sup>E</sup>*1, and <sup>θ</sup>*<sup>r</sup>* .



TABLE 3 | Comparison of the values of −θ ′ (0) and φ ′ (0) for different values of *Rd*, *Ec*, *Le*, *Nb*, *Nt*, *n*, *Pro*, *s*, α, and ǫ for Case B with Case A, respectively.

$$\begin{split} \nu\_{4} &= \boldsymbol{\theta}, \nu\_{5} = \boldsymbol{\theta}^{'}, \boldsymbol{\theta}^{''} = \boldsymbol{\nu\_{5}^{'}} = -\frac{Pr\_{o}}{(1+\frac{4}{3})Rd}(\boldsymbol{\nu\_{1}}\boldsymbol{\nu\_{5}} + \boldsymbol{N}\boldsymbol{b}\boldsymbol{\nu\_{5}}\boldsymbol{\nu\_{7}} \\ &+ \boldsymbol{Nt}\boldsymbol{\nu\_{5}^{2}} + \boldsymbol{M}\boldsymbol{\varepsilon}(\boldsymbol{\nu\_{2}} - \boldsymbol{E\_{1}})^{2} + \frac{2}{n+1}s\boldsymbol{\nu\_{4}}), \\ \nu\_{6} &= \boldsymbol{\phi}, \boldsymbol{\nu\_{7}} = \boldsymbol{\phi}^{'}, \boldsymbol{\phi}^{''} = \boldsymbol{\nu\_{7}^{'}} = -\boldsymbol{L}\boldsymbol{e}\boldsymbol{P}\boldsymbol{r\_{0}}\boldsymbol{\nu\_{1}}\boldsymbol{\nu\_{7}} - \frac{\boldsymbol{Nt}}{N\boldsymbol{b}}\boldsymbol{\nu\_{5}^{'}}. \end{split}$$

(b) Case B: The transformed ODEs for Case B are,

$$\begin{split} f &= u\_1, f' = u\_2, f'' = u\_3, f''' = u'\_3 = \frac{(u\_3u\_5)}{(u\_4 - \theta\_7)} \\ &+ \frac{(u\_4 - \theta\_7)}{\theta\_r} (-\frac{2n}{n+1}u\_2^2 + u\_1u\_3 + M(E\_1 - u\_2) - Kpu\_2), \\ u\_4 &= \theta, u\_5 = \theta', \theta'' = u'\_5 = \frac{-\epsilon u\_5^2}{1 + \epsilon u\_4} \\ &- \frac{Pr\_o}{(1 + \epsilon u\_4)(1 + \frac{4}{3}Rd)} (u\_1u\_5 + Nbu\_5u\_7 + Ntu\_5^2 + \\ &MEc(u\_2 - E\_1)^2 + \frac{2}{n+1}su\_4), \end{split}$$

$$
\mu\_6 = \phi, \mu\_7 = \phi^\prime, \phi^\prime = \boldsymbol{\mu\_7'} = -\boldsymbol{LePr\_o}\boldsymbol{\mu\_1}\boldsymbol{\mu\_7} - \frac{\text{Nt}}{\text{N}b}\boldsymbol{\mu\_5'}.
$$

#### 6. RESULT AND DISCUSSION

In this section, we present the outcomes of our results both in tabulated and graphical forms.

In **Table 1**, we compare our results with the literature for the skin friction coefficient against different values of n while fixing α = 0.25 and α = 0.5. The SFDM shows an excellent agreement with bvp4c and the literature. In summary, the skin friction coefficient is higher for Case B and lower values for Case A.

In **Table 2** we calculate the skin friction coefficient for various parameters like magnetic parameter M, power law index n, electric field E1, porosity parameter Kp, variable thickness α, and viscosity parameter θ<sup>r</sup> . Its value goes up by changing M, n, α, Kp, and θ<sup>r</sup> , while it gets lower by changing E1. **Table 3** shows the heat and mass transfer rates for various parameters.

An electric field parameter, E1, enhances the velocity of the fluid, as can be seen in **Figure 2**. Lorentz force is responsible for increasing velocity due to the fact that the skin friction coefficient (as shown in **Table 2**) decreases.

In **Figure 3**, we observe that the momentum boundary layer thickness thins with an increase in porosity parameter Kp. This decrease in velocity profile is due to an increase in skin friction for increasing values of porosity parameter Kp. Moreover, increasing

porosity provides resistance to the flow, which ultimately reduces the velocity of the fluid.

**Figure 4** describes the velocity profile for different values of viscosity parameter θ<sup>r</sup> . It is observed that the momentum boundary layer thins with an increase in fluid viscosity parameter θr . This can be related to **Table 2**, where we can see that increasing viscosity parameter θ<sup>r</sup> leads to the magnitude of the skin friction coefficient increasing, which causes the reduction in velocity. Increasing viscosity provides more resistance to the fluid motion since higher shear stress is required to move viscous fluids.

The effect of variable thickness parameter α on temperature can be seen in **Figure 5**. It is observed that only some energy is transmitted from the surface to the liquid when we raise the wall thickness parameter. Physically, it shows that as we enhance wall

FIGURE 11 | Concentration profile φ(η) for different values of *Nt*.

thickness parameter α, less heat is transferred from the sheet to the fluid. The temperature profile therefore decreases.

**Figure 6** is plotted to demonstrate the effect of thermal radiation parameter Rd on the temperature profile. It is found that with the rise in Rd, the temperature profile increases significantly, as an increase in the radiation parameter provides more energy to the fluid, which increases the thickness of the thermal boundary layer.

In **Figure 7**, it is observed that an increase in Prandtl number Pr<sup>o</sup> causes a reduction in the temperature profile. The reason for this decrease is that smaller values of Prandtl number Pr<sup>0</sup> are equivalent higher thermal conductivity. Since the thermal conductivity of air is higher, ultimately, the temperature is higher. However, a high Prandtl number corresponds to low thermal conductivity and lower temperature flow.

In **Figure 8**, we illustrate the influence of Biot number B<sup>i</sup> on the temperature profile. It is seen that for higher values of Biot number B<sup>i</sup> , the thermal boundary layer thickness increases. This increase in temperature profile is due to the heat transfer rate, which enhances for higher values of Biot number B<sup>i</sup> . Since the thermal conductivity is dominant compared to convection, heat transport increases as the Biot number increases.

To examine the effects of the Eckert number Ec on the temperature distribution, we plot **Figure 9**. For higher values of the Eckert number Ec, it is evaluated that somehow the temperature profile rises and the thermal boundary layer gets thinner. Eckert number Ec is the ratio of the kinetic energy of fluid and enthalpy. For increasing values of Eckert number Ec, the kinetic energy increases, which causes an enhancement in fluid temperature.

**Figure 10** is plotted to illustrate the effect of the Brownian motion parameter on the concentration profile. It is concluded that higher values of Brownian motion parameter Nb cause a reduction in the nanoparticle concentration profile.

**Figure 11** is presented to characterize the behavior of thermophoresis parameter Nt on the concentration profile. It is noted that by increasing the thermophoresis parameter, we find a reduction in the nanoparticle concentration profile.

In **Figure 12**, it is found that an increase in variable thermal conductivity parameter ǫ enhances the temperature profile. **Table 3** indicates that the Nusselt number decreases with increasing ǫ. Due to this, the heat transfer rate increases, and hence the temperature profile increases.

### 7. CONCLUSIONS

This analysis achieved two goals. Firstly, an assessment of distinctive features for constant and variable properties has been done. Secondly, we adopted a new numerical process, the SFDM, to compute solutions and compared its accuracy with bvp4c. The notable results for both cases, Case A and B, are as follows:


### REFERENCES


increases in porosity parameter Kp and fluid viscosity parameter θ<sup>r</sup> .


### AUTHOR CONTRIBUTIONS

MI and MF have jointly written the manuscript. The numerical part of bvp4c, as well as tables and graphs, have been completed by MI. TI investigated the SFDM for comparison. MF, MI, and TI have discussed results.

### ACKNOWLEDGMENTS

MF would like to thank Research Scientist S. Hussain for his valuable contributions in the numerical part of this work.

fluid towards a stretching sheet. Indian J Phys. (2013) 87:1121–4. doi: 10.1007/s12648-013-0339-8


Frontiers in Physics | www.frontiersin.org


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Irfan, Farooq and Iqra. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# Heat Transfer Analysis for Non-linear Boundary Driven Flow Over a Curved Stretching Sheet With a Variable Magnetic Field

#### Kehinde M. Sanni\*, Qumar Hussain and Saleem Asghar

*Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan*

A 2-D boundary-layer flow induced by non-linear (quadratic) stretching of a curved surface of an incompressible MHD viscous fluid is investigated. Heat transfer analysis is presented including viscous dissipation and thermal radiation. A radially variable magnetic field is applied that satisfies Maxwell's equation and incorporates the curvature effects. A new similarity variable and similarity transformation are introduced to reduce the governing PDE's into ODE's. A numerical procedure is adopted to find the solution of momentum and energy equations. The numerical scheme is validated with the existing data. The results are illustrated graphically and discussed physically. Comparison with the literature shows a significant improvement compared to existing studies.

#### Edited by:

*Sara I. Abdelsalam, National Autonomous University of Mexico, Mexico*

#### Reviewed by:

*Abdullah Zaher, Benha University, Egypt Muhammad Sohail, Institute of Space Technology, Pakistan*

\*Correspondence:

*Kehinde M. Sanni km.sanni@outlook.com*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *31 January 2020* Accepted: *23 March 2020* Published: *29 April 2020*

#### Citation:

*Sanni KM, Hussain Q and Asghar S (2020) Heat Transfer Analysis for Non-linear Boundary Driven Flow Over a Curved Stretching Sheet With a Variable Magnetic Field. Front. Phys. 8:113. doi: 10.3389/fphy.2020.00113*

Keywords: curved surface, non-linear stretching, MHD, variable temperature, viscous dissipation, Joule heating, radiation, CST and PSVT

## INTRODUCTION

Stretching is one of the most important mechanisms for boundary driven flows. Crane [1] was the first to present an exact analytical solution for linear stretching on a flat plate. Since then a lot of theoretical and numerical studies have been conducted with applications in the polymer industry and engineering processes. Linear stretching was extended to non-linear and exponential stretching velocities for plane surfaces and tubes in Newtonian and non-Newtonian fluids [2–13]. The stretching of curved surfaces is now being studied for its mathematical interest as a method for solving non-linear governing equations in curvilinear coordinates and for understanding boundary driven flow behavior and generalized flow geometry. As the body of literature about stretching is so large it cannot be cited here, we will focus on curved surfaces only. Sajid et al. [14] was first to introduce the concept of flow due to the linear stretching of the curved surface in a Newtonian fluid. They concluded that the velocity decreases as the radius of curvature increases, or the velocity and the boundary layer thickness increase for the curved surface in comparison with the flat surface. In addition, the pressure gradient is variable contrary to the constant pressure gradient for the flat surface. Sanni et al. [15] discussed non-linear power law stretching velocity. Magnetohydrodynamic (MHD) flow over a curved linear stretching surface with heat transferred to an electrically conducting fluid in the presence of a transversely applied magnetic field was presented by Abbas et al. [16]. We observe that the studies for MHD flow in the curvilinear geometry are normally undertaken using a uniform magnetic field [17–24]. However, we find that the magnetic field must be such that it satisfies the solenoidal property div **B** = 0 . Hence, the assumption of a uniform magnetic field is valid for a rectangular coordinate system but not for curvilinear coordinates. Variable magnetic fields are used for the treatment of peptic ulcers,

medical diagnosis and in medical therapeutic techniques. In industry, the growing of pure crystal semiconductors can be controlled using a variable magnetic field [25, 26]. Some other papers about curved structures also include: Reddy et al. [27] which analyzed a dual solution of nanofluid flow due to a curved stretching surface under the influence of non-linear radiation. The flow of a nanofluid with carbon nanotubes caused by a curved stretching surface, with internal heat generation, is examined by Saba et al. [28]. Naveed et al. [29] documented dual solutions of MHD viscous fluid flow past a shrinking curved surface. Hayat et al. [30] discussed a numerical solution for hydromagnetic fluid flow under Soret and Dufour effects. In the presence of variable viscosity and carbon nanotubes, Nadeem et al. [31] investigated an MHD nanofluid over a curved stretching surface. For the solution methodology, new similarities have been defined which take into account the effects of both linear and non-linear stretching velocities. In the literature, the similarity for the non-linear stretching velocity is defined in a way that takes care of the linear part only. We believe that it results in incomplete governing equations and incomplete results missing out the effects of the non-linear part. This motivates us to define a new similarity transformation, resulting in the complete set of governing equations and improved results which addresses the non-linear part of the velocity as well. Mathematically, the objective of the present study has been to formulate the Lorentz force for variable magnetic field in the curvilinear coordinates and to redefine the similarity transformation to improve upon the results of the non-linear stretching velocity for a flat surface. This is accomplished for the flow and heat transfer analysis over the non-linear stretching of a curved surface in an electrically conducting viscous fluid in the presence of a variable magnetic field. The important observations are that the velocity induced by the boundary decreases as the the magnetic field and the radius of curvature increases. Thus, the flow field and the boundary layer thickness can be maintained with the help of these parameters. The non-linear contribution of the boundary velocity has more significant effect than the linear part. Detailed consequences of this study are discussed in the last section.

#### PROBLEM FORMULATION

Consider the steady two-dimensional boundary-layer flow and heat transfer for an incompressible hydromagnetic viscous fluid moving over a curved surface. The flow is induced by a nonlinear (quadratic) stretching velocity of the form ax + bx<sup>2</sup> (a and b being the dimensional constants), and the energy equation includes the viscous dissipation and thermal radiation. A variable applied magnetic field, given as **B** (r) = RBo(R + r) −1 **e**ˆ**r** , is acting in the radial direction compared to the curved surface. The variable magnetic field is taken on purpose to make it consistent with Maxwell's equation (∇**.B** = 0). The governing equations are modeled using curvilinear coordinates. The Lorentz force **F** = **J** ×**B** and the current density **J** in the absence of an electrical current (**E** = 0) are expressed as:

$$J = \sigma \left( \mathbf{V} \times \mathbf{B} \right) \tag{1}$$

$$F = \left( -\sigma \, R B\_o (R+r)^{-1} u, 0, 0 \right). \tag{2}$$

We observe that an electrically conducting fluid transverses a curved path along the stretching surface (instead of in a linear direction), and the magnetic field is perpendicular to the flow direction. Using Equaion (1), the Lorentz force takes the form **F** = (−σRBo(R + r) −1 u, 0, 0) which takes a constant value as R goes to infinity. In the above equation, σ is the electrical conductivity of the fluid, **e**ˆ**<sup>r</sup>** is the unit vector in the radial direction and B<sup>o</sup> is the strength of the applied magnetic field; whereas u and v are the components of velocity field in the x− and r−directions. The geometry of the flow is given in **Figure 1**.

The boundary layer equations [16] in the presence of a variable magnetic field can be given as follows:

$$\partial\_r \left[ \left( R + r \right) \nu \right] = -R \partial\_\mathbf{x} \mu \tag{3}$$

$$\nu \partial\_r \mu + \frac{\kappa \mu}{\overline{R} + r} \partial\_\mathbf{x} \mu + \frac{\mu \nu}{\overline{R} + r} = -\frac{1}{\rho} \frac{\overline{R}}{\overline{R} + r} \partial\_\mathbf{x} P$$

$$\nu\_{\rm r}u + \frac{1}{\overline{\mathcal{R}} + r} \partial\_{\rm x} u + \frac{1}{\overline{\mathcal{R}} + r} = -\frac{1}{\rho} \frac{\rho}{\overline{\mathcal{R}} + r} \partial\_{\rm x} r$$

$$- + \nu \left\{ \left( \frac{1}{\overline{\mathcal{R}} + r} \partial\_{\rm r} [(\mathcal{R} + r) \, \partial\_{\rm r} u] \right) - \frac{u}{\left(\overline{\mathcal{R}} + r\right)^{2}} \right\} - \frac{\sigma \, \partial\_{0}^{2} R^{2}}{\rho \left(\overline{\mathcal{R}} + r\right)^{2}} u \tag{4}$$

$$\frac{\underline{u}^2}{R+r} = \frac{1}{\rho} \partial\_r P \tag{5}$$

$$\stackrel{R\underline{u}}{=} \partial\_r T = \stackrel{K}{=} \left\{ \begin{array}{c} \frac{1}{1-\rho} \partial\_r \left[ (R+r) \,\partial\_r T \right] \end{array} \right.$$

$$\nu \partial\_r T + \frac{R\mu}{R+r} \partial\_\mathbf{x} T = \frac{K}{\rho C\_p} \left\{ \frac{1}{R+r} \partial\_r [(R+r)\,\partial\_r T] \right\}$$

$$+ \frac{u}{\rho C\_p} \left( \partial\_r u - \frac{u}{R+r} \right)^2 - \frac{1}{\rho C\_p} \partial\_r q\_\mathbf{w} + \frac{\sigma R^2}{\rho C\_p (R+r)^2} u^2,\tag{6}$$

where u is the viscosity of the fluid, ρ is the fluid density, p is the pressure, T<sup>w</sup> is the temperature of the surface at γ = 0, T<sup>∞</sup> is the ambient temperature, K is the thermal conductivity of the fluid, C<sup>p</sup> is specific heat of the fluid at constant pressure, and q<sup>w</sup> is the heat flux.

The boundary conditions for all a, b∈ R satisfied by the velocity and the temperature fields are:

$$u\_{\mathbf{w}} = a\mathbf{x} + b\mathbf{x}^2, \quad \mathbf{v} = \nu\_{\mathbf{w}} = \mathbf{0}, \mathbf{T} = T\_{\mathbf{w}} \quad \text{at} \quad r = \mathbf{0} \tag{7}$$

$$u = \mathbf{0}, \quad \partial\_r \boldsymbol{u} = \mathbf{0}, \boldsymbol{T} \to T\_{\infty} \quad \text{as} \quad \boldsymbol{r} \to \infty. \tag{8}$$

In Equation (7), a and b determine the strength of the linear and non-linear parts of the stretching velocity, respectively.

The similarity variables given in [2–4] are revisited and modified for generalized curvilinear coordinates to include both Sanni et al. Non-linear Boundary Driven Flow

a linear and non-linear part. These are now defined as:

$$\gamma = r \left\{ \frac{a + b\mathbf{x}}{\upsilon} \right\}^{\frac{1}{2}}, = a\mathbf{x}\mathbf{g}'\left(\boldsymbol{\wp}\right) + b\mathbf{x}^2 h'\left(\boldsymbol{\wp}\right),$$

$$R = k \left\{ \frac{\boldsymbol{\wp}}{a + b\mathbf{x}} \right\}^{\frac{1}{2}}, \; \boldsymbol{\wp} = \rho u\_w^2 P\left(\boldsymbol{\eta}\right) \tag{9}$$

$$\nu = -\frac{R}{R+r} \left\{ \frac{\upsilon}{a+b\upsilon} \right\}^{\frac{1}{2}} \left\{ \left[ \log \left( \wp \right) + 2b\text{x}h \left( \wp \right) \right] \right.$$

$$+ \frac{d}{2 \left( a+b\upsilon \right)} \left\{ \left[ \wp \left[ \text{axg}' \left( \wp \right) + b\text{x}^2 h' \left( \wp \right) \right] \right. \right.$$

$$- \left[ \text{axg} \left( \wp \right) + b\text{x}^2 h \left( \wp \right) \right] \right\}. \tag{10}$$

Equation (3) is identically satisfied; however, Equations (4) and (5) together with Equations (9) and (10) yield

a 2 x − k h gg′′ − g ′ <sup>2</sup> <sup>−</sup> <sup>2</sup>p<sup>1</sup> (γ ) i k + γ − k(gg′ − 1 k g ′ ) k + γ 2 − g ′′ k + γ − g ′′′ + Hak<sup>2</sup> g ′ (k + γ ) 2 +abx<sup>2</sup> ( − k - h ′′g + 2hg′′ − 3h ′ g ′ − 6p<sup>2</sup> (γ ) k + γ − k 2hg′ + h ′ g k + γ 2 − h ′′ + g ′′ k + γ + h ′ + g ′ k + γ 2 − (h ′′′ + g ′′′) + Hak2h ′ (k + γ ) 2 + b 2 x 3 − 2k h hh′′ − g ′ <sup>2</sup> + 1 k h ′′ − 2p<sup>3</sup> (γ ) i k + γ − k - hh′ − 1 k h ′ k + γ 2 − h ′′′) = 0 (11) a 2 x " g ′ 2 k + γ − p<sup>i</sup> ′ (γ ) # + abx<sup>2</sup> 2h ′ g ′ k + γ − 2p<sup>j</sup> ′ (γ ) +b 2 x 3 " h ′ 2 k + γ − p<sup>k</sup> ′ (γ ) # = 0 (12)

where Ha = p σB<sup>0</sup> 2a 2/µ is the Hartman number. The boundary conditions in the dimensionless form are:

$$g\left(0\right) = 0, \ g'\left(0\right) = 1, \ h\left(0\right) = 0, \ h'\left(0\right) = 1 \quad \text{at} \quad \mathcal{Y} = 0 \tag{13}$$

$$h'(\infty) = 0, \ h'''(\infty) = 0, \ g'(\infty) = 0, \ g''(\infty) = 0 \quad \text{as } \chi \to \infty. \tag{14}$$

Equation (12) along with Equation (11) gives

$$\frac{2k}{k+\nu}P\_{\text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\text}}}}}}}}}}} \text{\text{\text{\tiny \text{\textdegree}}}} + \frac{\text{g}^{\prime\prime}}{k+\nu} - \frac{\text{g}^{\prime}}{\text{\text{\tiny \text{\tiny \text{\textdegree}}}} + \frac{k \text{gg}^{\prime}}{k+\nu}}{\text{\text{\tiny \text{\textdegree}}} + \nu}$$

$$+ \frac{k \text{gg}^{\prime}}{\text{\text{\tiny \text{\textgreater}}} + \nu} - \frac{k \text{\textquotedblleft} \text{g}^{\prime} \text{\textquotedblright}}{k+\nu} - \frac{Hak^{2}}{\text{\textquotedblleft} \text{k} + \text{\textquotedblleft} \text{\textquotedblright}} \text{\textquotedblright}\tag{15}$$

$$\begin{split} \frac{6k}{k+\mathscr{V}}\mathscr{P}\_{2}\left(\boldsymbol{\gamma}\right) &= \mathscr{g}^{\prime\prime\prime} + h^{\prime\prime} + \frac{h^{\prime\prime} + \mathscr{g}^{\prime\prime}}{k+\mathscr{V}} - \frac{h^{\prime} + \mathscr{g}^{\prime}}{\left(k+\mathscr{V}\right)^{2}} \\ &+ \frac{kh^{\prime\prime}\mathscr{g}}{\left(k+\mathscr{V}\right)} + \frac{2khg^{\prime\prime}}{\left(k+\mathscr{V}\right)} - \frac{3kh^{\prime}\mathscr{g}^{\prime}}{k+\mathscr{V}} \\ &+ \frac{2khg^{\prime\prime}}{\left(k+\mathscr{V}\right)^{2}} + \frac{kh^{\prime}\mathscr{g}}{\left(k+\mathscr{V}\right)^{2}} - \frac{Hak^{2}}{\left(k+\mathscr{V}\right)^{2}}h^{\prime} \\ &\frac{4k}{k+\mathscr{V}}\mathscr{P}\_{3}\left(\boldsymbol{\gamma}\right) = h^{\prime\prime\prime} + \frac{h^{\prime\prime}}{k+\mathscr{V}} - \frac{h^{\prime}}{\left(k+\mathscr{V}\right)^{2}} + \frac{2khh^{\prime\prime}}{k+\mathscr{V}} \\ &+ \frac{2khh^{\prime}}{\left(k+\mathscr{V}\right)^{2}} - \frac{2k\left(h^{\prime}\right)^{2}}{k+\mathscr{V}}. \end{split} \tag{17}$$

The pressure inside the boundary layer is now expressed as:

$$P\left(\boldsymbol{\nu}\right) = a^2 \boldsymbol{p}\_1\left(\boldsymbol{\nu}\right) + ab \boldsymbol{p}\_2\left(\boldsymbol{\nu}\right) + b \boldsymbol{p}\_3\left(\boldsymbol{\nu}\right). \tag{18}$$

Using the limit ξ → ∞, Equations (15)−(17) reduce to

$$\mathbf{g}^{\prime\prime\prime} + \mathbf{g}\mathbf{g}^{\prime} - \left(\mathbf{g}^{\prime}\right)^{2} - Ha^{2}\mathbf{g}^{\prime} = \mathbf{0} \tag{19}$$

$$h'''' + g'''' + h''g + 2hg'' - 3h'g' - Ha^2h' = 0\tag{20}$$

$$h''' + 2hh'' - 2\left(h'\right)^2 = 0\tag{21}$$

At this point, we make some observations of vital importance. One, the similarity transformation as defined in this paper considers the contribution of both linear and non-linear parts of the stretching velocity through the terms "a" and "b." The similarity used in the literature for non-linear stretching (η = y q a υ ) is deficient in that its only involves a which only corresponds to the linear part of the stretching velocity [2–4]. This omission leads to the omission of terms in the momentum equations and consequently results in an incomplete solution.

Eliminating the pressure from Equations (15), (16), and (17), we obtain self-similar equations as given below:

$$\begin{aligned} &g'' + \frac{2g'''}{k+\nu} - \frac{g''}{\left(k+\nu\right)^2} + \frac{g'}{\left(k+\nu\right)^3} + \frac{kgg''}{k+\nu} - \frac{kg'g'}{k+\nu} \\ &+ \frac{kgg''}{\left(k+\nu\right)^2} - \frac{k\left(g'\right)^2}{\left(k+\nu\right)^2} - \frac{kgg'}{\left(k+\nu\right)^3} - \frac{Ha^2k^2g''}{\left(k+\nu\right)^2} \\ &+ \frac{Ha^2k^2g'}{\left(k+\nu\right)^3} = 0 \\ &h'' + g' + \frac{2\left(h''' + g''''\right)}{k+\nu} - \frac{h'' + g''}{\left(k+\nu\right)^2} + \frac{h' + g'}{\left(k+\nu\right)^3} \\ &+ \frac{k\left(gh''' + 2hg''' - 2h''g' - h'g''\right)}{k+\nu} - \frac{k\left(2gh' + h'g\right)}{\left(k+\nu\right)^3} \\ &+ \frac{k\left(gh'' - 3h'g' + 2hg'\right)}{\left(k+\nu\right)^2} - \frac{Ha^2k^2h''}{\left(k+\nu\right)^2} + \frac{Ha^2k^2h'}{\left(k+\nu\right)^3} = 0 \quad \text{(23)} \end{aligned} \tag{23}$$

$$\begin{aligned} h'' + \frac{2h''}{k+\nu} - \frac{h''}{\left(k+\nu\right)^2} + \frac{h'}{\left(k+\nu\right)^3} + \frac{2khh''}{\left(k+\nu\right)^3} - \frac{2khh''}{\left(k+\nu\right)^3} \end{aligned}$$

$$1 + \frac{2khh''}{\left(k+\nu\right)^2} - \frac{2k\left(h'\right)^2}{\left(k+\nu\right)^2} - \frac{2khh'}{\left(k+\nu\right)^3} = 0.\tag{24}$$

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#### HEAT TRANSFER ANALYSIS

The thermal boundary layer Equation (6) is solved for a constant surface temperature and variable surface temperature, in sequence.

#### Constant Surface Temperature (CST)

In this case, T = T(r) and the Equation (6) reduces to

$$\begin{split} \nu \partial\_{r} T &= \frac{K}{\rho C\_{p}} \left\{ \frac{1}{R+r} \partial\_{r} \left[ \left( R+r \right) \partial\_{r} T \right] \right\} \\ &+ \frac{\mathbf{u}}{\rho C\_{p}} \left( \partial\_{r} \mu - \frac{\boldsymbol{u}}{R+r} \right)^{2} - \frac{1}{\rho C\_{p}} \partial\_{r} q\_{\text{w}} \\ &+ \frac{\sigma R^{2}}{\rho C\_{p} \left( R+r \right)^{2}} \boldsymbol{u}^{2} \end{split} \tag{25}$$

The boundary conditions are

$$T = \left. T \right|\_{\mathbb{W}\_{r=0}} \text{ and } T|\_{r \to \infty} = 0 \tag{26}$$

and the dimensionless temperature distributions is of the form,

$$\theta(\boldsymbol{\gamma}) = \frac{T - T\_{\infty}}{T\_{\text{w}} - T\_{\infty}}.\tag{27}$$

#### Variable Surface Temperature (PSVT)

Expressing T = T(x, r) in the form

$$T(\mathbf{x}, r) = T\_{\infty} + A \left(\frac{\boldsymbol{\varkappa}}{l}\right)^{n} \boldsymbol{\theta}(\boldsymbol{\wp}) \tag{28}$$

and the boundary conditions

$$T\_{\le \vert\_{r=0}} = T\_{\infty} + A \left(\frac{\chi}{l}\right)^n \text{ and } T\_{\vert r \to \infty} = 0 \tag{29}$$

in which n∈ R is the index of wall temperature parameter, and A is the dimensional wall constant.

The radiative heat flux q<sup>w</sup> under Rosseland's approximation is given by

$$q\_w = -\frac{4\sigma^\*}{3k^\*} \partial\_r T^4 \tag{30}$$

where k ∗ and σ ∗ are the mean absorption coefficient and the Stefan-Boltzmann constant, respectively.

Taylor's series is employed in the expansion of the temperature variation (T 4 ) about T∞, and we get

$$T^4 \approx 4T\_\infty^3 T - \Im T\_\infty^4. \tag{31}$$

Substituting Equation (31) in Equation (30), we have

$$
\partial\_{\boldsymbol{r}} q\_{\boldsymbol{w}} = -\frac{16\boldsymbol{\sigma}^\* T\_{\infty}^3}{3k^\*} \partial\_{\boldsymbol{r}\boldsymbol{r}} T. \tag{32}
$$

After using Equation (32), the energy Equation (25) and (6) for CST and PSVT cases reduce to

$$\nu \partial\_r T = \frac{K}{\rho C\_p} \left( 1 + \frac{16 \sigma^\* T\_\infty^3}{3k^\* K} \right) \partial\_{rr} T$$

$$\begin{split} &+\frac{\mu}{\rho C\_{p}}\left(\partial\_{r}\mu-\frac{u}{R+r}\right)^{2}+\frac{K}{\rho C\_{p}}\partial\_{r}T+\frac{\sigma R^{2}}{\rho C\_{p}(R+r)^{2}}u^{2} \text{ (33)}\\ &\nu\partial\_{r}T+\frac{Ru}{R+r}\partial\_{x}T=\frac{K}{\rho C\_{p}}\left(1+\frac{16\sigma^{\*}T\_{\infty}^{3}}{3k^{\*}K}\right)\partial\_{rr}T\\ &+\frac{\mu}{\rho C\_{p}}\left(\partial\_{r}u-\frac{u}{R+r}\right)^{2}+\frac{K}{\rho C\_{p}}\partial\_{r}T+\frac{\sigma R^{2}}{\rho C\_{p}(R+r)^{2}}u^{2} \text{ (34)} \end{split}$$

Equations (33) and (34) after using Equation (9), (10), (27), and (28) give

$$\begin{aligned} \left(1+Rd\right)\theta'' &+ \frac{\theta'}{\left(k+\mathcal{Y}\right)} + \frac{\text{Prk}\left(\text{g}+2h\right)\theta'}{\left(k+\mathcal{Y}\right)}\\ + \text{EcPr}\left(h''+\mathcal{g}''-\frac{h'+\mathcal{g}'}{k+\mathcal{Y}}\right)^2 &+ \frac{\text{wk}^2\left(h'+\mathcal{g}'\right)^2}{\left(k+\mathcal{Y}\right)^2} = 0\\ \left(1+Rd\right)\theta'' &+ \frac{\theta'}{\left(k+\mathcal{Y}\right)} + \frac{\text{Prk}\left[\left(\text{g}+2h\right)\theta'-n\left(h'+\mathcal{g}'\right)\theta\right]}{\left(k+\mathcal{Y}\right)}\\ + \text{EcPr}\left(h''+\mathcal{g}''-\frac{h'+\mathcal{g}'}{k+\mathcal{Y}}\right)^2 &+ \frac{\text{wk}^2\left(h'+\mathcal{g}'\right)^2}{\left(k+\mathcal{Y}\right)^2} = 0 \end{aligned} \tag{36}$$

The boundary conditions become

$$\left.\theta\right|\_{\mathcal{Y}=0} = 1 \text{ and } \left.\theta\right|\_{\mathcal{Y}\to\infty} = 0\tag{37}$$

where Pr = Cpµ/k<sup>o</sup> , Ec = U 2 /(C<sup>p</sup> (T<sup>w</sup> − T∞)) , Ec = U 2 /CpA (x) n , Rd = 16σ ∗T 3 <sup>∞</sup>/3k ∗K and w (= H<sup>2</sup> <sup>a</sup>EcPr) are Prandtl's number, Eckert's number, the modified Eckert number, Radiation and Joule heating parameters, respectively. Equation (36) is locally similar and corresponds to CST if n = 0.

The surface frictional drag and other important quantities experienced by the fluid flow at the surface are the skin-friction coefficients C<sup>f</sup> , Nusselt number Nu and Local Nusselt Nu<sup>∗</sup> . These are defined as follows:

$$C\_f = \frac{\tau\_{rx}|\_{r=0}}{\frac{1}{2}\rho u\_w^2}, \text{Nu} = \frac{xq\_w}{k^\*(T - T\_w)} \text{ and } \text{Nu}^\* = \frac{xq\_w}{k^\*B\left(\chi\right)^w} \tag{38}$$

such that

$$\tau\_{r\mathbf{x}}|\_{r=0} = \mu \left( \partial\_r u - \frac{u}{R+r} \right)\_{r=0}; R\_{\mathbf{e}\_\mathbf{x}}^{\frac{1}{2}} = \sqrt{\left( \frac{c+d\mathbf{x}}{\upsilon} \right)} \mathbf{x}^2;$$

$$q\_\mathbf{w} = -k^\* \ \partial\_r T|\_{r=0} \tag{39}$$

Equations (38) and (39) give

$$-\frac{1}{2}R\_{\mathfrak{e}\_{\mathbf{x}}}^{\frac{1}{2}}C\_{f} = \left(f^{\prime\prime}(0) + g^{\prime\prime}(0) - \frac{2}{k}\right) \tag{40}$$

$$N\mu \overset{\frac{1}{2}}{\epsilon} = -\theta'(0), N\mu^\* R\_{\mathfrak{e}\_\mathcal{X}}^{\frac{1}{2}} \tag{41}$$

We notice that Equation (41) is subjected to the heat conditions defined in Equation (37).

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TABLE 1 | Comparison of the present results of the Skin-friction coefficient and the Local Nusselt number.

### COMPUTATIONAL METHODOLOGY

In this present work, our focus is to present physical and plausible solutions for the three momentum equations in response to a curved structure through a numerical approach. Substituting Equation (22) into Equation (23), differentiating the resulting equation and accommodating Equation (24) gives

$$\begin{split} &g'' + \frac{2g''' + k \text{gg'''} - kg'g''}{\left(k + \gamma\right)} - \frac{g'' - k \text{gg''} + k \left(g'\right)^2 + (Ha)^2 \, k^2 \text{g''}}{\left(k + \gamma\right)} \\ &\left(k + \gamma\right)^2 + \frac{g' - k \text{gg'} - (Ha)^2 \, k^2 \text{g'}}{\left(k + \gamma\right)^3} = 0 \\ &h'' - \frac{7h'''}{\left(k + \gamma\right)^2} + \frac{5h''}{\left(k + \gamma\right)^3} - \frac{5h'}{\left(k + \gamma\right)^4} \\ &+ \frac{4khh'}{\left(k + \gamma\right)^4} - \frac{2k\left(2hh''' - 2h'h''\right)}{\left(k + \gamma\right)^2} \\ &- \frac{2k\left[2hh'' - 2\left(h'\right)^2\right]}{\left(k + \gamma\right)^3} + \frac{k\left[3\text{gg''} - \left(g'\right)^2\right]}{\left(k + \gamma\right)^3} \\ &- \frac{3k\text{gg'}}{\left(k + \gamma\right)^4} - \frac{k\left[\text{gg''} - \left(g''\right)^2\right]}{\left(k + \gamma\right)} \end{split}$$

$$\begin{aligned} &+\frac{k\left(2h\mathbf{g}^{i\boldsymbol{\nu}}+h'\mathbf{g}^{\boldsymbol{\nu}\prime\prime}-h'\mathbf{g}'-3h'\mathbf{g}'\right)}{\left(k+\boldsymbol{\nu}\right)}+\frac{kgh^{i\boldsymbol{\nu}}}{\left(k+\boldsymbol{\nu}\right)}\\&-\frac{k\left(3h''\mathbf{g}+6hg''-3h'\mathbf{g}'\right)}{\left(k+\boldsymbol{\nu}\right)^{3}}+\frac{3k\left(2h\mathbf{g}'+h'\mathbf{g}\right)}{\left(k+\boldsymbol{\nu}\right)^{4}}\\&-\frac{\left(Ha\right)^{2}k^{2}\left(h''\mathbf{\nu}+\mathbf{g}'''\right)}{\left(k+\boldsymbol{\nu}\right)^{2}}-\frac{\left(Ha\right)^{2}k^{2}\left(h''+\mathbf{g}''\right)}{\left(k+\boldsymbol{\nu}\right)^{3}}\\&+\frac{\left(Ha\right)^{2}k^{2}\left(3h'+\mathbf{g}'\right)}{\left(k+\boldsymbol{\nu}\right)^{4}}=0\end{aligned}\tag{43}$$

At this point, the solution of the non-linear coupled system of differential Equations (35), (36), (42), and (43) as subject to boundary conditions from Equations (13), (14), and (37), is obtained by using the shooting method with Runge-Kutta algorithms in MATLAB. The initial expression of the higher order system into first order differential equations are transformed into an initial value problem by considering g, g ′ , g ′′ , g ′′′ , h, h ′ , h ′′ , h ′′′ , h ′ v, θ, θ ′ <sup>T</sup> = (s1, s2,s3,s4,s5,s6s7,s8,s9,s10,s11) T . The implementation of our numerical technique into the above system of equations gives the following.

FIGURE 3 | (A) Effects of *k* on velocity field *u*(γ). (B) Effects of *Ha* on velocity field *u*(γ). (C) Effects of *a* on velocity field *u*(γ). (D) Effects of *b* on velocity field *u*(γ).

 s1 ′ s2 ′ s3 ′ s4 ′ s5 ′ s6 ′ s7 ′ s8 ′ s9 ′ s10 ′ s11 ′ = s2 s3 s4 − z 2s<sup>4</sup> + ks1s<sup>4</sup> − ks2s<sup>3</sup> − z 2 s<sup>3</sup> − ks1s<sup>3</sup> + ks<sup>2</sup> <sup>2</sup> + (Ha) 2 k 2 s3 +z 3 s<sup>2</sup> − ks1s<sup>2</sup> − (Ha) 2 k 2 s2 s6 s7 s8 s9 − −7z 2 s<sup>8</sup> + 5z 3 s<sup>7</sup> − 5z 4 s<sup>6</sup> + 4kz<sup>4</sup> s5s<sup>6</sup> − 4kz<sup>2</sup> s5s<sup>8</sup> + 4kz<sup>2</sup> s6s<sup>7</sup> −4kz<sup>3</sup> s5s<sup>7</sup> + 4kz<sup>3</sup> s 2 <sup>6</sup> + 3kz<sup>3</sup> s1s<sup>3</sup> − kz<sup>3</sup> s 2 <sup>2</sup> − 3kz<sup>4</sup> s1s<sup>2</sup> − kzs1s<sup>4</sup> ′ −kzs<sup>2</sup> <sup>3</sup> + 2kzs1s<sup>4</sup> ′ + kzs6s<sup>4</sup> − kzs2s<sup>8</sup> − 3kzs3s<sup>7</sup> + kzs1s<sup>9</sup> ′ − 3kz<sup>3</sup> s1s<sup>7</sup> −6kz<sup>3</sup> s3s<sup>5</sup> + 3kz<sup>3</sup> s2s<sup>6</sup> + 6kz<sup>4</sup> s2s<sup>5</sup> + 3kz<sup>4</sup> s1s<sup>6</sup> − (Ha) 2 k 2 z 2 (s<sup>8</sup> + s4) − (Ha) 2 k 2 z 3 (s<sup>7</sup> + s3) + (Ha) 2 k 2 z 4 (3s<sup>6</sup> + s2) s11 −z<sup>k</sup> zs<sup>11</sup> + kzPr (s1s<sup>10</sup> + 2s5s<sup>10</sup> − ns<sup>10</sup> [s<sup>6</sup> + s2]) + EcPr (s<sup>7</sup> + s<sup>3</sup> − z [s<sup>6</sup> + s2]) +wk<sup>2</sup> z 2 (s<sup>6</sup> + s2) , (44) s<sup>1</sup> (0) 0 where z = 1 (k+γ ) and z<sup>k</sup> = 1 (1+Rd) . The unknown initial values N1, N2, N3, N4, N<sup>5</sup> and N<sup>6</sup> are approximated with the

where  $z = \frac{1}{(k+\mathcal{Y})}$  and  $z\_k = \frac{1}{(1+\mathcal{R}d)}$ . The unknown initial values  $N\_1$ ,  $N\_2$ ,  $N\_3$ ,  $N\_4$ ,  $N\_5$  and  $N\_6$  are approximated with the help of Newton's method till the required conditions ( $h'(\mathcal{Y}) = 0$ ,  $h''(\mathcal{g}) = 0$ ,  $g'(\mathcal{Y}) = 0$ ,  $g''(\mathcal{Y}) = 0$ ,  $g''(\mathcal{Y}) = 0$ ,  $\theta'(\mathcal{Y}) = 0$ ) are satisfied as  $\mathcal{Y} \to \infty$ . The initial guesses are given by

$$\begin{array}{c} \mathbf{g}^{\prime\prime} = \mathbf{N}\_1 \\ \mathbf{g}^{\prime\prime} = \mathbf{N}\_2 \\ \mathbf{h}^{\prime\prime} = \mathbf{N}\_3 \\ \mathbf{h}^{\prime\prime} = \mathbf{N}\_4 \\ \mathbf{h}^{\prime} \boldsymbol{\nu} = \mathbf{N}\_5 \\ \boldsymbol{\theta}^{\prime} = \mathbf{N}\_6 \end{array} \tag{46}$$

 

s<sup>2</sup> (0) s<sup>3</sup> (0) s<sup>4</sup> (0) s<sup>5</sup> (0) s<sup>6</sup> (0) s<sup>7</sup> (0) s<sup>8</sup> (0) s<sup>9</sup> (0) s<sup>10</sup> (0) s<sup>11</sup> (0)

 

=

 

1 N1 N2 0 1 N3 N4 N5 1 N6

 

, (45)

Expanding Equation (46) about γ = ∞

$$\begin{aligned} \mathbf{g}^{\prime\prime}(\boldsymbol{\nu}, \ N\_{1} + \Delta N\_{1}, N\_{2} + \Delta N\_{2}, N\_{3} + \Delta N\_{3}, N\_{4} \\ + \Delta N\_{4}, N\_{5} + \Delta N\_{5}, N\_{6} + \Delta N\_{6}) \\ = \mathbf{g}^{\prime\prime}(\boldsymbol{\nu}, \ N\_{1}, N\_{2}, N\_{3}, N\_{4}, N\_{5}, N\_{6}) + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{1}} \Delta N\_{1} + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{2}} \Delta N\_{2} \\ + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{3}} \Delta N\_{3} + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{4}} \Delta N\_{4} + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{5}} \Delta N\_{5} + \frac{\partial \mathbf{g}^{\prime\prime}}{\partial N\_{6}} \Delta N\_{6}. \end{aligned}$$

The remaining conditions (g ′′′ , h ′′ , h ′′′ , h ′ v, θ ′ ) are subsequently expressed in the form of Equation (47). The required Jacobian Matrix computed (for γ = ∞) after several processes is given as

$$
\hat{A}\hat{X} = \hat{B},
\tag{48}
$$

where Xˆ = (N1, N2, N3, N4, N5, N6) T , Bˆ = (B1, B2, B3, B4, B5, B6) T and the iterations that generate the above Matrix takes the form:

$$\begin{aligned} \text{For } \begin{aligned} i &= 1 \ (1) \ 6 \\ j &= 1 \ (1) \ 6 \end{aligned} \\ \hat{A} \begin{pmatrix} i,j \end{pmatrix} &= s \ (11i+3) \\ \times s \left(11j+3\right) + s \left(11i+4\right) \times s \left(11j+4\right) \\ + s \left(11i+7\right) \times s \left(11j+7\right) \\ + s \left(11i+8\right) \times s \left(11j+8\right) + s \left(11i+9\right) \\ \times s \left(11j+9\right) + s \left(11i+11\right) \times s \left(11j+11\right) \left(50\right) \\ \hat{B} \left(i\right) &= -\left(s\ (3)\times s \left(11i+3\right) + s\left(4\right)\right) \\ \times s \left(11i+4\right) + s \left(7\right) \times s \left(11i+7\right) + s \left(8\right) \end{aligned} $$
  $\times s \left(11i+8\right) + s \left(9\right) \times s \left(11i+9\right) + s \left(11i+11\right) \left(.51\right)$ 

The final point of the boundary layer region is determined successfully when no changes occur at s = 1 to a tolerance

FIGURE 5 | (A) Effects of *k* on pressure profile *P*(γ ). (B) Effects of *Ha* on pressure profile *P*(γ ). (C) Effects of *a* on pressure profile *P*(γ ). (D) Effects of *b* on pressure profile *P*(γ ).

value of 10−<sup>8</sup> . Our interest focuses in investigating the flow characteristics: velocity, temperature, momentum and thermal boundary layer thickness over a curved surface under certain physical parameters.

### RESULT AND DISCUSSION

In this section, we present the effects of characterizing parameters on flow and thermal behavior. **Table 1** gives the surface drag force and heat transfer rate for CST/PSVT cases. **Figures 2A,B** establish the patterns of fluid trajectory which decreases as the radius of curvature, k, increases. **Figures 3A,B** examine the behavior of the velocity, u (γ ), and momentum boundary layer for an increasing radius of curvature, k, and the Lorentz force. The fluid velocity and the momentum boundary layer are found to decrease as these parameters increase. This helps to control the fluid flow by means of curvature (**Figure 3A**) and the Lorentz force (**Figure 3B**). Thus, besides the well-known behavior of the Lorentz force, the curvature plays an important role in reducing the velocity. This alternate way of reducing the velocity field through the radius of curvature (for curved structures) has been established for the first time. The effects of the linear, a, and non-linear, b, parts of the stretching velocity are presented in **Figures 3C,D**. It is noted that fixing either a


TABLE 2 | Numerical values of −*R* 1 2 *<sup>e</sup><sup>x</sup> C<sup>f</sup>* and rate of heat transfer at fixed *a* = 0.5, and *b* = 0.5.

or b and varying the other parameter increases the velocity field and the boundary layer thickness; inferring that both parameters are indispensable and equally important. The temperature profile is found to decrease/increase for increasing/decreasing Prandtl/Eckert numbers for both CST (**Figure 4A**) and PSVT (**Figure 4B**) according to the physics of heat flow. Increasing the radiation parameter, Rd, increases the fluid temperature and the thermal boundary layer thickness (radiation servers as additional source for heat generation) as shown in **Figure 4C**. This effect is more significant in CST than PSVT. **Figure 4D** gives a comparison of the temperature distribution between CST (n = 0) and PSVT (n > 0). It is observed that the thermal kinetics profile is maintained over the surface for CST. However, the temperature and thermal boundary layer decreases as the temperature index, n, increases. **Figure 4E** expresses the effect of Lorentz force in the generation of surface heating. Thus, the application of Lorentz force increases the heat flow characteristic in both CST/PSVT. The effects of a magnetic parameter on the temperature and thermal boundary layer thickness are presented in **Figure 4F** and show a slight increase with Ha. It is further observed that heat transfer from the surface to the fluid is more significant for a constant surface temperature than a variable surface temperature. This shows an additional effect of magnetic fields (hitherto unknown) is raising the temperature of the fluid flow over the curved surface. The pressure gradient P (γ ) in the boundary layer region for the curved surface cannot be neglected; whereas it is neglected for the straight surface. However, the effect of increasing curvature, k, and Ha on the pressure is shown in **Figures 5A,B**. We observe from **Figure 5A** that the pressure rises from the start of the curved surface and decreases subsequently. The observation conforms with the velocity behavior which decreases for large k, while the pressure approaches zero for the flat surface as (k → ∞). In **Figure 5B** the pressure decreases significantly along the curved surface due to an opposing Lorentz force that suppresses the bulk movement of the fluid. This agrees with the behavior of the velocity as explained in the figure above. **Figures 5C,D** show the effects of stretching strengths a and b on the pressure. The pressure increases when either a (**Figure 5C**) or b (**Figure 5D**) is increased. This increase is more significant at the start of the curved surface for b, proving that the strength of non-linear stretching contributes more effectively compared to the linear strength of the stretching velocity. This phenomenal observation is presented for the first time. We further notice that the flow field characteristic decreases for linear stretching in **Figure 5C** while it increases for non-linear stretching in **Figure 5D**. The surface drag force for varying curvature and magnetic field parameters is shown in **Figures 6A,B**. **Figure 6A** shows that the drag force increases with k for increasing Ha, while in **Figure 6B** it decreases with Ha as a consequence of increasing k. **Table 1** is presented to show the impacts of a variable magnetic field in comparison with the constant magnetic input on surface drag force and heat transfer rate in view of possible engineering applications. The differences raise a slight concern due to improvements in the geometry (curvilinear) of the magnetic field rather that using a constantly applied field as in the existing literature. **Table 2** is computed to tabulate the numerical values of the skin friction coefficient and the heat transfer rate (Nusselt/local Nusselt numbers) for CST/PSVT under varying values of the characterizing parameters.

### CONCLUSION

The flow and heat transfer analysis of a two-dimensional steady hydromagnetic viscous fluid flow due to non-linear (quadratic) stretching of the curved surface is investigated. The energy equation contains viscous dissipation, linear radiation and joule heating effects. The similarity transformation is improved to contain both the effects of the linear and non-linear parts of the stretching velocity on the velocity field. The expression of Lorentz force is modified for the curved surface. The heat flow is discussed for the cases of constant surface temperature (CST) and variable surface temperature (PSVT). The reduced boundary layer equations are solved numerically using Runge-Kutta (RK) fourth order algorithms. The salient features of this work are: (i) Correct modeling of the quadratic stretching is presented by redefining the similarity transformation. (ii) An accurate expression of the Lorentz force is obtained for an applied magnetic field on the curved structure by considering the variable magnetic field that depends on the radial direction. (iii) The velocity field and the momentum boundary layer thickness can be maintained by the curvature and the Lorentz force. (iv) The effects of the strengths of the linear and non-linear parts of the stretching velocity are investigated for controlling the flow over the curved surface. (v) For both CST/PSVT cases the magnetic field increases slightly due to quantum heat generation caused by the Lorentz force. (vi) A decrease of the dimensionless radius of curvature (increasing the curvature) gives a decrease in the heat transfer from the curved surface to the fluid as compared

#### REFERENCES


to a flat surface. (vii) Increasing the Eckert/Local Eckert number enhances the temperature field and thermal boundary layer thickness. (viii) Low thermal conductivity due to an increasing Prandtl number consequently diminishes the temperature field and thermal boundary layer thickness. (ix) A high radiation parameter increases the heat flow from the surface to the fluid. (x) Variation of the wall temperature (PSVT) index reduces the heat flow characteristics, consequently it helps in regulating the heat flow rate generated over a curved sheet. (xi) The pressure decreases for a large radius of curvature, k, and Ha and increases due to the non-linear part of the stretching velocity.

### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### AUTHOR CONTRIBUTIONS

The corresponding author is a PhD student under supervision of SA and co-guidance QH. All authors work and mediated on, the technicality, physical intuition, and mathematical significance contributions of the manuscript content to the research world.


with convective heat and mass conditions. Results Phys. (2017) **7**:2601– 6. doi: 10.1016/j.rinp.2017.08.030


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Sanni, Hussain and Asghar. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Particle–Fluid Suspension of a Non-Newtonian Fluid Through a Curved Passage: An Application of Urinary Tract Infections

Arshad Riaz <sup>1</sup> \* and Muhammad Adil Sadiq<sup>2</sup>

*<sup>1</sup> Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan, <sup>2</sup> Department of Mathematics, Dammam Community College, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia*

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Maryiam Javed, Institute of Space Technology, Pakistan Noreen Akbar, National University of Sciences and Technology (NUST), Pakistan Arash Asadollahi, Southern Illinois University Carbondale, United States Anwar Shahid, Nanjing University of Aeronautics and Astronautics, China*

> \*Correspondence: *Arshad Riaz arshad-riaz@ue.edu.pk*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *08 January 2020* Accepted: *23 March 2020* Published: *06 May 2020*

#### Citation:

*Riaz A and Sadiq MA (2020) Particle–Fluid Suspension of a Non-Newtonian Fluid Through a Curved Passage: An Application of Urinary Tract Infections. Front. Phys. 8:109. doi: 10.3389/fphy.2020.00109*

The current investigation deals with the inclusion of solid particles in the flow of a non-Newtonian incompressible fluid passing through a symmetric, curved channel admitting flexible walls and exhibiting wavy characteristics for the passage of fluid. This analysis reflects the disease of white particles occurring in the flow of urine. The problem formulation is structured under the constraints of lubrication approach. The flow is considered to be laminar and steady by transforming the unsteady coordinates into wave frame coordinates. The governing equations have been formulated with the help of similarity transformations. The solution of boundary value problems has been handled by perturbation procedure. The analytical solutions for fluid and particulate phase velocities, mean flow rates, and pressure gradient profile have been presented, while a numerical treatment has been carried out for pressure rise. Analyses of fluid velocity and particulate suspension velocity, pressure gradient, and pressure rise curves under the variations of material parameters have been discussed by graphs. It is observed from this investigation that solid particles are curtailing the velocity and pressure of the liquid. It is also procured that the curvature of the channel also reduces the movement of the fluid and that the particulate suspension is occurring at the bottom of the container. It is very considerable that the increase in peristaltic pumping causes a decrease in the solid particle concentration. This theoretical analysis can help in curing the diseases like urinary tract infections (UTIs). The analysis may also be pertinent to the flow of other physiological liquids and industrial solicitation where peristaltic pumping is concerned.

Keywords: analytical solution, eyring-powell model, pumping phenomenon, solid particles, two-phase flow

### INTRODUCTION

Peristaltic flows are produced by spreading waves along the exorable membranes of a conduit. These flows provide an efficient means for fluid transport and are therefore used in the physical simulation. In clinical and medical contexts, peristaltic flows are meant for the blood transport within tiny blood vessels or fabricated blood instruments. Fluid trapping and material reflux are the two wonderful aspects of peristaltic passages. They describe the development and flow of free transport, called bolus supply. These two factors are of major importance, as they can be responsible for blood circulation and transport of viruses. From the point of view of mechanical engineering, these phenomena highlight the complexity of the chosen apparatus, but also encourage the fundamental study of such flows. Studies of peristaltic phenomena have been reported in Bhatti et al. [1], Hussain et al. [2], and Riaz [3].

It is noticed that the instant flow studies are congested to Newtonian fluids. The non-Newtonian behavior of fluids is of greater concern in many areas of science and technology. In applications for electroosmosis, for example, test accumulation, discovery, blending, and division of different natural and synthetic species on a chip coordinated with fluidic siphons and valves, the liquid rheological conduct for the most part should be considered. Major comprehension of the non-Newtonian job in fluid transport through microchannels is imperative in accurately foreseeing the exhibition and qualities of microfluidic gadgets. Numerous specialists have researched the entry of non-Newtonian liquid through peristaltic component [4, 5].

The progression of particles in a liquid is a part of multiphase mechanism. Such studies are significant in different physical issues, for example, sedimentation, barometril aftermath, powder innovation, vaporized filtration, fluidization [6], debris and lunar streams, and so forth. Moreover, with the assistance of the continuum hypothesis of blends, it is anything but difficult to look at different assorted subjects, for example, the rheology of blood [7], dissemination of proteins, demeanor of particles in a respiratory tract, and swimming of microorganisms [8]. Besides, molecule portrayal is likewise a significant part in a generation of molecule, preparing, taking care of, producing, and in different modern scientific applications [9]. Molecule portrayal is an essential and starting approach that aids in a procedure concerning solid particles. Such a depiction not only includes the natural static parameters—for example, volume, morphology, recision, dimensions, and so on—but also their dynamic frame of mind related to the liquid stream for example maximum speed and drag constant. Yao et al. [10] considered the multiphase course across the penetrable porous passage with walls impact. He prepared the perturbation solutions by considering the slip boundary conditions and observed that the slip limit condition essentially improved the speed of the liquid and a reduction of slip factor will in general increase the speed through a channel. Additionally, with an augmentation in volume portion thickness, liquid axis speed climbs. Mekheimer and Abd Elmaboud [11] evaluated the viscous fluid and particle mixture in uniform and non-uniform inlets for peristaltic concept and exact solutions are structured. Kamel et al. [12] explored the wave stream of molecule liquid adultration considering a planar channel having boundary slip and exhibited an arrangement utilizing perturbation technique. Lozano et al. [13] presented the peristaltic flow of incompressible Newtonian fluid with alike, solid particles of spherical shape distribution. They have found that the pressure in the wrinkled part of the ureter is enhanced accordingly with larger particle volume fraction.

Experimental work demonstrates that there is no check on basic speed for a liquid coursing through a curved channel. If the channel is straight, the loss of the head increases suddenly as the speed reaches its base value. The head loss varies below the basic frequency of the speed, but over it approximately as the following energy. But, through a curved channel, there is no impression of such an unexpected change at any speed of the stream. One plausibility is that movement through a curved channel is streamlined at speeds much more noteworthy than the basic for a straight channel, however testing seems to indicate that the basic speed is less in a curved channel than in a straight one. The mathematical examination of the peristaltic flow of hyperbolic tangent liquid in a curved channel has been explored by Nadeem and Maraj [14]. Narla et al. [15] have disclosed the peristaltic transport of Jeffrey nanofluid in curved channels. They discussed the dissemination of velocity, temperature, and nanoparticles fixation for different parameters overseeing the stream with the concurrent impacts of Brownian movement and thermophoretic dispersion of nanoparticles.

The urinary mechanism explains the homeostatic regulation of water and ion content in the blood and the disposal of waste products of metabolism. The kidneys receive blood from the renal artery, process it, and return the processed blood to the body through the renal vein. Urine produced in the kidneys passes into the urethra. Under normal conditions, peristalsis in the upper urinary tract begins with the origin of electrical activity at pacemaker sites located in the proximal part of the urinary collecting system. This electrical activity spreads distally, triggering the mechanical event of peristalsis and renal pelvic and ureteral contractions, which push urine from the kidney into the bladder [16]. Urine is expelled through the urethra into the outer body. Likoudis and Roos [17] studied the fluid flow in the ureter under lubrication approximation and focused their analysis on the pressure profile in the contracted part. Griffiths [18] studied the ureter with a one-dimensional lubrication approach and emphasized the relationship between low and high flow rates, pressure fields, and peristaltic contractions. Peristaltic flow in the ureter presents as an important application of peristalsis; the parameters are reasonably known, and the fluid being transported is fundamentally non-neutron and incompressible. Geometrically, however, the problem is complex. Peristaltic waves in the ureter can occur in multiple forms, either isolated or periodic, with complete occlusion throughout the cycle. Although the ureter itself is a tubular duct, the configuration of the lumen during peristalsis can be altered because its inner layer is made up of mucosa lined by the transitional epithelium. In this study, the geometry of having a two-dimensional curved shape is considered as it is of immense importance in the sense of applications.

As far as we could possibly know, no endeavor is made for peristaltic system of Eyring—Powell tensor within the sight of solid particles coursing through a curved channel. This examination is uncovered to fill this void in the literature and present the analytical and numerical examination of the model chosen. Right off the bat, we have transformed the conservation of mass and momentum into segment structure of velocity field and afterward changed over them by presenting wave outline. After this progression, physical demonstrated conditions have been diminished into a dimensionless structure by receiving some new dimensionless parameters. We have assembled the problem more comprehensively by lubrication constraints. To assess the nonlinear coupled differential conditions, perturbation strategy is applied on Eyring–Powell parameter A. The outflows

of liquid and particulate suspensions, stream rates, pressure slope, and pressure rise have been revealed. At last, physical ailments have been outlined in different diagrams under the changing estimations of appropriate parameters.

#### MODELING AND FORMATTING OF THE PROBLEM

Let us assume the creeping transport through a curved passage with small solid particles. We have adopted curvilinear cylindrical coordinates in a three-dimensional curved passage where R¯ and X¯ rays are selected to be normal and parallel to the flow, respectively. Moreover, the surfaces of the container are supposed to be flexible and executing sinusoidal waves propagating at the lower and upper surfaces at a fixed pace "c."

The boundary of the panel is expressed mathematically as

$$
\bar{H}\left(\bar{X},\bar{t}\right) = \tilde{a} + \tilde{b}\cos\left(\frac{2\pi}{\tilde{\lambda}}(\bar{X} - c\bar{t})\right). \tag{1}
$$

The symbols, like a˜ and ˜b, represent the radius of the channel and wave amplitude, accordingly. Moreover, λ˜ is the wavelength, and t¯ executes time characteristics (see **Figure 1**). Here, we write the continuity and momentum conservation relations of the fluid and particle phases.

#### Fluid Phase

For fluid phase, the physical conservation laws of mass and momentum can be described in component form as where C is the partial volume fraction parameter, µ<sup>s</sup> is the solvent viscosity, U1<sup>f</sup> and U2<sup>f</sup> represent the fluid velocities, τ ij exhibits the stress tensor components whose general form is defined as [19]:

$$\bar{\mathbf{r}} = \mu \,\,\partial\_i \mathbf{V}\_i + \frac{1}{\beta} \sinh^{-1} \left( \frac{\partial\_i \mathbf{V}\_i}{l} \right), \tag{5}$$

where ∂<sup>i</sup> **V**<sup>i</sup> gives the gradient tensor of velocity vector, the dynamic viscosity is measured by µ, and flow constants are represented by β and l.

#### Particulate Phase

For particle phase, the above defined equation will take the following form:

$$\frac{\partial \overline{U}\_{1p}}{\partial \overline{R}} + \frac{R\_1}{R\_1 + \overline{R}} \frac{\partial \overline{U}\_{2p}}{\partial \overline{X}} + \frac{\overline{U}\_{1p}}{R\_1 + \overline{R}} = 0,\tag{6}$$

$$\rho\_p C \left(\frac{\partial \overline{U}\_{2p}}{\partial \overline{t}} + \overline{U}\_{2p} \frac{\partial \overline{U}\_{2p}}{\partial \overline{R}} + \frac{\overline{U}\_{1p} R\_1}{R\_1 + \overline{R}} \frac{\partial \overline{U}\_{2p}}{\partial \overline{X}} + \frac{\overline{U}\_{1p}}{R\_1 + \overline{R}}\right)$$

$$= -C \frac{\partial \overline{P}}{\partial \overline{R}} + CS \left(\overline{U}\_{2f} - \overline{U}\_{2p}\right),\tag{7}$$

$$\rho\_p C \left(\frac{\partial \overline{U}\_{1p}}{\partial \overline{t}} + \overline{U}\_{2p} \frac{\partial \overline{U}\_{1p}}{\partial \overline{R}} + \frac{\overline{U}\_{1p} R\_1}{R\_1 + \overline{R}} \frac{\partial \overline{U}\_{1p}}{\partial \overline{X}} + \frac{\overline{U}\_{2p} \overline{U}\_{1p} R\_1}{R\_1 + \overline{R}}\right)$$

$$= -C \frac{R\_1}{R\_1 + \overline{R}} \frac{\partial \overline{P}}{\partial \overline{X}} + CS \left(\overline{U}\_{1f} - \overline{U}\_{1p}\right). \tag{8}$$

In above relations, ρp, U2p, U1p, and S represent the density of solid particles, their velocities and drag coefficient, respectively. The drag coefficient term and the empirical expression for the suspension viscosity are defined as [1]

∂X

$$\text{S} = \frac{4.5\mu'\_0}{R\_0^2} \bar{\lambda} \text{(C)}, \,\mu\_s = \frac{\mu'\_0}{1 - \bar{m}C},$$

$$\overline{\lambda} \text{(C)} = \frac{4 + \left(8C - 3C^2\right)^{1/2} + 3C}{4 + 9C^2 - 12C},$$

$$\overline{m} = 0.70e^{\left[\frac{249}{100}C + \frac{1107}{7}\exp\left(-\frac{169}{100}C\right)\right]}$$

Now suggesting the following lab and wave framework transformations

∂ ∂R¯ (R<sup>1</sup> <sup>+</sup> <sup>R</sup>¯)U2<sup>f</sup> + R<sup>1</sup> ∂U1<sup>f</sup> ∂X¯ = 0, (2) ρ (1 − C) ∂U2<sup>f</sup> ∂t + U2<sup>f</sup> ∂U2<sup>f</sup> ∂R R<sup>1</sup> + U1<sup>f</sup> R<sup>1</sup> R<sup>1</sup> + R ∂U2<sup>f</sup> ∂X R<sup>1</sup> − U1<sup>f</sup> 2 R1 R<sup>1</sup> + R ! = − (1 − C) ∂P ∂R +µ<sup>s</sup> (1 − C) R1 R<sup>1</sup> + R ∂ ∂R R<sup>1</sup> + R τ <sup>11</sup> + R1 R<sup>1</sup> + R ∂τ <sup>21</sup> ∂X R<sup>1</sup> − R1 R<sup>1</sup> + R <sup>τ</sup> <sup>22</sup> + CS U2<sup>p</sup> − U2<sup>f</sup> , (3) ρ (1 − C) ∂U1<sup>f</sup> ∂t + U2<sup>f</sup> ∂U1<sup>f</sup> ∂R R<sup>1</sup> + U1<sup>f</sup> R<sup>1</sup> R<sup>1</sup> + R ∂U1<sup>f</sup> ∂X R<sup>1</sup> + U2<sup>f</sup> U1<sup>f</sup> R<sup>1</sup> R<sup>1</sup> + R ! = (1 − C) − R1 R<sup>1</sup> + R ∂P ∂X + µsR<sup>1</sup> R<sup>1</sup> + R ∂τ <sup>22</sup> ∂X + µs R<sup>1</sup> + R R<sup>1</sup> + R ∂τ <sup>12</sup> ∂R <sup>+</sup> <sup>τ</sup> <sup>12</sup> + CS U1<sup>p</sup> − U1<sup>f</sup> , (4)

$$
\overline{\mathfrak{X}} = \overline{X} - c\overline{t}, \overline{r} = \overline{R}, \overline{u}\_{f,p} = \overline{U}\_{1f,1p} - c, \overline{v}\_{f,p} = \overline{U}\_{2f,2p}, \overline{p} = \overline{P}.
(9)
$$

In new frame of reference, Equations (3), (4), (7), and (8) transformed into the subsequent form

$$\frac{\partial p}{\partial y} = 0,\tag{19}$$

$$-\frac{\partial \rho}{\partial \mathbf{x}} + \frac{1}{k\_1} \frac{\partial}{\partial \boldsymbol{\chi}} \left( \begin{pmatrix} k\_1 + \boldsymbol{\chi} \end{pmatrix} \begin{pmatrix} \boldsymbol{\pi}\_{12} \end{pmatrix} + \frac{N\_1 C \begin{pmatrix} k\_1 + \boldsymbol{\chi} \end{pmatrix} \begin{pmatrix} \boldsymbol{\mu}\_{\mathcal{P}} - \boldsymbol{u}\_f \end{pmatrix}}{\left(1 - C\right) \ k\_1} = \boldsymbol{0}, \text{(20))}$$

$$\rho\_f(1-C)\left(\overline{\boldsymbol{\tau}}\_f \frac{\partial \overline{\boldsymbol{\tau}}\_f}{\partial \overline{\boldsymbol{\tau}}} \boldsymbol{R}\_1 + \frac{\left(\overline{\boldsymbol{\mu}}\_f + \boldsymbol{c}\right)}{\boldsymbol{R}\_1 + \overline{\boldsymbol{\tau}}} \frac{\partial \overline{\boldsymbol{\tau}}\_f}{\partial \overline{\boldsymbol{\omega}}} \boldsymbol{R}\_1 - \frac{\left(\overline{\boldsymbol{\mu}}\_f + \boldsymbol{c}\right)^2 \boldsymbol{R}\_1}{\boldsymbol{R}\_1 + \overline{\boldsymbol{\tau}}}\right) = -\left(1-C\right)\frac{\partial \overline{\boldsymbol{p}}}{\partial \overline{\boldsymbol{\tau}}} \boldsymbol{R}\_1$$

$$\boldsymbol{\tau} + \mu \boldsymbol{s} \left(1-C\right) \left(\frac{\boldsymbol{R}\_1}{\boldsymbol{R}\_1 + \overline{\boldsymbol{\tau}}} \frac{\partial}{\partial \overline{\boldsymbol{\tau}}} \left(\left(\boldsymbol{R}\_1 + \overline{\boldsymbol{r}}\right) \boldsymbol{\tau}\_{11}\right) + \frac{\boldsymbol{R}\_1}{\boldsymbol{R}\_1 + \overline{\boldsymbol{r}}} \frac{\partial \tau\_{21}}{\partial \overline{\boldsymbol{\omega}}} \boldsymbol{R}\_1 - \frac{\boldsymbol{R}\_1}{\boldsymbol{R}\_1 + \overline{\boldsymbol{r}}} \overline{\boldsymbol{\tau}}\_{22}\right) + CS \left(\overline{\boldsymbol{v}}\_{\overline{\boldsymbol{r}}} - \left(\overline{\boldsymbol{v}}\_{\overline{\boldsymbol{r}}}\right)\right),\tag{10}$$

$$\boldsymbol{\tau}(1, \dots, \zeta) = \left(\overline{\boldsymbol{v}}\_{\overline{\boldsymbol{r}}} \frac{\partial \overline{\boldsymbol{u}}\_f}{\boldsymbol{R}\_1 + \underline{\boldsymbol{r}}} \left(\overline{\boldsymbol{u}}\_f + \boldsymbol{c}\right) \boldsymbol{R}\_1 \,\partial \overline{\boldsymbol{u}}\_f \right$$

$$\rho(1-\mathcal{C})\_{\rho\overline{f}}\left(\overline{\eta}\,\bigg|\,\frac{\overline{\tau\eta}}{\partial\overline{\tau}}\,R\_{1}+\frac{\mathsf{V}^{\gamma}\overline{\mathsf{V}}\cdot\mathsf{V}\cdot\mathsf{r}\_{1}}{R\_{1}+\overline{\tau}}\frac{\overline{\tau\eta}}{\partial\overline{\varpi}}\,R\_{1}+\frac{\mathsf{V}^{\gamma}\left(\overline{\gamma\mathsf{V}}\cdot\mathsf{V}\cdot\mathsf{r}\_{1}\right)}{R\_{1}+\overline{\tau}}\right)=(1-\mathcal{C})$$

$$\left(-\frac{R\_{1}}{R\_{1}+\overline{\tau}}\frac{\partial\overline{\rho}}{\partial\overline{\varpi}}+\frac{\mu\_{\mathsf{s}}R\_{1}}{R\_{1}+\overline{\tau}}\frac{\partial\overline{\tau}\_{22}}{\partial\overline{\varpi}}+\frac{\mu\_{\mathsf{s}}}{R\_{1}+\overline{\tau}}\left((R\_{1}+\overline{\tau})\cdot\frac{\partial\overline{\tau}\_{12}}{\partial\overline{\tau}}+\overline{\tau}\_{12}\right)\right)+\mathrm{CS}\left(\overline{\mu}\_{\mathbb{P}}-\overline{\mu}\_{\mathbb{f}}\right),\tag{11}$$

$$\rho\_{\rho} \mathcal{C} \left( \overline{\nu}\_{\overline{p}} \frac{\partial \overline{\nu}\_{\overline{p}}}{\partial \overline{r}} + \frac{\left( \overline{u}\_{\overline{p}} + c \right)}{R\_1 + \overline{r}} \frac{\partial \overline{\nu}\_{\overline{p}}}{\partial \overline{x}} - \frac{\left( \overline{u}\_{\overline{p}} + c \right)^2 R\_1}{R\_1 + \overline{r}} \right) = -\mathcal{C} \frac{\partial \overline{\overline{p}}}{\partial \overline{r}} + \mathcal{C} \mathcal{S} \left( \overline{\nu}\_{\overline{f}} - \overline{\nu}\_{\overline{p}} \right), \tag{12}$$

$$\rho\_{\rho} \mathcal{C} \left( \overline{\boldsymbol{v}}\_{\overline{p}} \frac{\partial \overline{\boldsymbol{u}}\_{\overline{p}}}{\partial \overline{\boldsymbol{r}}} + \frac{\left( \overline{\boldsymbol{u}}\_{\overline{p}} + \boldsymbol{c} \right)}{R\_1 + \overline{\boldsymbol{r}}} \frac{R\_1}{\partial \overline{\boldsymbol{x}}} \frac{\partial \overline{\boldsymbol{u}}\_{\overline{p}}}{\partial \overline{\boldsymbol{x}}} + \frac{\overline{\boldsymbol{v}}\_{\overline{p}} \left( \overline{\boldsymbol{u}}\_{\overline{p}} + \boldsymbol{c} \right) R\_1}{R\_1 + \overline{\boldsymbol{r}}} \right) = -\mathcal{C} \frac{R\_1}{R\_1 + \overline{\boldsymbol{r}}} \frac{\partial \overline{\boldsymbol{p}}}{\partial \overline{\boldsymbol{x}}} + \mathcal{C} \mathcal{S} \left( \overline{\boldsymbol{u}}\_{\overline{f}} - \overline{\boldsymbol{u}}\_{\overline{p}} \right), \tag{13}$$

Now we introduce the following dimensionless quantities for further simplification

$$
\mu\_{\mathcal{P}} = \mu\_f - \frac{1}{N\_1} \frac{k\_1}{k\_1 + \mathcal{Y}} \frac{\partial \mathcal{P}}{\partial \mathbf{x}},\tag{21}
$$

$$\mu\_{\boldsymbol{f},\boldsymbol{p}} = \frac{\overline{u}\_{\boldsymbol{f},\boldsymbol{p}}}{c}, \; \nu\_{\boldsymbol{f},\boldsymbol{p}} = \frac{\overline{v}\_{\boldsymbol{f}}}{c\delta}, \; h = \frac{\overline{H}}{\tilde{a}}, \; \boldsymbol{p} = \frac{\tilde{a}^2}{\tilde{\lambda}\tilde{c}\mu\_s}\overline{p},$$

$$R\_{\varepsilon} = \frac{\rho\tilde{a}c}{\mu\_s}, \; \boldsymbol{y} = \frac{\overline{r}}{\tilde{a}}, \; \boldsymbol{x} = \frac{2\pi\overline{\boldsymbol{x}}}{\tilde{\lambda}}, \; k\_1 = \frac{R\_1}{\tilde{a}},$$

$$h' = \frac{H'}{\tilde{a}}, \; \boldsymbol{\varphi} = \frac{\tilde{b}}{\tilde{a}}, \; \tau\_{\boldsymbol{ij}} = \frac{\tilde{a}}{\mu c}\overline{\tau}\_{\boldsymbol{ij}}, \; B = \frac{1}{\beta\mu l},$$

$$A = \frac{Bc^2}{6l^2\tilde{a}^2}, \; \delta = \frac{2\pi\tilde{a}}{\tilde{\lambda}}, N\_1 = \frac{S\tilde{a}^2}{\mu\_s}.\tag{14}$$

Injecting the above revealed factors, Equations (10) through (13) become

where the stress component τ<sup>12</sup> for Eyring–Powell fluid is found as [19]

$$\pi\_{12} = -\left(1 + B\right)\left(u\_{\hat{\mathcal{V}}} + \frac{1 + u\_{\hat{f}}}{k\_1 + \chi}\right) + A\left(u\_{\hat{\mathcal{V}}} + \frac{1 + u\_{\hat{f}}}{k\_1 + \chi}\right)^3. \tag{22}$$

After proper substitution, Equation (22) becomes

$$\begin{aligned} & -\frac{dp}{dx} + \frac{1}{k\_1} \frac{\partial}{\partial \boldsymbol{\chi}} \left( (\boldsymbol{k}\_1 + \boldsymbol{\chi}) \left( -(1+B) \left( \boldsymbol{u}\_{\boldsymbol{f}\boldsymbol{\chi}} + \frac{1+\boldsymbol{u}\_f}{k\_1+\boldsymbol{\chi}} \right) \right) \right) \\ & + A \left( \boldsymbol{u}\_{\boldsymbol{f}\boldsymbol{\chi}} + \frac{1+\boldsymbol{u}\_f}{k\_1+\boldsymbol{\chi}} \right)^3 \Bigg) \end{aligned}$$

$$R\_{\varepsilon}\delta \left(1 - C\right) \left(\nu\_{\prime}\frac{\partial \nu\_{\prime}}{\partial \boldsymbol{y}} + \frac{k\_{1}\nu\_{\prime}}{k\_{1} + \boldsymbol{y}}\delta \frac{\partial \nu\_{\prime}}{\partial \boldsymbol{x}} - \frac{\left(\mu\_{\prime} + 1\right)^{2}}{k\_{1} + \boldsymbol{y}}\right) = -\left(1 - C\right)\frac{\partial p}{\partial \boldsymbol{y}}$$

$$+ (1 - C) \left(\frac{\delta}{k\_{1} + \boldsymbol{y}}\tau\_{11} + \delta \frac{\partial \tau\_{21}}{\partial \boldsymbol{x}} - \delta \frac{\tau\_{22}}{k\_{1} + \boldsymbol{y}}\right) + \delta C N\_{1} \left(u\_{p} - u\_{f}\right), \tag{15}$$

$$R\_{\varepsilon}(1 - C) \left(\nu\_{\prime}\frac{\partial u\_{\prime}}{\partial \boldsymbol{y}} + \delta \frac{k\_{1}\left(u\_{f} + 1\right)}{k\_{1} + \boldsymbol{y}}\frac{\partial u\_{f}}{\partial \boldsymbol{x}} - \frac{\nu\_{f}\left(u\_{f} + 1\right)}{k\_{1} + \boldsymbol{y}}\right) = -\left(1 - C\right)\frac{k\_{1}}{k\_{1} + \boldsymbol{y}}\frac{\partial \rho}{\partial \boldsymbol{x}}$$

$$+\left(1-\mathcal{C}\right)\left(\delta\frac{k\_1}{k\_1+\mathcal{Y}}\frac{\partial\tau\_{22}}{\partial x}+\frac{1}{k\_1+\mathcal{Y}}\frac{\partial}{\partial \mathcal{Y}}\left(\begin{pmatrix}k\_1+\mathcal{Y} \ \mathsf{r}\_{12}\end{pmatrix}\right)+N\_1\mathcal{C}\begin{pmatrix}u\_p-u\_f\end{pmatrix},\tag{16}$$

$$\operatorname{Re}\mathcal{S}\mathcal{C}\left(\delta^{2}\nu\_{\mathcal{P}}\frac{\partial\nu\_{\mathcal{P}}}{\partial\mathcal{y}} + \frac{\delta^{2}}{\lambda}\frac{k\_{1}}{k\_{1}+\mathcal{y}}\nu\_{\mathcal{P}}\frac{\partial\nu\_{\mathcal{P}}}{\partial\mathbf{x}} - \frac{\mu\_{\mathcal{P}}+1}{k\_{1}+\mathcal{y}}\right) = -\mathcal{C}\frac{\partial\mathcal{p}}{\partial\mathbf{y}} + \mathcal{CN}\_{\mathbf{I}}\delta\left(\nu\_{\mathcal{P}}-\nu\_{\mathcal{I}}\right),\tag{17}$$

$$R\_{\mathbf{r}}C\left(\delta\nu\_p \frac{\partial u\_p}{\partial \mathbf{y}} + \frac{1}{\lambda} \frac{k\_1 \left(u\_p + 1\right)}{k\_1 + \mathbf{y}} \frac{\partial u\_p}{\partial \mathbf{x}} + \delta \frac{\nu\_p \left(u\_p + 1\right)}{k\_1 + \mathbf{y}}\right) = -C \frac{k\_1}{k\_1 + \mathbf{y}} \frac{\partial \mathbf{p}}{\partial \mathbf{x}} + N\_1 C \left(u\_f - u\_p\right). \tag{18}$$

Now inserting assumptions of long wavelength (δ ≈ 0) and low Reynolds number (R<sup>e</sup> ≈ 0), we arrive at

$$+\frac{N\_1 C}{1-C} \left(\frac{k\_1+\wp}{k\_1}\right) \left(-\frac{1}{N\_1} \frac{k\_1}{k\_1+\wp} \frac{\partial p}{\partial \varkappa}\right) = 0.\tag{23}$$

We apply no-slip at the walls and the corresponding boundary conditions are manufactured as

$$U\_f\left(H'\right) = 0 \text{ and } U\_f\left(-H'\right) = 0. \tag{24}$$

In dimensionless form, using wave frame, we have

$$
\mu\_f \left( h' \right) = -1 \text{ and } \mu\_f \left( -h' \right) = -1,\tag{25}
$$

where dimensionless form of the channel height in wave frame is disclosed as ±h ′ = ± (1 + ϕ cos x).

#### METHODS AND RESULTS

This section has produced regular perturbation solutions for small values of A. So, we will use the following series expansion as a proposed solution for u<sup>f</sup>

$$u\_f = \sum\_{i=0}^{\infty} A^i u\_i. \tag{26}$$

The system generated by equating coefficients of exponent A 0

$$\begin{split} \frac{d\boldsymbol{p}}{d\boldsymbol{\omega}} + \frac{1}{k\_1} \frac{\partial}{\partial \boldsymbol{\chi}} \left( (\boldsymbol{k}\_1 + \boldsymbol{\chi}) \left( -(1+\boldsymbol{B}) \left( \boldsymbol{u}\_{0\boldsymbol{\upgamma}} + \frac{1+\boldsymbol{u}\_0}{k\_1+\boldsymbol{\upgamma}} \right) \right) \right) \\ + \frac{N\_1 \mathcal{C}}{1-\mathcal{C}} \left( \frac{k\_1+\boldsymbol{\upgamma}}{k\_1} \right) \left( \boldsymbol{u}\_{\boldsymbol{\uprho}} - \boldsymbol{u}\_0 \right) = \mathbf{0}, \end{split} \tag{27}$$

with corresponding B.Cs

$$
\mu\_0 \left( h' \right) = -1 \text{ and } \mu\_0 \left( -h' \right) = -1 \tag{28}
$$

and the first order system (comparing coefficients of A 1 ) is achieved as

$$\frac{1}{k\_1}\frac{\partial}{\partial \boldsymbol{\chi}}\left( (k\_1+\boldsymbol{\chi})\left(-(1+B)\left(\boldsymbol{u}\_{1\boldsymbol{\chi}}+\frac{1+\boldsymbol{u}\_1}{k\_1+\boldsymbol{\chi}}\right)\right)\right.\tag{29}$$

$$+\left(\boldsymbol{u}\_{0\boldsymbol{\chi}}+\frac{1+\boldsymbol{u}\_0}{k\_1+\boldsymbol{\chi}}\right)^3\right)\frac{N\_1\mathcal{C}}{1-\mathcal{C}}\left(\frac{k\_1+\boldsymbol{\chi}}{k\_1}\right)+\left(\boldsymbol{u}\_{\boldsymbol{\mathcal{P}}}-\boldsymbol{u}\_1\right)=0,$$

with

$$
\mu\_1\left(h'\right) = 0 \text{ and } \mu\_1\left(-h'\right) = 0. \tag{30}
$$

After handling the above obtained problems by executing builtin commands of the computer software, Mathematica, we finally get the following results

$$u\_0 = -1 + \frac{k\_1(-h'^2 + \wp^2) \, dp/dx}{2(1+\mathcal{B}) \, (C-1) \, (k\_1 + \wp)}.\tag{31}$$

$$u\_1 = \frac{k\_1^3 \left(-h' \left(h' - y\right) \left(h' + y\right) \left(h'^2 \left(k\_1 + y\right) - k\_1^2 \left(3k\_1 + y\right)\right)\right) \left(\frac{dp}{dx}\right)^3}{2(1+B)^4 \left(-1+C\right)^3 h' \left(h' - k\_1\right) \left(h' + k\_1\right) \left(k\_1 + y\right)^2}$$

$$-\frac{1}{2(1+B)^4 \left(-1+C\right)^3 h' \left(h' - k\_1\right) \left(h' + k\_1\right) \left(k\_1 + y\right)^2} (3k\_1^3 \left(\left(h' - k\_1\right) + k\_1\right) \left(h' + k\_1\right)) \left(\frac{dp}{dx}\right)^3 \left(\frac{dp}{dx}\right)^4}$$

$$k\_1^2 \left(h' + k\_1\right) \left(k\_1 + y\right) \left(\left(h' - y\right) \log\left(-h' + k\_1\right) + \left(h' + y\right) \log\left(h' + k\_1\right)\right) \right) \left(\frac{dp}{dx}\right)^3}{2(1+B)^4 \left(-1+C\right)^3 h' \left(h' - k\_1\right) \left(h' - k\_1\right) \left(h' + k\_1\right) \left(k\_1 + y\right)^2}. \tag{32}$$

Hence,

$$\begin{split} u\_{f} &= -1 + \frac{k\_{1} \left( -h'^{2} + \wp^{2} \right)}{2 \left( 1 + B \right) \left( -1 + C \right) \left( k\_{1} + \wp \right)} \\ &+ \frac{Ak\_{1}^{3} \left( -h' \left( h' - \wp \right) \left( h' + \wp \right) \left( h'^{2} \left( k\_{1} + \wp \right) - k\_{1}^{2} \left( 3k\_{1} + \wp \right) \right) \right)}{2 \left( 1 + B \right)^{4} (-1 + C)^{3} h' \left( h' - k\_{1} \right) \left( h' + k\_{1} \right) \left( k\_{1} + \wp \right)^{2}} \\ &\left( \frac{dp}{dx} \right)^{3} - \frac{1}{2(1 + B)^{4} (-1 + C)^{3} h' \left( h' - k\_{1} \right) \left( h' + k\_{1} \right) \left( k\_{1} + \wp \right)^{2}} \\ &\left( 3Ak\_{1}^{3} \left( \left( h' - k\_{1} \right) \right. \left. k\_{1}^{2} \left( h' + k\_{1} \right) \left( k\_{1} + \wp \right) \right) \right. \\ &\left( \left( h' - \wp \right) \log \left( -h' + k\_{1} \right) + \left( h' + \wp \right) \log \left( h' + k\_{1} \right) \right) \right) \\ &\left( \left( \frac{dp}{dx} \right)^{3} \right) - \frac{Ak\_{1}^{3} \left( 2h' \log \left( k\_{1} + \wp \right) \right) \left( \frac{dp}{dx} \right)^{3}}{2(1 + B)^{4} (-1 + C)^{3} h' \left( h' - k\_{1} \right$$

From Equation (21), we get the solution of particulate velocity, up, which is displayed below

u<sup>p</sup> = −1 + k1 −h ′<sup>2</sup> + y 2 dp dx 2 (1 + B) (−1 + C) k<sup>1</sup> + y + Ak<sup>3</sup> 1 −h ′ h ′ − y h ′ + y h ′2 k<sup>1</sup> + y − k 2 1 3k<sup>1</sup> + y 2(1 + B) 4 (−1 + C) 3h ′ h ′ <sup>−</sup> <sup>k</sup><sup>1</sup> h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> k<sup>1</sup> + y 2 dp dx3 − 1 2(1 + B) 4 (−1 + C) 3h ′ h ′ <sup>−</sup> <sup>k</sup><sup>1</sup> h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> k<sup>1</sup> + y 2 (3Ak<sup>3</sup> 1 h ′ <sup>−</sup> <sup>k</sup><sup>1</sup> k 2 1 h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> k<sup>1</sup> + y h ′ − y log −h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> + h ′ + y log h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> dp dx3 ! − Ak<sup>3</sup> 1 2h ′ log k<sup>1</sup> + y dp dx 3 2(1 + B) 4 (−1 + C) 3h ′ h ′ <sup>−</sup> <sup>k</sup><sup>1</sup> h ′ <sup>+</sup> <sup>k</sup><sup>1</sup> k<sup>1</sup> + y 2 − k1 dp dx k<sup>1</sup> + y N1 . (34)

Mathematical form of total mean volume flow rate due to fluid and particles is recognized as

$$Q = Q\_f + Q\_{\mathcal{P}},\tag{35}$$

where

$$Q\_f = (1 - C) \int u\_f dy,\tag{36}$$

$$Q = \frac{4\left(1+B\right)\left(-1+C\right)h' - h'\left(h'-2k\_1\right)k\_1\frac{dp}{dx} + 2k\_1\left(-h'^2 + k\_1^2\right)\frac{dp}{dx}\left(\log\left(k\_1\right) - \log\left(h'+k\_1\right)\right)}{4\left(1+B\right)}\tag{37}$$

$$\text{and} \qquad \qquad \text{Q}\_{\text{p}} = \text{C} \int u\_{\text{p}} d\text{y},\tag{38}$$

$$\mathbf{Q}\_{\mathbb{P}} = \frac{4\left(1+B\right)\left(-1+\mathcal{C}\right)\operatorname{C}k\_{1}\frac{dp}{dx}\left(\log\left(k\_{1}\right) - \log\left(h'+k\_{1}\right)\right)}{4\left(1+B\right)\left(-1+\mathcal{C}\right)N\_{1}} + \frac{\mathcal{C}\left(-4\left(1+B\right)\left(-1+\mathcal{C}\right)h'\right)}{4\left(1+B\right)\left(-1+\mathcal{C}\right)N\_{1}}$$

$$\frac{-4\left(h'\left(h'-2k\_{1}\right)k\_{1}\frac{dp}{dx} + 2\left(h'-k\_{1}\right)k\_{1}\left(h'+k\_{1}\right)\frac{dp}{dx}\left(\log\left(k\_{1}\right) - \log\left(h'+k\_{1}\right)\right)\right)}{4\left(1+B\right)\left(-1+\mathcal{C}\right)}.\tag{39}$$

Hence, we conclude

$$Q = \frac{4\left(1+B\right)\left(-1+C\right)h'-h'\left(h'-2k\_1\right)k\_1\frac{dp}{dx}+2k\_1\left(-h'^2+k\_1^2\right)\frac{dp}{dx}\left(\log\left(k\_1\right)-\log\left(h'+k\_1\right)\right)}{4\left(1+B\right)\left(-1+C\right)N\_1}$$

$$+\frac{4\left(1+B\right)\left(-1+C\right)Ck\_1\frac{dp}{dx}\left(\log\left(k\_1\right)-\log\left(h'+k\_1\right)\right)}{4\left(1+B\right)\left(-1+C\right)N\_1}+\frac{C\left(-4\left(1+B\right)\left(-1+C\right)h'\right)}{4\left(1+B\right)\left(-1+C\right)N\_1}+$$

$$\frac{-4\left(h'\left(h'-2k\_1\right)k\_1\frac{dp}{dx}+2\left(h'-k\_1\right)k\_1\left(h'+k\_1\right)\frac{dp}{dx}\left(\log\left(k\_1\right)-\log\left(h'+k\_1\right)\right)}{4\left(1+B\right)\left(-1+C\right)}.\tag{40}$$

From the above described Equations (35) through (40), we can find the value of pressure gradient dp/dx, which is achieved as shown below:

$$\frac{dp}{dx} = \frac{4\left(1+B\right)\left(-1+C\right)\left(h'+Q\right)N\_1}{4\left(1+B\right)\left(-1+C\right)\operatorname{C}k\_1\left(\log\left(k\_1\right)-\log\left(h'+k\_1\right)\right)+k\_1}.$$

$$\frac{1}{\left(h'\left(h'-2k\_1\right)+2\left(h'-k\_1\right)\left(h'+k\_1\right)\left(\log\left(k\_1\right)-\log\left(h'+k\_1\right)\right)\right)N\_1}.\tag{41}$$

#### GRAPHICAL ANALYSIS

*Q* = 1, φ = 0.1, *k*<sup>1</sup> = 1, *B* = 0.01, *C* = 0.12.

In the above section, we have solved the obtained governing equations for velocity, pressure gradient, and pressure rise by regular perturbation technique. The observing systems of differential equations have been handled on a mathematical software, Mathematica, via built-in DSolve commands. The more

clarified results can be shown by plotting the graphs of aboveobtained important quantities to see the effect of various physical parameters on them. The graphs will give a clearer picture of what is happening to the velocity, pressure gradient, and pressure rise when changes are made to the values of affecting parameters. To imagine these theoretical aspects, we have plotted the profiles of velocities u<sup>f</sup> and u<sup>p</sup> against the radial coordinate y in **Figures 2**– **5**, the pressure gradient dp dx vs. the coordinate x in **Figures 6**, **7**, and pressure rise along the flow rate Q in **Figures 8**, **9**. The trapping bolus mechanism has been provoked in **Figures 10**, **11**. It is observed from **Figures 2**, **3** that when we increase the numerical values of curvature parameter, k1, and solid particle concentration, N1, the fluid velocity, u<sup>f</sup> , is decreasing its height in most part of the channel for both the parameters expect the lower part where the velocity is showing almost a constant behavior with k1. They are usually included in systems that allow solids to settle to the bottom of the channel without any interruption. This is showing the physical fact that when channel is more curved and there are some solid particles placed in front of fluid flow,

FIGURE 10 | Trapping variation for fluid phase when *N*<sup>1</sup> = 3, φ = 0.04, *Q* = 1, *B* = 0.1, *A* = 0.1, *C* = 0.03. (A) *k* = 3.1, (B) *k* = 3.2, and (C) *k* = 3.3.

the velocity lowers, which is very much in agreement with the true experimental and physical facts. **Figures 4**, **5** are plotted for velocity of solid particles, up, with the variation of parameters k<sup>1</sup> and N1. From these figures, it is captured that the velocity of solid particles, up, is showing almost a similar character as we have measured in the graphs of fluid velocity, u<sup>f</sup> , but the height of the parabolic path of velocity is less than that of fluid velocity, which admits that the velocity of particulate phase is less than that of the fluid phase. This is because the increase in curvature will slow down the particle's movement and because the large amount of particles will affect the motion and suppresses the fluid. **Figures 6**, **7** have been drawn to estimate the behavior of pressure gradient dp dx for different values of curvature parameter, k<sup>1</sup> and N1. It is very obvious from these figures that pressure gradient profile is decreasing with the increasing magnitudes of both the parameters, and maximum change in axial pressure is depicted at the central part of the channel as compared to the both side corners. **Figures 8**, **9** shows the variation of pressure rise quantity, 1p, against the flow rate parameter, Q, to find the influence of k<sup>1</sup> and N1. These two plots can be divided into two portions, namely Region-I 1p > 0, η < 0 and Region-II 1p < 0, η > 0 , and we can observe that point of intersection

of all the lines is almost, the origin. In Region-I, it can be seen that pressure rise curves are showing inverse behavior with the variation of k1, but in Region-II, the situation is completely opposite (see **Figure 8**). From **Figure 9**, it is quite clear that 1p rises proportionally to the increasing values of N<sup>1</sup> in Region-I, while in Region-II, the curves are showing inverse relation.

The most important phenomenon of peristaltic flows is circulating bolus trapping. The scenario is mentioned in **Figures 10**, **11**. **Figure 10** is developed for fluid phase under the variation of curvature parameter, k1. It is measured here that boluses expand against the increasing values of curvature, which shows that curvature affects the bolus shape directly. **Figure 11** also depicts the same results for particulate phase streamlines, but, in this case, the number of boluses has been reduced to one.

#### CONCLUSIONS

In the above study, we have obtained the analytical solutions of peristaltic flow Eyring–Powell fluid model in a curved twodimensional channel in the presence of solid particles. This study can contribute to the curing of diseases like urinary tract infections (UTIs). The problem is maintained simple under the implementation of lubrication approach. Analytical solutions have been achieved by applying the perturbation technique. The graphs have been plotted to show the behavior of some prominent quantities under the variation of pertinent parameters. From all of the above discussion, the following key points have been measured:


#### REFERENCES


#### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### AUTHOR CONTRIBUTIONS

AR produced the methodology and formal analysis. MS wrote and edited the paper.

#### FUNDING

The authors wish to express their thanks for the financial support received from King Fahd University of Petroleum and Minerals, Saudi Arabia.

#### ACKNOWLEDGMENTS

The authors are grateful to the University of Education, Lahore Pakistan for providing suitable facilities to perform this research.


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Riaz and Sadiq. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# A Novel Investigation and Hidden Effects of MHD and Thermal Radiations in Viscous Dissipative Nanofluid Flow Models

Naveed Ahmed<sup>1</sup> , Adnan<sup>2</sup> , Umar Khan<sup>3</sup> , Syed Tauseef Mohyud-Din<sup>4</sup> , Ilyas Khan<sup>5</sup> \*, Rashid Murtaza<sup>2</sup> , Iftikhar Hussain<sup>2</sup> and El-Sayed M. Sherif 6,7

*<sup>1</sup> Department of Mathematics, Faculty of Sciences, HITEC University, Taxila, Pakistan, <sup>2</sup> Department of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif, Pakistan, <sup>3</sup> Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan, <sup>4</sup> Department of Mathematics, University of Multan, Multan, Pakistan, <sup>5</sup> Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>6</sup> Center of Excellence for Research in Engineering Materials (CEREM), King Saud University, Riyadh, Saudi Arabia, <sup>7</sup> Electrochemistry and Corrosion Laboratory, Department of Physical Chemistry, National Research Centre, Cairo, Egypt*

#### Edited by:

*Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan*

#### Reviewed by:

*Taseer Muhammad, King Khalid University, Saudi Arabia Usman Masud, University of Engineering and Technology, Pakistan*

> \*Correspondence: *Ilyas Khan ilyaskhan@tdtu.edu.vn*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *04 October 2019* Accepted: *03 March 2020* Published: *12 May 2020*

#### Citation:

*Ahmed N, Adnan, Khan U, Mohyud-Din ST, Khan I, Murtaza R, Hussain I and Sherif E-SM (2020) A Novel Investigation and Hidden Effects of MHD and Thermal Radiations in Viscous Dissipative Nanofluid Flow Models. Front. Phys. 8:75. doi: 10.3389/fphy.2020.00075* Hidden effects of MHD and thermal radiations for a viscous dissipative nanofluids (*Al*2*O*<sup>3</sup> − *H*2*O* and γ*Al*2*O*<sup>3</sup> − *H*2*O*) are taken under consideration. The models are formulated by implementing the suitable similarity transformations. Then, two models are discussed mathematically by using RK scheme together with shooting method. The results for flow regimes, coefficient of skin friction, thermophysical characteristics, and heat transfer coefficient are pictured and discussed comprehensively by changing the pertinent flow parameters. It is observed that the nanofluids velocity increases abruptly for higher Hartree pressure gradient. For assisting flow situation, the velocity *F* ′ (η) increases and abrupt decreasing behavior is examined for opposing flow case. The composition of *Al*2*O*3−*H*2*O*, and γ*Al*2*O*3−*H*2*O* becomes more dense for high volume fraction therefore, drops in the velocity field is noted. The temperature of *Al*2*O*<sup>3</sup> − *H*2*O*, and γ*Al*2*O*<sup>3</sup> − *H*2*O* rises rapidly by varying opposing flow parameter γ < 0 and high volume fraction φ. Also, the temperature β(η) declines abruptly for parameter λ.

Keywords: wedge, host fluid, magnetic field, thermal radiation, viscous dissipation, γAl2O<sup>3</sup> nanoparticles, RK scheme

### INTRODUCTION

The analysis of the host fluids saturated by various sort of nanoparticles over a wedge geometry is of the essential and interesting topic in fluid dynamics and heat transfer phenomena. Currently, the hidden effects of significant flow parameters like magnetic number, thermal radiation, and viscous dissipation for regular and nanofluids models becomes important.

The flow and entropy generation analysis in magnetized nanofluid by considering the impacts of porosity described in Ellahi et al. [1]. The study of effective dynamic viscosity in non-Newtonian fluids in porous medium reported in Eberhard et al. [2]. The analysis of Roselands heat flux over non-linear stretchable surface with slip flow conditions was examined in Majeed et al. [3].

Keeping in mind the importance and popularity of the wedge type flow, Falkner and Skan [4, 5] focused on this particular direction of fluid mechanics and presented earlier study. They analyzed the boundary layer model mathematically and extend the case for stretching walls of the wedge. The work of Falkner and Skan provided a new direction in the fluid dynamics. Rajagopal et al. [6] inspired by the work of Falkner and Skan [4, 5] and extended the boundary layer model for non-Newtonian fluid and found fascinating results. They analyzed the flow characteristics over a fixed wedge in the fluid. Later on, Lin et al. [7] inspired by the concept forced convection and extend the model by incorporating the various Prandtl values. Hartree [8] discussed the boundary layer model for approximate solutions and highlighted important results. Watanabe [9] and Watanabe and Pop [10] extended the Falkner Skan model by considering the phenomena of free and forced convection and explored the hidden effects on the flow characteristics, respectively. They presented the impacts of magnetic field, suction and injection in the flow field.

The porosity of the wedge walls affects the flow characteristics significantly. In the light of this fact, Koh and Hartnett [11] explored the results for skin friction and local coefficient of heat transfer and observed significant variations due to porosity parameter. Similarly, Kumari et al. [12] discussed the phenomena of mixed convection over a porous wedge. In 2003, Chamkha et al. [13] reported the radiative Falkner Skan flow and presented its characteristics over a semi-infinite domain. El-Dabe et al. [14] prolonged the Falkner Skan model for Casson fluid and treated the respective non-linear flow model numerically and incorporated the influences of magnetic field in the flow behavior and the heat transfer phenomena. The flow characteristics of Casson fluid over a symmetric wedge was presented by Mukhopadhyay et al. [15] in 2013. A novel analysis comprising the impacts of ohmic heating, applied magnetic field, and mixed convection on radiative flow over a stretchable wedge reported in Su et al. [16]. The alterations in the flow pattern of micropolar and Newtonian fluids over a wedge moving in the fluid and Viscoelastic fluid flow in the presence Lorentz force reported in Ishak et al. [17, 18] and Rashidi et al. [19], respectively.

Kandasamy et al. [20] prolonged the Falkner Skan flow by comprising the impacts of chemical reaction and found the results for suction or blowing on the radiative flow over a porous wedge. Hussanan et al. [21] highlighted the influences of Joule heating in the flow past over an oscillating plate. They also highlighted the alterations in the flow characteristics due to convective flow condition and resistive heat phenomena. A comprehensive analysis of Falkner Skan flow in the presence of velocity slip phenomena and applied Lorentz force described by Su et al. [22]. Recently, Ullah et al. [23] contributed the Falkner Skan model for non-Newtonian nature of the fluid.

A prominent fact that the regular fluids have less heat transfer characteristics. For many productions in various industries required considerable level of heat transfer and base liquids fail to provide such amount heat. However, researchers focused and thought to overcome this issue. Finally, a new class of the fluid developed and titled as Nanofluid. Basically, the nanofluid is a compound fluid composed by base liquid with the nanoparticles. These nanoparticles obtained from various metals and their oxides. In nanofluids, the volume fraction of the nanoparticles plays the role of back bone for the heat transfer enhancement. The development of the nanofluids reduces the problems and issues faced by the industrialist and engineers. Thus, the analysis of the nanofluids became an orbit for the researchers and engineers and explored new and fascinating characteristics of the nanofluids.

Recently, Rafique et al. [24] reported the numerical study of Casson nanofluid over an inclined surface and found the results for flow field. Impact of magnetic field by considering second slip flow condition on the flow Casson nanofluid explored by Majeed et al. [25]. In 2019, Bibi et al. [26] examined the flow model in the presence of convective boundary condition. The significant analysis for different sort of nanofluids under various flow conditions are examined in Saba et al. [27] and Srinivasacharya et al. [28].

The nanofluid models (Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O) considering the phenomena of magnetic field, thermal radiation, and viscous dissipation is taken over a wedge geometry in the Cartesian coordinates. Two types of thermal conductivities are incorporated in the energy equation to enhance the heat transfer rate in Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids. The model is described in section Model Formulation and treated mathematically in section Mathematical Analysis. The fascinating role of magnetic field and thermal radiation in the flow regimes explored and explain in section Physical Interpretation of Results. The quantities related to engineering interest (Skin friction and local Nusselt number) are presented and analyzed for varying flow parameters. In the end, major effects of under consideration model are incorporated.

### MODEL FORMULATION

### Statement and Geometry

Steady, laminar and viscous incompressible flow of H2O saturated by Al2O<sup>3</sup> and γAl2O<sup>3</sup> nanoparticles is taken over a wedge. The effects of magnetic field and thermal radiation are taken into account. The velocity at the wedge surface is u˘<sup>w</sup> = U b wx <sup>m</sup> and at the free stream is U˘ <sup>e</sup> = U b ∞x <sup>m</sup> and are functions of x. Here, U b <sup>w</sup> and U b ∞ are constants at the surface and away from the surface. Furthermore, corresponding to wedge angle λ = /π, the Hartree pressure parameter is λ = 2m/(m+1). Induced magnetic field produced due to the motion of Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids is neglected through the analysis. The temperature at the wedge surface is <sup>T</sup>ˆ<sup>w</sup> <sup>=</sup> <sup>T</sup>ˆ<sup>∞</sup> <sup>+</sup> <sup>A</sup>/<sup>x</sup> <sup>2</sup><sup>m</sup> and is a function of x. The temperature at the surface and at the free stream is T<sup>w</sup> and T∞, respectively. The physical theme of the model comprising the role of Al2O<sup>3</sup> and γAl2O<sup>3</sup> nanoparticles is demonstrated in **Figure 1** over a semi-infinite region.

### Governing Equations and Non-dimensionalization

In the light of above highlighted assumptions, the following is the model which govern the flow of nanofluids over a wedge [23, 28]:

$$\frac{\partial \hat{u}}{\partial \mathbf{x}} + \frac{\partial \hat{v}}{\partial y} = \mathbf{0} \tag{1}$$

$$
\begin{split}
\hat{u}\frac{\partial\hat{u}}{\partial\mathbf{x}} + \hat{\boldsymbol{\nu}}\frac{\partial\hat{u}}{\partial\boldsymbol{\nu}} &= \,^\*\tilde{U}\_{\varepsilon}\,(\mathbf{x})\,\frac{d\tilde{U}\_{\varepsilon}\,(\mathbf{x})}{d\mathbf{x}} + \frac{\hat{\mu}\_{\eta\mathbf{f}}}{\hat{\rho}\_{\eta\mathbf{f}}} \left(\frac{\partial^2\hat{u}}{\partial\boldsymbol{\nu}^2}\right) \\ &- \,^\*\frac{\hat{\sigma}\_{\eta\mathbf{f}}}{\hat{\rho}\_{\eta\mathbf{f}}}\,B\_0^2\left(\mathbf{x}\right)\,\big(\boldsymbol{\nu}-U\_{\varepsilon}\,(\mathbf{x})\big)
\end{split}
\tag{2}
$$

$$
\begin{split}
\hat{u}\frac{\partial\hat{T}}{\partial\boldsymbol{\chi}} + \hat{\boldsymbol{\nu}}\frac{\partial\hat{T}}{\partial\boldsymbol{\chi}} &= \frac{\hat{k}\_{\boldsymbol{n}\boldsymbol{f}}}{\left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\boldsymbol{n}\boldsymbol{f}}} \left(\frac{\partial^{2}\hat{T}}{\partial\boldsymbol{\chi}^{2}}\right) + \frac{1}{\left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\boldsymbol{n}\boldsymbol{f}}} \left(\frac{\partial\hat{u}}{\partial\boldsymbol{\chi}}\right)^{2} \\ &- \frac{16\sigma^{\*}T\_{\infty}^{3}}{3k\left(\hat{\rho}\,\mathbf{C}\_{\mathcal{P}}\right)\_{\boldsymbol{n}\boldsymbol{f}}} \left(\frac{\partial^{2}\hat{T}}{\partial\boldsymbol{\chi}^{2}}\right)
\end{split} \tag{3}$$

The law of conservation of mass, momentum and energy shown in Equations (1–3), respectively. The velocities in x and y directions are uˆ and vˆ, respectively. The velocity at the free stream, temperature, effective dynamic viscosity, density, electrical conductivity, thermal conductivity, and heat capacity are represented by U b <sup>e</sup>, <sup>T</sup>ˆ, <sup>µ</sup><sup>ˆ</sup> nf , <sup>ρ</sup>ˆnf , <sup>σ</sup>ˆnf , <sup>ˆ</sup>knf , and ρc<sup>p</sup> nf , respectively. Mean absorption coefficient and Stefan Boltzmann constants are denoted by k and σ ∗ , respectively.

The conditions on the flow at the boundaries are defined as Ullah et al. [23]:

$$\begin{array}{c} \text{At the surface} \\ \hat{\boldsymbol{u}} = \hat{\boldsymbol{u}}\_{\text{w}} \left( \boldsymbol{\alpha} \right), \\ \hat{\boldsymbol{v}} = \boldsymbol{0}, \\ \hat{\boldsymbol{T}} = \hat{\boldsymbol{T}}\_{\infty} + \frac{\boldsymbol{A}}{\boldsymbol{x}^{-2m}} \\ \text{At the free stream} \\ \hat{\boldsymbol{u}} \to \breve{\boldsymbol{U}}\_{\varepsilon} \left( \boldsymbol{\alpha} \right), \\ \hat{\boldsymbol{T}} \to \hat{\boldsymbol{T}}\_{\infty} \end{array} \tag{4}$$

The similarity variables defined in the following way for the non-dimensionalization of the governing flow model [23]:

$$\begin{aligned} \hat{u} &= \frac{\frac{\partial \hat{\psi}}{\partial \boldsymbol{\gamma}}}{\partial \boldsymbol{\gamma}}\\ \hat{\boldsymbol{\psi}} &= -\frac{\frac{\partial \hat{\psi}}{\partial \boldsymbol{\chi}}}{\partial \boldsymbol{\chi}}\\ \boldsymbol{\eta} &= \sqrt{\frac{\frac{2\boldsymbol{\gamma}\_{\boldsymbol{\ell}} \boldsymbol{x} \hat{\mathcal{U}}\_{\boldsymbol{\ell}}(\boldsymbol{x})}{(m+1)}}{2\boldsymbol{\gamma}\_{\boldsymbol{\ell}}\boldsymbol{x}}} \boldsymbol{\varchi}(\boldsymbol{\eta})\\ \boldsymbol{\eta} &= \sqrt{\frac{(m+1)\hat{\mathcal{U}}\_{\boldsymbol{\ell}}(\boldsymbol{x})}{2\boldsymbol{\eta}\_{\boldsymbol{\ell}}\boldsymbol{x}}} \boldsymbol{\upchi} \end{aligned} \tag{5}$$

The following models are used to enhance the performance of the particular model [29]:

$$\hat{\rho}\_{nf} = \left\{ (1 - \phi) + \frac{\phi \hat{\rho}\_{\mathcal{I}}}{\hat{\rho}\_{\mathcal{I}}} \right\} \hat{\rho}\_{\mathcal{I}} \tag{6}$$

$$\begin{split} \hat{\mu}\_{nf} &= \hat{\mu}\_{\mathcal{I}} (1 - \phi)^{-2.5} \\ \hat{\mu}\_{nf} &= \hat{\mu}\_{\mathcal{I}} (123 \phi^2 + 7.3 \phi + 1) \end{split} \tag{7}$$

$$\hat{\mu}\_{nf} = \hat{\mu}\_{\mathcal{I}} (123 \phi^2 + 7.3 \phi + 1)$$

$$\text{For Al}\_2\text{O}\_3 - \text{H}\_2\text{O} \text{ and } \gamma\text{Al}\_2\text{O}\_3 - \text{H}\_2\text{O}, \text{ (7)}$$

$$\begin{aligned} \left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\eta\mathcal{f}} &= \left\{ (1-\phi) + \frac{\phi\left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\mathcal{s}}}{\left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\mathcal{f}}} \right\} \left(\hat{\rho}c\_{\mathcal{P}}\right)\_{\mathcal{f}}\\ \hat{k}\_{\eta\mathcal{f}} &= \hat{k}\_{\mathcal{f}} \left\{ \frac{\hat{k}\_{i} + 2\hat{k}\_{\mathcal{V}} - 2\phi\left(\hat{k}\_{\mathcal{V}} - \hat{k}\_{i}\right)}{\hat{k}\_{i} + 2\hat{k}\_{\mathcal{V}} + \phi\left(\hat{k}\_{\mathcal{V}} - \hat{k}\_{i}\right)} \right\} \\ \hat{k}\_{\eta\mathcal{I}} &= \hat{k}\_{\mathcal{f}} \{ 4.97\phi^{2} + 2.72\phi + 1 \} \end{aligned} \tag{8}$$

$$\text{For Al}\_2\text{O}\_3-\text{H}\_2\text{O and }\gamma\text{Al}\_2\text{O}\_3-\text{H}\_2\text{O} \quad \text{(9)}$$

$$\hat{\sigma}\_{\eta f} = \hat{\sigma}\_f \left\{ 1 + \frac{3\left(\frac{\hat{\sigma}\_i}{\hat{\sigma}\_f} - 1\right) \phi}{\left(\frac{\hat{\sigma}\_i}{\hat{\sigma}\_f} + 2\right) - \left(\frac{\hat{\sigma}\_i}{\hat{\sigma}\_f} - 1\right) \phi} \right\}. \tag{10}$$

The particular values of thermophysical characteristics embedded in Equations (6–10) are given in **Table 1**.



After performing the suitable differentiation and incorporating the effective nanofluids models given in Equations (6–10) in the dimensional model, the following two models of nanofluids are obtained:

## Al2O<sup>3</sup> − H2O Model

$$\begin{aligned} &F'''' + \frac{1-\phi+\frac{\phi\dot{\phi}}{\dot{\phi}}}{(1-\phi)^{-2.5}} \left(F'' + \lambda\left(1-F'^{2}\right)\right) \\ &+ \frac{(1-\phi)^{2.5}}{\left(1+\frac{3\left(\frac{\dot{\phi}}{\dot{\phi}\_{f}}-1\right)\phi}{\left(\frac{\dot{\phi}\_{f}}{\dot{\phi}\_{f}}+2\right)-\left(\frac{\dot{\phi}}{\dot{\phi}\_{f}}-1\right)\phi}\right)^{-1}}M^{2}(1-F') = 0,\end{aligned} \tag{11}$$
 
$$\begin{bmatrix} 1+\frac{Rd}{\frac{\dot{\lambda}\_{i}+2\dot{\lambda}\_{f}-2\phi\left(\dot{\lambda}\_{i}-\dot{\lambda}\_{i}\right)}{\tilde{k}\_{i}+2\dot{k}\_{f}+\phi\left(\dot{k}\_{f}-\dot{k}\_{i}\right)} \\ \frac{1}{\tilde{k}\_{i}+2\dot{k}\_{f}+\phi\left(\dot{k}\_{f}-\dot{k}\_{i}\right)} \end{bmatrix}\beta^{\prime\prime} \\ &+\frac{1}{\frac{\dot{k}\_{i}+2\dot{k}\_{f}-2\phi\left(\dot{k}\_{f}-\dot{k}\_{i}\right)}{\tilde{k}\_{i}+2\dot{k}\_{f}+\phi\left(\dot{k}\_{f}-\dot{k}\_{i}\right)} \left[\frac{\left(PrF\beta^{\prime}-2\lambda PrF^{\prime}\beta\right)}{\left(1-\phi\right)+\frac{\phi\left(\alpha\_{\mathcal{E}}\right)}{\left(\alpha\_{\mathcal{E}}\right)}\right]^{-1}}+Pr\text{EcF}^{\prime\prime}2 \\ &= 0,\end{aligned} \tag{12}$$

γAl2O<sup>3</sup> − H2O Model

$$\begin{aligned} &F^{\prime\prime} + \frac{\left(1 - \phi + \frac{\phi\hat{\rho}\_{\flat}}{\hat{\rho}\_{f}}\right)}{123\phi^{2} + 7.3\phi + 1} \left(F^{\prime\prime} + \lambda \left(1 - F^{\prime}2\right)\right) \\ &+ \frac{\left(1 + \frac{3\left(\frac{\phi}{\hat{\sigma}\_{f}} - 1\right)\phi}{\left(\frac{\phi}{\hat{\sigma}\_{f}} + 2\right) - \left(\frac{\phi}{\hat{\sigma}\_{f}} - 1\right)\phi}}{123\phi^{2} + 7.3\phi + 1} M^{2}(1 - F^{\prime}) = 0, \\ &\left[\frac{1}{14} + \frac{Rd}{r} \right]\_{\theta^{\prime\prime}} \end{aligned} \tag{13}$$

$$\begin{aligned} & \left[ 1 + \frac{\epsilon \omega}{4.97 \phi^2 + 2.72 \phi + 1} \right] \beta'' \\ &+ \frac{1}{4.97 \phi^2 + 2.72 \phi + 1} \left[ \frac{\left( PrF \beta' - 2 \lambda PrF' \beta \right)}{\left( (1 - \phi) + \frac{\phi \left( \hat{\beta} \epsilon\_\beta \right)\_s}{\left( \hat{\beta} \epsilon\_\beta \right)\_f} \right)^{-1}} + PrEcF' 2 \right] \\ &= 0. \end{aligned} \tag{14}$$

The conditions at the boundaries at the surface and at the free stream are as under:

$$\begin{array}{c} \text{At the surface } \eta = 0\\ \qquad \begin{array}{c} F \left( \eta \right) = 0\\ F \left( \eta \right) = \eta \\ \beta \left( \eta \right) = 1 \end{array} \\ \text{At the free surface } \eta \to \infty\\ \qquad \begin{array}{c} \text{At the free surface } \eta \to \infty\\ F^{\left( \eta \right)} \to 1\\ \beta \left( \eta \right) \to 0 \end{array} \end{array} \tag{15}$$

The parameters embedded in the models are Eckert number,

$$\begin{aligned} \frac{\hat{U}^2\left(\mathbf{x}\right)}{\left(c\_{\mathcal{P}}\right)\_f \left(T\_w - T\_\infty\right)}, \Pr &= \frac{\hat{\mu}\_f \left(\hat{c}\_{\mathcal{P}}\right)\_f}{\hat{k}\_f}, M^2 = \frac{\hat{\sigma}\_f B\_0^2 \nu\_f}{\mathcal{U}\_\infty(m+1)},\\ Rd &= \frac{16\sigma^\* T\_\infty^3}{3\hat{k}\_f k}, and \mathcal{V} = \frac{\tilde{U}\_w}{\tilde{U}\_\infty}. \end{aligned}$$

#### Quantities of Engineering Interest

Skin friction and local heat transfer phenomena are of great importance from engineering point of view. Mathematical dimensional expressions for these quantities are as under:

$$\mathsf{\heartsuit}\_{F} = \frac{\hat{\mu}\_{\mathfrak{nf}}}{\hat{\rho}\_{\mathfrak{nf}} \, \hat{U}^{2}(\mathbf{x})} \left(\frac{\partial \hat{u}}{\partial \boldsymbol{\chi}}\right) \downarrow\_{\mathfrak{\boldsymbol{\nu}}=\mathbf{0}} \tag{16}$$

$$\mathfrak{N}u\_{\mathbf{x}} = \left[\frac{-\varkappa \hat{k}\_{\eta f}}{\hat{k}\_{f}\left(\hat{T}\_{\mathrm{w}} - \hat{T}\_{\infty}\right)}\right] \left(\frac{\partial \hat{T}}{\partial \boldsymbol{\nu}}\right) \downarrow\_{\boldsymbol{\mathcal{V}} = \boldsymbol{0}}.\tag{17}$$

These expressions reduced in the following non-dimensional form by implementing the suitable differentiation and nanofluids models:

$$\mathcal{K}\_{\text{F}}\sqrt{\text{Re}\_{\text{x}}} = \frac{123\phi^2 + 7.3\phi + 1}{\left((1 - \phi) + \frac{\phi\hat{\phi}}{\hat{\phi}\_{\text{f}}}\right)} F''(0),\tag{18}$$

$$\begin{array}{ll} \widetilde{N}u\_{\rm x} \left(Re\_{\rm x}\right)^{-\frac{1}{2}} \\ = & -\frac{\widehat{k}\_{i} + 2\widehat{k}\_{f} - 2\phi \left(\widehat{k}\_{f} - \widehat{k}\_{i}\right)}{\widehat{k}\_{i} + 2\widehat{k}\_{f} + \phi \left(\widehat{k}\_{f} - \widehat{k}\_{i}\right)} \beta'(0) \text{ For } Al\_{2}O\_{3} - H\_{2}O \\ & - \left(4.97\phi^{2} + 2.72\phi + 1\right)\beta'(0) \text{ For } \gamma Al\_{2}O\_{3} - H\_{2}O \end{array} \tag{19}$$

here, Re<sup>x</sup> = xU b (x) ν b f is denotes the local Reynold number.

#### MATHEMATICAL ANALYSIS

Shooting technique [30, 31] is adopted for the mathematical analysis of the particular nanofluids flow models. The reason behind this choice is the non-linearity (for instance see [32–40]) of the models over semi-infinite region. To initiate the technique, the following substitution are made:

$$
\widehat{\mathbf{y'}\mathbf{y}}\_1 = \mathbf{F}, \widehat{\mathbf{y'}\mathbf{y}}\_2 = \mathbf{F}', \widehat{\mathbf{y'}\mathbf{y}}\_3 = \mathbf{F''}, \widehat{\mathbf{y'}\mathbf{y}}\_4 = \boldsymbol{\beta}, \widehat{\mathbf{y'}\mathbf{y}}\_5 = \boldsymbol{\beta'}.\text{(20))}
$$

## Al2O<sup>3</sup> − H2O Model

The flow model Al2O<sup>3</sup> − H2O can be reduced into the following form:

$$F^{\prime\prime} = -\frac{1-\phi + \frac{\phi\hat{\rho}\_{\dot{s}}}{\hat{\rho}\_f}}{(1-\phi)^{-2.5}} \left(F^{\prime\prime} + \lambda \left(1 - F^{\prime}2\right)\right)$$

$$-\frac{(1-\phi)^{2.5}}{\left(1 + \frac{3\left(\frac{\phi}{\hat{\sigma}\_f} - 1\right)\phi}{\left(\frac{\phi}{\hat{\sigma}\_f} + 2\right) - \left(\frac{\phi}{\hat{\sigma}\_f} - 1\right)\phi}}M^2(1 - F^{\prime}), \quad \text{(21)}$$

$$\begin{aligned} \beta^{\prime\prime} &= \frac{-1}{1 + \frac{\frac{Rd}{\hat{k}\_{\hat{r}} + 2\hat{k}\_{\hat{r}} - 2\phi \left(\hat{k}\_{\hat{r}} - \hat{k}\_{\hat{s}}\right)}}} \\\\ \left[\frac{1}{\frac{\hat{k}\_{\hat{r}} + 2\hat{k}\_{\hat{r}} - 2\phi \left(\hat{k}\_{\hat{r}} - \hat{k}\_{\hat{s}}\right)}{\hat{k}\_{\hat{r}} + 2\hat{k}\_{\hat{f}} + \phi \left(\hat{k}\_{\hat{f}} - \hat{k}\_{\hat{s}}\right)}} \left\{\frac{\left(PrF\beta^{\prime} - 2\lambda PrF^{\prime}\beta\right)}{\left(1 - \phi\right) + \frac{\phi \left(\hat{\mu}\_{\hat{f}}\right)\_{\hat{r}}}{\left(\hat{\mu}\_{\hat{f}} + \hat{k}\_{\hat{f}}\right)}}\right\}^{-1} + PrEcF^{\prime\prime}2\right). \end{aligned} \tag{22}$$

By implementing the transformations made in Equation (19), the system of Equations. (21, 22)

TABLE 2 | Reliability of the study by comparing with existing scientific literature for *F* ′′(0).


$$\text{Where, } \checkmark\_1 = \frac{-1}{1 + \frac{R\mathfrak{s}}{\hat{k}\_{\delta} + 2\hat{k}\_f - 2\phi\left(\hat{k}\_f - \hat{k}\_\delta\right)}}$$

$$\widetilde{A}\_2 = \frac{1}{\frac{\hat{k}\_s + 2\hat{k}\_f - 2\phi\left(\hat{k}\_f - \hat{k}\_s\right)}{\hat{k}\_s + 2\hat{k}\_f + \phi\left(\hat{k}\_f - \hat{k}\_s\right)}}$$

and the set of conditions at η = 0 are as:

$$
\begin{bmatrix}
\overbrace{\mathcal{V}\_1}^{\cdot} \\
\overbrace{\mathcal{V}\_2}^{\cdot} \\
\overbrace{\mathcal{V}\_3}^{\cdot} \\
\overbrace{\mathcal{V}\_4}^{\cdot} \\
\overbrace{\mathcal{V}\_5}^{\cdot}
\end{bmatrix} = \begin{bmatrix} 0 \\ \nu \\ \pi\_1 \\ 1 \\ \widetilde{\pi\_2} \end{bmatrix} \tag{24}
$$

transformed into the following initial value problem:

Here, n b <sup>1</sup> and n b <sup>2</sup> are unknown and the accuracy is set as 10<sup>−</sup> <sup>6</sup> .

$$
\begin{bmatrix}
\widehat{\boldsymbol{\gamma}^{\*}}\_{1} \\
\widehat{\boldsymbol{\gamma}^{\*}}\_{2} \\
\widehat{\boldsymbol{\gamma}^{\*}}\_{3} \\
\widehat{\boldsymbol{\gamma}^{\*}}\_{4} \\
\widehat{\boldsymbol{\gamma}^{\*}}\_{5}
\end{bmatrix} = \begin{bmatrix}
\widehat{\boldsymbol{\gamma}^{\*}}\_{2} \\
\frac{\boldsymbol{\gamma}^{\*}\_{2}}{\left(\widehat{\boldsymbol{\gamma}^{\*}\_{1}}\right)\left(\boldsymbol{\gamma}^{\*}\_{2}\right)} \\
+\underbrace{\left(\frac{\boldsymbol{\gamma}^{\*}\_{2}}{\left(\widehat{\boldsymbol{\gamma}^{\*}\_{1}}\right)^{-2}\right)}\left(\frac{\widehat{\boldsymbol{\gamma}^{\*}\_{2}}}{\left(1-\phi\right)^{-2}}\right)}\_{\left(1-\phi\right)^{-2}}M^{2}(1-\widehat{\boldsymbol{\gamma}^{\*}}\_{1})}\boldsymbol{M}^{2}(1-\widehat{\boldsymbol{\gamma}^{\*}}\_{2}) \\

## γAl2O<sup>3</sup> − H2O Model

The model γAl2O<sup>3</sup> − H2O reduced into the following pattern:

momentum of the fluid drops. Due to drop in momentum, the

$$F'' = -\frac{\left(1 - \phi + \frac{\phi \hat{\rho}\_b}{\hat{\rho}\_f}\right)}{123\phi^2 + 7.3\phi + 1} \left(\text{FF}'' + \lambda \left(1 - \text{F}' 2\right)\right) - \frac{\left(1 + \frac{3\left(\frac{\phi}{\hat{\rho}\_f} - 1\right)\phi}{\left(\frac{\phi}{\hat{\rho}\_f} + 2\right) - \left(\frac{\phi}{\hat{\rho}\_f} - 1\right)\phi}}{123\phi^2 + 7.3\phi + 1} M^2 (1 - \text{F}'), \tag{25}$$

$$\boldsymbol{\beta}^{''} = \frac{-1}{1 + \frac{Rd}{4.97\phi^2 + 2.72\phi + 1}} \left[ \frac{1}{4.97\phi^2 + 2.72\phi + 1} (\frac{\left(Pr\mathcal{F}\beta' - 2\lambda Pr\mathcal{F}'\beta\right)}{\left(1 - \phi\right) + \frac{\phi\left(\hat{\kappa}\epsilon\_0\right)\_i}{\left(\hat{\kappa}\epsilon\_0\right)\_f}})^{-1} + Pr\text{Ec}\mathcal{F}'\mathcal{Q}} \right]. \tag{26}$$

By using transformations, the following system is obtained:

 z}|{ y ′ 1 z}|{ y ′ 2 z}|{ y ′ 3 z}|{ y ′ 4 z}|{ y ′ 5 = z}|{ y <sup>2</sup> z}|{ y <sup>3</sup> − (1−φ+ φρs ρf ) z}|{ y <sup>1</sup> z}|{ y <sup>3</sup>+<sup>λ</sup> 1− z}|{ y <sup>2</sup> 2 !! 123φ <sup>2</sup>+7.3φ+1 − 1+ 3 σˆs σˆ f −1 ! φ σˆs σˆ f +2 ! − σˆs σˆ f −1 ! φ 123φ <sup>2</sup>+7.3φ+<sup>1</sup> <sup>M</sup><sup>2</sup> (1 − z}|{ y <sup>2</sup> ) z}|{ y <sup>5</sup> −A b 11 A b 12 Pr z}|{ y <sup>1</sup> z}|{ y <sup>5</sup>−2λPr z}|{ y <sup>2</sup> z}|{ y <sup>4</sup> ( (1−φ)+ <sup>φ</sup>(ρˆcp)<sup>s</sup> (ρˆcp)<sup>f</sup> )−<sup>1</sup> <sup>+</sup> Ec z}|{ y <sup>3</sup> 2 , (27) b 1

$$\text{where}, \overline{A}\_{11} = \frac{1}{1 + \frac{Rd}{4.97\phi^2 + 2.72\phi + 1}}$$

$$\overline{A}\_{12} = \frac{1}{4.97\phi^2 + 2.72\phi + 1} \tag{28}$$

The initial conditions are same as in Equation (24).

#### PHYSICAL INTERPRETATION OF RESULTS

The flow parameters like magnetic parameter, thermal radiation, and viscous dissipation play fascinating role in the flow regimes. The influences of afore mentioned flow parameters on the flow field explored graphically and discussed comprehensively in this section. Moreover, the results for the quantities of engineering interest are taken into account and discussed. The results plotted for two cases of the wedge according to the wedge movement. It is important to mention the cases of flow depending on the value of parameter γ . The wedge and fluid move in opposite direction for negative γ and move in alike direction for positive γ .

#### Velocity Field

**Figure 2** interprets the behavior of the nanofluids velocity (Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O) for γ > 0 and γ < 0, respectively. From **Figure 2A**, it is obvious that the velocity increases when the wedge and the nanofluids move in alike way (γ > 0). The velocity of Al2O<sup>3</sup> − H2O nanofluids increases abruptly in comparison with γAl2O<sup>3</sup> − H2O nanofluid. The reason behind this is the difference between the effective models of dynamic viscosity for Al2O<sup>3</sup> and γAl2O3. The nanofluid γAl2O<sup>3</sup> −H2O becomes more dense due to the dynamic viscosity containing high volume fraction of the nanoparticles and the velocity starts decreasing. Near the wedge surface, momentum of the nanofluids declines due to the friction between the wedge surface and the nanofluids. The effects of the pressure parameter are very prominent in the region 2 ≤ η ≤ 4. These effects are elaborated in **Figure 2A**. **Figure 2B** highlights the alterations in the nanofluids velocity for opposing case. The negative values of γ shows that the nanofluids and wedge move in opposite direction. For opposing case, the asymptotic region increases for γAl2O<sup>3</sup> − H2O nanofluid.

**Figure 3** elaborates the alterations in the velocity F ′ (η) for varying wedge parameter γ . Due to altering wedge parameter, very interesting variations in the velocity field are observed. For assisting case γ > 0, the velocity upturns abruptly. When the nanofluids and wedge move in alike direction then the movement of wedge in the direction of nanofluids provide extra momentum to the nanofluids. Therefore, the velocity positively increases. In the vicinity of the wedge, the velocity increase abruptly for both sort of nanofluids. For Al2O<sup>3</sup> − H2O nanofluid, the velocity shows asymptotic behavior quickly in comparison with γAl2O3− H2O. These results are plotted in **Figure 3A**. **Figure 3B** shows that the velocity of the nanofluids drops very quickly over the domain of interest. The opposite movement of the wedge and the nanofluids cause the declines in the velocity profile. Due to the opposite movement, friction between the wedge surface and the nanofluids slow down the momentum of the nanofluids. Consequently, the velocity drops.

The volume fraction of the nanoparticles is very key ingredients which alters the nanofluid characteristics affectively. These effects are portrayed in **Figures 4A,B** for assisting and opposing case, respectively. The behavior of the velocity for assisting case elaborated in **Figure 4A**. For increasing φ, the nanofluids velocity drops rapidly. For γAl2O<sup>3</sup> − H2O, the prompt decrement in the velocity occurs due to high volume fraction incorporated in the dynamic viscosity. The velocity vanishes asymptotically beyond η ≥ 4. **Figure 4B** portrays the velocity profile for opposing flow case. For opposing flow, the velocity declines very abruptly in comparison with assisting flow. The opposite movement and high volume fraction of the nanoparticles opposes the opposes the motion. Therefore, the velocity rapidly drops and asymptotically vanishes beyond η ≥ 6.

The influences in the velocity behavior by altering the magnetic parameter are depicted in **Figure 5**. From **Figure 5A**, it is observed that the velocity of the nanofluids F ′ (η) positively increases for assisting flow. Due to less dense composition of Al2O<sup>3</sup> − H2O nanofluid, the velocity profile increases promptly as compared to γAl2O<sup>3</sup> −H2O nanofluid. Similarly, for opposing case, the velocity field portrayed in **Figure 5B**. **Figure 5B** shows the prompt increasing behavior of the velocity for both sort of nanofluids. In the region 1 ≤ η ≤ 3.5, these effects of M on the velocity F ′ (η) are very rapid.

#### Temperature Field

The alterations in temperature fields of Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids by varying the wedge parameter γ presented in **Figure 6**. **Figure 6A** shows that for assisting flow, the temperature β(η) drops. The drops in the temperature is due to the alike motion of the nanofluids and wedge. For Al2O<sup>3</sup> − H2O nanofluids, decreasing pattern of the temperature is quite rapid and prominent in the region 1 ≤ η ≤ 3. The temperature β(η) vanishes asymptotically at the free stream. An interesting impacts of γ are observed for opposing flow case. These alterations are depicted in **Figure 6B**. When the wedge moves in the opposite direction of the nanofluids, then due to the force of friction between the wedge surface and the molecules of the nanofluids heat produces which favors the temperature β(η). The temperature increases abruptly near the wedge. The reason is that the more friction between the molecules of the nanofluids and the wedge surface. For γAl2O<sup>3</sup> −H2O nanofluids, the temperature arises rapidly than Al2O<sup>3</sup> − H2O nanofluid.

**Figure 7** highlights the influences of pressure parameter λ on the temperature of Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids. The pressure parameter λ opposes the nanofluids temperature. For assisting flow, the temperature is decreasing function of λ and the decrement in Al2O<sup>3</sup> − H2O nanofluid is rapid. Due to high dynamic viscosity of γAl2O<sup>3</sup> − H2O nanofluid, the temperature drops slowly than Al2O<sup>3</sup> − H2O nanofluid. It is observed that the temperature vanishes at the thermal boundary layer which starts beyond η > 3. These influences are shown in **Figure 7A**. It is investigated that the opposite motion of the wedge and nanofluids reduces the temperature very rapidly. Due to opposite motion, the velocity of the momentum drops. Consequently, the velocity declines which cause the rapid decrement in the temperature. The role of λ on β(η) is very cleared. Moreover, for opposing case, thermal boundary layer increases and the temperature decreases beyond η > 5. This behavior of the temperature is portrays in **Figure 7B**.

**Figures 8**–**10** depicted the behavior of temperature β(η) for volumetric fraction φ, Eckert number Ec and thermal radiation parameter Rd, respectively. The temperature patterns for both assisting and opposing case are plotted.

In the study of nanofluids, the importance of volume fraction cannot be neglected. The volume fraction alters the temperature effectively and plays vibrant role. The variations in the temperature for assisting and opposing flow due to altering φ are plotted in **Figures 8A,B**, respectively. From these, it is inspected that the volume fraction favors the temperature positively. For γ > 0, the increasing pattern of the temperature is quite slow than γ < 0. The main reason of this phenomena is the force of friction produces between the wedge surface and molecules of the nanofluids. For assisting case, thermal boundary layer decrease and in the case of opposing flow it starts increases and the temperature is vanishes beyond η > 3 and η > 4, respectively. For γAl2O<sup>3</sup> − H2O<sup>3</sup> the temperature β(η) increases very promptly due to the dynamic viscosities of Al2O<sup>3</sup> − H2O<sup>3</sup> and γAl2O<sup>3</sup> − H2O<sup>3</sup> nanofluids.

The effects of Eckert number which appears due to the viscous dissipation are plotted in **Figures 9A,B** for alike and opposing flow cases, respectively. The temperature varies almost inconsequentially for γ > 0 for both sort of nanofluids. On the other hand, it is inspected that for more dissipative nanofluids, the temperature β(η) arises rapidly. Thermal boundary layer decreases for Al2O<sup>3</sup> − H2O nanofluid and increases for γAl2O<sup>3</sup> − H2O nanofluid. From **Figures 10A,B**, it is obvious that thermal radiation parameter increases the nanofluids temperature for both the cases. For opposing case, the temperature β(η) arises quite rapid than alike flow case.

#### Skin Friction and Local Nusselt Number

This subsection highlights the behavior of skin friction and local Nusselt number for different values of the flow parameters for assisting and opposing flow. It is observed that the skin friction is directly proportional to the pressure parameter λ. For opposing flow, skin friction increases slowly in comparison with opposing flow case. These are depicted in **Figure 11**. **Figure 12** shows that the magnetic parameter M favors the skin friction for both assisting and opposing flows. For assisting case, it varies very slowly for assisting case while abrupt alterations are observed for opposing flow.

The alterations in local heat transfer coefficient (Nusselt number) for different parameters incorporating in **Figures 13**– **15** for alike and opposing flow cases. It is observed that heat transfer decreases at the wedge surface for alike flow case for radiative flow. The heat transfers for γAl2O<sup>3</sup> −H2O nanofluids is rapidly drops than Al2O<sup>3</sup> − H2O nanofluids. On the other hand, decrement in the heat transfer is observed for opposing flow. The heat transfer drops abruptly for γAl2O<sup>3</sup> − H2O nanofluid. These variations are portrayed in **Figure 14**. **Figure 15** depicts the influences of dissipation phenomena on the heat transfer. The heat transfer drops abruptly for more dissipative γAl2O<sup>3</sup> − H2O nanofluid and in Al2O<sup>3</sup> − H2O nanofluid, these effects are quite slow.

#### Thermophysical Properties

Thermophysical characteristics contribute vibrantly in the flow regimes of nanofluids. These properties effectively alter the nanofluid characteristics. The effects of volume fraction φ on the effective dynamic viscosity, thermal conductivity, electrical conductivity and density are plotted in **Figures 16**–**19**. From **Figure 16**, it is clear that the dynamic viscosity of γAl2O<sup>3</sup> − H2O increases exponentially. The for Al2O<sup>3</sup> − H2O nanofluids, these are very slow. Due to this improvement in the dynamic viscosity, the characteristics of the nanofluids affects. Similarly, thermal conductivity of γAl2O<sup>3</sup> − H2O nanofluid is quite rapid. However, no major difference between thermal conductivities of Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids is observed. Furthermore, electrical conductivity and density of the nanofluid arises by increasing the volume fraction in feasible domain. These are elaborated in **Figures 18**, **19**, respectively.

#### Comparison With Scientific Literature

**Table 2** elaborating the reliability of the presented results with existing scientific literature for F ′′(0). It is detected that by setting different physical parameters equal to zero, our results meets the existing scientific results in the literature that show the reliability of the presented physical results and applied numerical technique.

#### CONCLUSIONS

A novel radiative and dissipative study on Al2O<sup>3</sup> − H2O and γAl2O<sup>3</sup> − H2O nanofluids heat transfer model in the presence of applied magnetic field is considered over wedge. Two nanofluids models are obtained corresponding to two different sort of nanoparticles together with host liquid water. Then the models are treated mathematically by implementing RK scheme coupled with shooting method. Finally, the results for the flow regimes, heat transfer and thermophysical characteristics are plotted and found the following major outcomes:


#### REFERENCES


#### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

#### AUTHOR CONTRIBUTIONS

The formulation of the problem was done by UK, A and SM-D. Non-dimensionalization of the nanofluid models by using invertible transformations done by RM, IH, E-SS, and NA. Mathematical analysis and the graphical results plotted and discussed by SM-D and IK. The revision and editing was done by UK, E-SS, SM-D, and IK. All the authors have equal contributions.

#### ACKNOWLEDGMENTS

Researchers Supporting Project number (RSP-2019/33), King Saud University, Riyadh, Saudi Arabia.


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Ahmed, Adnan, Khan, Mohyud-Din, Khan, Murtaza, Hussain and Sherif. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


## NOMENCLATURE


# Novel Microstructural Features on Heat and Mass Transfer in Peristaltic Flow Through a Curved Channel

Raheel Ahmed1†, Nasir Ali 1†, Sami Ullah Khan2†, A. M. Rashad3†, Hossam A. Nabwey 4,5† and Iskander Tlili 6,7 \* †

*<sup>1</sup> Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan, <sup>2</sup> Department of Mathematics, COMSATS University Islamabad, Sahiwal, Pakistan, <sup>3</sup> Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt, <sup>4</sup> Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia, <sup>5</sup> Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom, Egypt, <sup>6</sup> Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>7</sup> Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam*

#### Edited by:

*Sara I. Abdelsalam, National Autonomous University of Mexico, Mexico*

#### Reviewed by:

*Abdullah Zaher, Benha University, Egypt Sohail Nadeem, Quaid-i-Azam University, Pakistan*

> \*Correspondence: *Iskander Tlili iskander.tlili@tdtu.edu.vn*

*†These authors have contributed equally to this work*

#### Specialty section:

*This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics*

Received: *10 March 2020* Accepted: *27 April 2020* Published: *09 June 2020*

#### Citation:

*Ahmed R, Ali N, Khan SU, Rashad AM, Nabwey HA and Tlili I (2020) Novel Microstructural Features on Heat and Mass Transfer in Peristaltic Flow Through a Curved Channel. Front. Phys. 8:178. doi: 10.3389/fphy.2020.00178*

Recently, significant interest has been developed by researchers toward the peristaltic transport of fluid, as this phenomenon involves a variety of applications in the biomechanics, bioengineering, and biomedical industries. In the present contribution, we investigate the effect of heat and mass transfer on magnetically influenced micropolar flow induced by peristaltic waves. The fundamental laws regarding current flow problem are employed by using curvilinear coordinates. A reduction of these equations is made based on lubrication approximation. The solution algorithm is based on the implementation of the famous finite difference method. The fundamental impacts of coupling number, micropolar parameter, Hartmann number, Brinkman number, rate of chemical reaction, and curvature parameter on longitudinal velocity, pressure rise, temperature, and mass concentration are analyzed in detail. The flow patterns in the channel illustrating the effects of several involved parameters are also displayed.

Keywords: micropolar fluid, hartmann number, heat and mass transfer, curved channel, implicit finite difference method

### INTRODUCTION

The theory of fluids has gained the attention of scientists, engineers, biologists, and mathematicians in recent times. Generally, fluids are categorized as Newtonian or non-Newtonian. Newtonian fluids are those in which viscous stresses sustain a linear relationship between strain rates at every point. The viscous fluids are referred to as a simple linear model that reports the viscosity. Examples of Newtonian fluids are water, glycerol, alcohol, thin motor oil, and air. Another class of fluids is defined as a fluid which fails to follow Newton's viscosity model. A number of fluids are non-Newtonian in nature. Examples are custard, ketchup, shampoo, starch, paint, blood, and suspension. Recently, many researchers have been concentrating on the flows of non-Newtonian fluids. This is due to the applications of non-Newtonian fluids in polymer processing, biofluid mechanics, and complex mathematical non-linear constitutive equations.

**178**

The motivating implication of the Peristalsis phenomenon has attained a valuable attraction of scientists in the last few years as it involves fundamental industrial and bioscience applications. Several investigations are reported regarding the peristaltic transport of non-Newtonian materials which encountered many physiology applications. Raju and Devanathan [1] worked on the peristaltic study of the power law model configured by a tube, where it was assumed that a sinusoidal with lower amplitude traveled down along the channel wall. Mekheimer [2] studied the transport of magnetohydrodynamic viscous and incompressible peristaltic flow in an inclined planar channel. Hakeem et al. [3] modeled the peristaltic flow motion equations for Carreau fluid by using long wavelength assumptions in a uniform tube. The peristaltic study of Johnson–Segalman liquid influenced by magnetic force in a 2D flexible channel has been depicted by Elshahed and Haroun [4]. Hakeem et al. [5] examined the significance of magnetic force in trapping evolved regarding the peristaltic flow generalized viscous fluid. The rheological justification of non-Newtonian Burger's fluid due to peristaltic movement encountered by a planar channel was reported by Hayat et al. [6]. The reported flow model was based on long wavelength theory, and later on, an exact solution is developed for a formulated problem. The theoretical model developed by Haroun [7] signified the rheological consequences in third-order liquid for a peristaltic phenomenon configured by the asymmetric channel. Wang et al. [8] studied the magnetohydrodynamic peristaltic motion of a Sisko fluid in a symmetric or asymmetric channel. An investigation with peristaltic aspects for third-grade fluid encountered by a circular cylindrical tube was performed by Ali et al. [9]. The peristaltic mechanism of Prandtl–Eyring fluid along with heat transfer features in a curved channel was evaluated by Hayat et al. [10]. Rafiq et al. [11] investigated ion-slip and Hall features in the peristaltic flow of nanoparticles with biomedical applications. Ahmed and Javed [12] used finite element technique to model Navier–Stokes expressions for the peristaltic phenomenon in the presence of a porous medium. The influence of electromagnetic features in the peristaltic flow of Eyring-Powell nanofluid. Asha

and Sunitha [13] involved the Hall effects in peristaltic transport of nanoparticles in the asymmetric channel.

Non-Newtonian fluids are characterized through a number of models due to their complexity. Among these models, a micropolar fluid model has been gaining attention of a number of researchers. In this model, stiff particles cramped into a relatively minute-volume element are able to move about the center of the element. The rotational attribute of fluid particles is carried out through a vector called micro-rotation expressions. It is not worth remarking that those intrinsic rotation features of the fluid are associated with rigid body movement for a whole-volume element which depends upon various factors. Such features are referred to the micro-rotational effects, and the role of these factors at macro-scale cannot be taken into account. However, such effects become important when flow is considered in narrow gaps, i.e., when geometric dimensions of the flow domain are very small [14]. In peristaltic flows, fluid is usually pushed through nano-size vessels, and therefore, it is expected that microrotation of particles reflects some diverse and distinguished flow characteristics. Important effects cannot be captured by the Navier–Stokes theory. The fundamental work on the micropolar fluid theory was first introduced by Eringen [15, 16] to describe the suspensions of neutrally buoyant rigid particles in a viscous fluid. The work on micropolar fluid was initiated by Eringen by involving the micro-rotation features in the classical Navier– Stokes theory. Ariman et al. [17] studied the application of micro-continuum fluid mechanics in a broader prospect. The assessment of a boundary layer for the micropolar fluid has been successfully examined by Na and Pop [18]. Srinivasacharya et al. [19] reported a closed-form relation for peristaltic aspects of micropolar fluid flow due to a circular tube. The model problem was based on the famous long wavelength and small Reynolds number assumptions. The peristaltic movement under the influence of wall properties in 2D flow of micropolar liquid has been revealed by Muthu et al. [20]. Lok et al. [21] investigated the steady mixed convective due to vertical moving geometry for flow of micropolar fluid. Hayat et al. [22] examined the impact of different waveforms in a peristaltic flow of a micropolar fluid. The flow of micropolar liquid followed by a peristaltic pattern in the asymmetric channel has been focused by Ali and Hayat [23]. Another continuation made by Hayat and Ali [24] reported the endoscope consequences for micropolar fluid flow in a concentric tube. Another peristaltic phenomenon based on an exploration for micropolar liquid with implementation of magnetic field impact was estimated by Mekheimer [25]. Ishak et al. [26] studied the magnetohydrodynamic flow of a micropolar fluid toward a stagnation point on a vertical surface. Mekheimer and El Kot [27] used the micropolar fluid model for blood flow through a stenosed tapered artery. Sajid et al. [28] employed the homotopy analysis method to discuss boundary layer flow micropolar fluid through a porous channel. Ashraf et al. [29] examined numerical solutions portraying the appliance of micropolar material in a channel having porous walls. Rashidi et al. [30] applied the differential transform method to get a semi-analytical solution of micropolar flow in a porous channel with mass injection. Ali et al. [31] comprehensively studied the peristaltic flow of a micropolar fluid in a curved channel. The unsteady peristaltic prospective

for 2D channel flow of micropolar fluid in the presence of heat and mass transportation has been examined by El-Dabe and Zeid [32]. The investigation for micropolar fluid in contacting a wall channel additionally featuring an isotropic porous space was assessed by Abd Elmaboud [33]. Sui et al. [34] reported a constitutive diffusion model to investigate the heat transfer performances in micropolar fluid encountered by a moving geometry. Waqas et al. [35] numerically predicted the join

FIGURE 2 | (A,B) Impact of *Ha* on *u* (η) with γ = 2.5, *N*<sup>1</sup> = 0.5, *N*<sup>2</sup> = 1.2, λ = 0.4, and 2 = 1.5. (B) Impact of *Ha* on *w* (η) with γ = 2.5, *N*<sup>1</sup> = 0.5, *N*<sup>2</sup> = 1.2, λ = 0.4, and 2 = 1.5.

FIGURE 3 | (A,B) Impact of *N*<sup>1</sup> on *u* (η) with γ = 2.5, *Ha* = 2, *N*<sup>2</sup> = 0.2, λ = 0.4, and 2 = 1.5. (B) Impact of *N*<sup>1</sup> on *w* (η) with γ = 2.5, *N*<sup>2</sup> = 0.2, λ = 0.4, and 2 = 1.5.

features of Maxwell viscoelasticity–based nanofluid additionally featuring a porous medium. In another contribution, Waqas and co-workers [36] utilized the bioconvection phenomenon in the flow of micropolar nanofluid with additional thermal radiation features. Ali et al. [37] intended the micropolar liquid rheological significance compiled in calendaring geometry. Ahmed et al. [38] examined the effects of heat and mass transfer on peristaltic flow of Sisko fluid through a curved channel. Mekheimer et al.

FIGURE 5 | (A) Impact of *Ha* on 1*p* with *N*<sup>1</sup> = 0.5,*N*<sup>2</sup> = 1.2, γ = 2.5, and λ = 0.4. (B) Impact of *N*<sup>1</sup> on 1*p* with γ = 2.5, *N*<sup>2</sup> = 1.2, and λ = 0.4. (C) Impact of *N*<sup>2</sup> on 1*p* with *N*<sup>1</sup> = 0.5, γ = 2.5, and λ = 0.4.

[39] studied the effect of gold nanoparticle third-grade fluid on peristaltic blood flow. Elkhair et al. [40] considered the impact of heat transfer on oscillatory flow of a dielectric fluid through a porous medium. Recently, Mekheimer et al. [41] investigated the behavior of a blood confined by stenotic arterial walls. In another useful attempt, Mekheimer et al. [42] performed the features of heat transfer additionally featuring AC current. Nadeem et al. [43, 44] studied hybrid-based nanofluid flow over a curved surface in different scenarios. Abbas et al. [45] observed transportation of micropolar hybrid nanomaterial which was externally impacted by magnetic influence. Sadaf et al. [46] discussed the effect of heat transfer on fluid motion generated by cilia and a pressure gradient in a curved channel. In a most recent study, Nadeem et al. [47] investigated the effect of heat transfer on micropolar fluid flow over a Riga plate. Some more recent investigations on this topic are seen in references [48–50].

From the literature cited above, it is noted that the hydrodynamic flow of micropolar fluid through a curved channel with peristalsis is studied but less attention is paid to hydromagnetic aspects of micropolar liquid along with heat and mass transportation aspects. The prime objective of this study is to investigate the effects of coupling number, micropolar parameter, Hartmann number, Brinkman number, and dimensionless radius of curvature on flow, heat, and mass transfer characteristics. To this end, the associated equations for velocity, temperature, and mass concentration are constituted. The modeled system is numerically interpolated with assistance of finite difference scheme. The fluid velocity, temperature, and concentration fields are analyzed for several values of the involved parameters. It is important to mention that governing equations for heat and mass transfer for the flow of micropolar fluid in a curved peristaltic channel are derived for the first time in the literature.

### GOVERNING EQUATIONS

For micropolar fluid, the mathematical expressions in presence of heat/mass influences are given by [37].

Continuity equation:

$$U\_{i, \dot{i}} = \mathbf{0} \tag{1}$$

Momentum equation:

$$
\rho \, \dot{U}\_k = \mathfrak{r}\_{lk,l} + \rho f\_k \tag{2}
$$

Moment of momentum equation:

$$
\rho j \dot{w}\_k = m\_{lk,l} + e\_{kij} r\_{ij} \tag{3}
$$

Energy equation:

$$
\rho c\_p \dot{T} = kT\_{,ii} + \tau\_{kl} a\_{kl} - m\_{kl} b\_{kl} \tag{4}
$$

*N*<sup>2</sup> with *Br* = 2, *N*<sup>1</sup> = 0.5, λ = 0.4, and γ = 2.

Concentration equation:

$$
\dot{C} = DC\_{\text{,}i} + \frac{Dk\_T}{T\_m} T\_{\text{,}i} - k\_1 C \tag{5}
$$

In the above equations, U<sup>k</sup> is the velocity, C is the mass concentration, f<sup>k</sup> is the body force, T is the symbolized temperature, τkl is the Cauchy stress tensor, mkl is the moment stress tensor, p is the pressure, ρ is the fluid density, w<sup>k</sup> is the micro-rotation vector, c<sup>p</sup> is the specific heat at constant pressure, D is the coefficient of mass diffusivity, K<sup>T</sup> is the thermal diffusivity, T<sup>m</sup> is the mean temperature, k<sup>1</sup> is the rate of chemical reaction, k is the thermal conductivity, j is the micro moment of inertia, and dot indicates the material time derivative. Moreover, τkl, mkl, akl and bkl are given by

$$\begin{aligned} \tau\_{kl} &= -p\delta\_{kl} + \left(\mu + k\_2\right)a\_{kl} + \mu a\_{lk}, \\ m\_{kl} &= \alpha \operatorname{tr}(b\_{mm})\delta\_{kl} + \beta b\_{kl} + \gamma^\* b\_{lk}, \\ a\_{kl} &= \nu\_{l,k} + \varepsilon\_{lkm} \omega\_m, \\ b\_{kl} &= \omega\_{k,l}, \end{aligned} \tag{6}$$

where µ is the viscosity, k<sup>2</sup> is the dynamic micro-rotation viscosity, elkm is the permutation symbol, and α, β, γ ∗ are the constants called coefficient of angular viscosity. It is remarked that Equation (2) has been diminished into a Navier–Stokes expression when k<sup>2</sup> = α = β = γ <sup>∗</sup> = 0. It is further emphasized that if k<sup>2</sup> = 0, both micro-rotation and velocity are unyoked and micro-rotation does not play to alter the global motion. Following Eringen [51], the following relations hold for µ, k2,α, β, and γ ∗

$$2\mu + k\_2 \ge 0,\\ k\_2 \ge 0,\\ 3\alpha + \beta + \gamma^\* \ge 0,\\ \alpha \ge |\beta|\dots$$

#### MATHEMATICAL MODELING

Consider a curved channel of width 2w coiled in circle having radius R<sup>0</sup> and center O. An incompressible micropolar fluid flows inside the channel. The fluid flows due to the wall of the channel which deforms uniformly. Let T0, T1, C0, and C<sup>1</sup> represent the upper wall temperature, lower wall temperature, upper

*N*<sup>2</sup> = 1.2,λ = 0.4, and γ = 2.

wall concentration, and lower wall concentration, respectively. The fluid movement is described by following the curvilinear coordinate system (R, χ, Z). It is emphasized that χ is specified in the flow direction and R is radially oriented while Z is assumed normal to the plane. The flow visualization for the current problem can be described by sketching **Figure 1**. The shape of both walls is described mathematically as [31, 38]

$$\begin{aligned} H\_1 \left( \chi, t \right) &= \quad \le +a \sin \left( \left( \frac{2\pi}{\lambda^\*} \right) (\chi - ct) \right), \quad &\text{Upper wall}, \text{(7)},\\ H\_2 \left( \chi, t \right) &= -\omega - a \sin \left( \left( \frac{2\pi}{\lambda^\*} \right) (\chi - ct) \right), \quad &\text{Lower wall}, \text{(8)}.\end{aligned}$$

where λ ∗ is the wavelength, c is the wave speed, a is the amplitude, and t is the time. The present work is based on the following assumptions:

where B ∗ is the limiting value of B when R˜ → ∞. Thus, by generalizing Ohm's law, the body force term in Equation (2) becomes

$$
\rho f\_k = (\mathbb{J} \times B)\_{\mathbb{k}^\*}
$$

where J = σ(V × B). Here, we neglect the electric field and invoke the low magnetic Reynolds number assumption. Using the velocity, temperature, concentration, and micro-rotation fields defined by

$$\begin{aligned} \mathbf{U} &= \left[ U\_1 \left( \boldsymbol{\chi}, \boldsymbol{R}, t \right), U\_2 \left( \boldsymbol{\chi}, \boldsymbol{R}, t \right), 0 \right], \; T = T \left( \boldsymbol{\chi}, \boldsymbol{R}, t \right), \\ \mathbf{C} &= \left[ C \left( \boldsymbol{\chi}, \boldsymbol{R}, t \right), \mathbf{w} = \left[ 0, 0, -\boldsymbol{w} \left( \boldsymbol{\chi}, r \right) \right] \end{aligned} \right] $$

the set of Equations (1)–(5) in component form becomes [31, 38]

$$
\frac{\partial}{\partial R} \left\{ \left( R + \bar{R} \right) U\_1 \right\} + \bar{R} \frac{\partial U\_2}{\partial \chi} = 0,\tag{9}
$$

$$\frac{\partial U\_1}{\partial t} + (U\_1 \cdot \nabla) U\_1 - \frac{U\_2^2}{R + \tilde{R}} = -\frac{1}{\rho} \frac{\partial P}{\partial R} + \frac{1}{\rho} \left(\mu + k\_2\right) \left[\nabla^2 U\_1 - \frac{U\_1}{\left(R + \tilde{R}\right)^2} - \frac{2\tilde{R}}{\left(R + \tilde{R}\right)^2} \frac{\partial U\_2}{\partial \chi}\right] + \frac{k\_2 \tilde{R}}{\rho \left(R + \tilde{R}\right)} \frac{\partial w}{\partial \chi},\tag{10}$$

$$\frac{\partial U\_{2}}{\partial t} + (\text{U}\_{1}\nabla)U\_{2} + \frac{U\_{1}U\_{2}}{\mathbb{R} + \bar{\mathbb{R}}} = -\frac{\bar{\mathbb{R}}}{\rho \left(\mathbb{R} + \bar{\mathbb{R}}\right)} \frac{\partial P}{\partial \chi} + \frac{1}{\rho} \left(\mu + kz\right) \left[\nabla^{2}U\_{2} - \frac{U\_{2}}{\left(\mathbb{R} + \bar{\mathbb{R}}\right)^{2}} + \frac{2\bar{\mathbb{R}}}{\left(\mathbb{R} + \bar{\mathbb{R}}\right)^{2}} \frac{\partial U\_{1}}{\partial \chi}\right] - \frac{k\_{2}}{\rho} \frac{\partial \mathbf{w}}{\partial \mathbf{R}} - \frac{\sigma B^{2} \bar{\mathbb{R}}^{2}}{\rho \left(\mathbb{R} + \bar{\mathbb{R}}\right)^{2}} U\_{2}. \tag{11}$$

$$(U\_1 \nabla) \le -\frac{\mathcal{V}}{\rho j} \left[ \frac{\partial^2 w}{\partial R^2} + \frac{1}{R + \bar{R}} \frac{\partial w}{\partial R} + \left( \frac{\bar{R}}{R + \bar{R}} \right)^2 \frac{\partial^2 w}{\partial \chi^2} \right] + \frac{k\_2}{\rho j} \left[ 2w - \frac{\bar{R}}{R + \bar{R}} \frac{\partial U\_1}{\partial \chi} + \frac{\partial U\_2}{\partial R} + \frac{U\_2}{R + \bar{R}} \right]. \tag{12}$$
 
$$\rho \mathcal{L}\_p \left( \frac{\partial T}{\partial t} + U\_1 \frac{\partial T}{\partial R} + \frac{U\_2 \bar{R}}{R + \bar{R}} \frac{\partial T}{\partial \chi} \right) = k \left( \frac{\partial^2 T}{\partial R^2} + \frac{1}{R + \bar{R}} \frac{\partial T}{\partial R} + \frac{\bar{R}^2}{\left( R + \bar{R} \right)^2} \frac{\partial^2 T}{\partial \chi^2} \right) + \frac{\partial U\_1}{\partial R} \left( -p + 2\mu \frac{\partial U\_1}{\partial R} + k\_2 \frac{\partial U\_1}{\partial R} \right)$$

 R + R˜ 2 + R˜ R + R˜ ∂U<sup>1</sup> ∂χ − U2 R + R˜ − w ! µR˜ R + R˜ ∂U<sup>1</sup> ∂χ + µ ∂U<sup>2</sup> ∂R − µU<sup>2</sup> R + R˜ + kR˜ R + R˜ ∂U<sup>1</sup> ∂χ − kU<sup>2</sup> R + R˜ − k2w ! + ∂U<sup>2</sup> ∂R + w µR˜ R + R˜ ∂U<sup>1</sup> ∂χ + µ ∂U<sup>2</sup> ∂R − µU<sup>2</sup> R + R˜ + k<sup>2</sup> ∂U<sup>2</sup> ∂R + k2w ! + R˜ R + R˜ ∂U<sup>2</sup> ∂χ + U1 R + R˜ ! −<sup>p</sup> <sup>+</sup> 2µR˜ R + R˜ ∂U<sup>2</sup> ∂χ + 2µU<sup>1</sup> R + R˜ + kR˜ R + R˜ ∂U<sup>2</sup> ∂χ + kU<sup>1</sup> R + R˜ ! +γ ∗ ∂w ∂R 2 + R˜ R + R˜ ∂w ∂χ !2 , (13)

$$\frac{\partial \mathcal{C}}{\partial t} + (U\_1, \nabla) \mathcal{C} = D \nabla^2 \mathcal{C} + \frac{D K\_T}{T\_m} \left( \frac{\partial^2 T}{\partial R^2} + \frac{\tilde{R}}{R + \tilde{R}} \frac{\partial T}{\partial R} + \left( \frac{\tilde{R}}{R + \tilde{R}} \right)^2 \frac{\partial^2 T}{\partial \chi^2} \right) - k\_1 \mathcal{C},\tag{14}$$


It is further assumed that the flow is subjected to the radial magnetic field of the form

$$\mathbf{B} = \left(\frac{B^\*\tilde{R}}{R+\tilde{R}}\right)\mathbf{e}\_{R\*}$$

where

$$U\_1 \cdot \nabla = U\_1 \frac{\partial}{\partial R} + \frac{\tilde{R} U\_2}{R + \tilde{R}} \frac{\partial}{\partial \chi},\tag{15}$$

and

$$\nabla^2 = \frac{1}{R + \tilde{R}} \frac{\partial}{\partial R} \left\{ \left( R + \tilde{R} \right) \frac{\partial}{\partial R} \right\} + \left( \frac{\tilde{R}}{R + \tilde{R}} \right)^2 \frac{\partial^2}{\partial \chi^2} \tag{16}$$

The boundary conditions associated with Equations (9)–(14) are [8]

$$U\_2 = \begin{array}{c} \text{0, } U\_1 = \ \frac{\partial H\_1}{\partial t}, \; w = 0, T = T\_o, \; C = C\_o \text{ at } R = H\_1(\chi, t), \; \text{(17)} \\\ U\_2 = \ \text{0, } U\_1 = \ \frac{\partial H\_2}{\partial t}, \; w = 0, T = T\_1, \; C = C\_1 \text{at} \; R = H\_2(\chi, t). \; \text{(18)} \end{array}$$

0) reduces to

The following transformations are suggested to transmute the fixed wave from (R, χ) to new wave from (r, x)

$$x = \chi - c \ t, \ r = \text{ } \text{R, } p = P, \ u\_1 = \text{ } U\_1, \ u\_2 = \text{ } U\_2 - c. \text{ (19)}$$

The governing flow equation is transmuted into a wave frame while defining the following appropriate variables [31, 38]:

$$\begin{aligned} \bar{\boldsymbol{\chi}} &= \, ^{2\pi \mathbf{x}} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \; \boldsymbol{\eta} = \, ^{\prime} \acute{\boldsymbol{\uptriangleleft}}\_{a}, \; \boldsymbol{\upgamma} = \frac{\ddot{\boldsymbol{R}}}{a}, \; \bar{\boldsymbol{u}}\_{1} = \, ^{\mathrm{u}\_{1}} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \bar{\boldsymbol{u}}\_{2} = \, ^{\mathrm{u}\_{2}} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \\ \bar{\boldsymbol{w}} &= \, ^{\mathrm{a} \mathbf{w}} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \; \bar{\boldsymbol{p}} = \, ^{2\pi a^{2} \mathfrak{p}} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \; \mathrm{Re} = \, ^{\rho \mathbf{c} a} \acute{\boldsymbol{\uptriangleleft}}\_{\mu}, \; \theta = \, ^{(T-T\_{1})} \acute{\boldsymbol{\uptriangleleft}}\_{(T\_{0}-T\_{1})}, \\ \boldsymbol{\uptriangleleft} &= \, ^{(C-\mathrm{C}\_{1})} \acute{\boldsymbol{\uptriangleleft}}\_{(\mathrm{C}\_{0}-\mathrm{C}\_{1})}, \; \delta = \, ^{2\pi a} \acute{\boldsymbol{\uptriangleleft}}\_{\*}, \bar{\mathbf{j}} = \, ^{\circ} \acute{\boldsymbol{\uptriangleleft}}\_{a}, \; N\_{1} = ^{k\_{2}} \acute{\boldsymbol{\uptriangleleft}}\_{\mu}, \; N\_{2} = ^{\nu^{\prime}} \acute{\boldsymbol{\uptriangleleft}}\_{a^{2} \mu} \end{aligned}$$

Moreover, invoking the lubrication approximations (δ ≈ 0, Re ≈

The transmuted boundary assumptions (17)–(18) are

$$\begin{aligned} \psi &= -\frac{q}{2}, \,\frac{\partial \psi}{\partial \eta} = 1, \,\, w = 0, \,\theta = 0, \,\, \phi = 0, \,\, \text{at} \\ \eta &= h\_1 = 1 + \lambda \text{sin}\chi, \\ \psi &= \frac{q}{2}, \,\, \frac{\partial \psi}{\partial \eta} = 1, \,\, w = 0, \,\, \theta = 1, \,\, \phi = 1, \,\, \text{at} \\ \eta &= \,\, h\_2 = -1 - \lambda \text{sin}\chi. \end{aligned} \tag{28}$$

In the above set of equations, λ = <sup>a</sup><sup>w</sup> symbolizes the amplitude ratio. Our objective is to compute the solution of Equations (22)–(24) and (26) subject to boundary conditions (27) and (28).

We summarized the following relations for pressure rise per wavelength 1p , heat transfer coefficients zi(i = 1, 2), and expressions for Sherwood number Shi(i = 1, 2) at the upper and lower wall surfaces in the following forms:

$$
\Delta p = \int\_0^{2\pi} \frac{dp}{d\mathbf{x}} d\mathbf{x},\tag{29}
$$

$$\begin{aligned} \frac{\partial p}{\partial \eta} &= 0, \\\\ \frac{\partial p}{\partial \eta} &= 1, \quad \Gamma \partial \eta \{ \eta \} &= \partial^2 \eta \{ \eta \} + 1, \quad \left( \begin{array}{ccccc} \partial \eta \end{array} \right) \left( \begin{array}{c} \partial \eta \end{array} \right) \left( \begin{array}{c} \partial \eta \end{array} \right) \left( \begin{array}{c} \partial \eta \end{array} \right) \end{aligned} \tag{20}$$

$$-\frac{\partial p}{\partial x} - \frac{1}{\chi (1 - N\_1)} \left[ \frac{\partial}{\partial \eta} \left\{ (\eta + \chi) \frac{\partial^2 \psi}{\partial \eta^2} \right\} + \frac{1}{\eta + \chi} \left( 1 - \frac{\partial \psi}{\partial \eta} \right) - N\_1 \left( \eta + \chi \right) \frac{\partial w}{\partial \eta} \right] - \frac{\chi H a^2}{\eta + \chi} \left( 1 - \frac{\partial \psi}{\partial \eta} \right) = 0,\tag{21}$$

$$
\left(\frac{2 - N\_1}{N\_2}\right) \left[\frac{\partial^2 w}{\partial \eta^2} + \frac{1}{\eta + \chi} \frac{\partial w}{\partial \eta}\right] - 2w + \frac{\partial^2 \psi}{\partial \eta^2} - \frac{1}{\eta + \chi} \left(1 - \frac{\partial \psi}{\partial \eta}\right) = 0,\tag{22}
$$

$$\begin{aligned} \frac{\partial^2 \theta}{\partial \eta^2} + \frac{1}{\left(\eta + \chi\right)} \frac{\partial \theta}{\partial \eta} + Br \left(\frac{1}{\eta + \chi} \left(1 - \frac{\partial \psi}{\partial \eta}\right) + \mathbf{w}\right) \\ \left(\frac{\partial^2 \psi}{\partial \eta^2} + \frac{1}{\eta + \chi} \left(1 - \frac{\partial \psi}{\partial \eta}\right) (1 + N\_1) + N\_1 \mathbf{w}\right) + \\ \dots \end{aligned} \tag{23}$$

$$\begin{split} & \left( \partial^{\eta} \eta + \frac{\eta + \gamma}{\beta \eta^{2}} \right) \left( \frac{\partial^{2} \psi}{\partial \eta^{2}} + \frac{1}{\eta + \gamma} \left( 1 - \frac{\partial \psi}{\partial \eta} \right)^{\prime} + N\_{1} \left( \frac{\partial^{2} \psi}{\partial \eta^{2}} - w \right) \right) + N\_{2} \text{Br} \left( \frac{\partial w}{\partial \eta} \right)^{2} = 0, \\ & \frac{\partial^{2} \phi}{\partial \eta^{2}} + \frac{1}{(\eta + \gamma)} \frac{\partial \phi}{\partial \eta} - R\_{\ell} \phi = -\text{Scfc} \left( \frac{\partial^{2} \theta}{\partial \eta^{2}} + \frac{1}{(\eta + \gamma)} \frac{\partial \theta}{\partial \eta} \right) + R\_{\ell}. \tag{24} \end{split} \tag{25}$$

In the above equations N1, N2, Re, δ, γ , K ∗ , and R<sup>c</sup> represent the coupling number, micropolar constant, Reynolds number, wave number, radius of curvature, dimensionless permeability parameter, and dimensionless rate of the chemical reaction parameter, respectively. Here, the coupling number presents the coupling between the vortex viscosity and shear viscosity coefficients, while micropolar parameter is the ratio between the coefficient of angular viscosity and the shear viscosity coefficient. Further, the stream function ψ and velocity components u<sup>1</sup> and u<sup>2</sup> are related through the expressions

$$
\mu\_1 \, \, \, = \, \, \delta \frac{\nu}{\eta + \nu} \frac{\partial \, \psi}{\partial \mathbf{x}}, \, \, \, \, \nu\_2 \, \, \, = -\frac{\partial \, \psi}{\partial \eta}. \tag{25}
$$

$$\begin{array}{ll}\text{Combining Equation (20) with Equation (21), one gets} \\ \frac{1}{1-N\_{1}} \left[ \frac{\partial^{2}}{\partial \eta^{2}} \left\{ (\eta+\boldsymbol{\gamma}) \frac{\partial^{2} \boldsymbol{\psi}}{\partial \eta^{2}} \right\} + \frac{\partial}{\partial \boldsymbol{\eta}} \left( \frac{1}{\eta+\boldsymbol{\gamma}} \left( 1 - \frac{\partial \boldsymbol{\psi}}{\partial \eta} \right) \right) \right] & - & N\_{1} \frac{\partial}{\partial \boldsymbol{\eta}} \\ \left( (\boldsymbol{\eta}+\boldsymbol{\gamma}) \frac{\partial \boldsymbol{w}}{\partial \eta} \right) \left[ -\frac{\boldsymbol{\nu}^{2} H a^{2}}{\boldsymbol{\eta}+\boldsymbol{\gamma}} \left( 1 - \frac{\partial \boldsymbol{\psi}}{\partial \eta} \right) = 0. & \end{array}$$

$$z\_i = \left. \frac{\partial h\_i}{\partial \mathbf{x}} \frac{\partial \theta}{\partial \eta} \right|\_{\eta = h\_i}, \ i = 1, 2. \tag{30}$$

$$Sh = \left. \frac{\partial h\_i}{\partial \mathbf{x}} \frac{\partial \phi}{\partial \eta} \right|\_{\eta = h\_i}. \quad i = 1, 2. \tag{31}$$

#### NUMERICAL SOLUTION

In this, we numerically address the solution procedure of Equations (27), (28), and (30) subject to boundary conditions given in Equations (31) and (32). On this end, we adopted a famous finite difference procedure to perform such numerical simulations [52–54]. According to this method, simulations are performed by the following steps:

(i) As a first step, an iterative procedure has been compiled by transmuting non-linear flow equations into linear equations at the (m + 1)th iterative step. Adopting such iterative process, we get

$$\begin{split} &\frac{\partial^{2}\psi^{(m+1)}}{\partial\eta^{4}} + \frac{2}{\left(\eta+\boldsymbol{\gamma}\right)}\frac{\partial^{3}\psi^{(m+1)}}{\partial\eta^{3}} - \frac{1}{\left(\eta+\boldsymbol{\gamma}\right)^{2}}\frac{\partial^{2}\psi^{(m+1)}}{\partial\eta^{2}} + \left\{\frac{1}{\left(\eta+\boldsymbol{\gamma}\right)^{3}} + \frac{\boldsymbol{\gamma}^{2}H\mathbf{d}^{2}\left(1-\mathbf{N}\_{1}\right)}{\left(\eta+\boldsymbol{\gamma}\right)^{2}}\right\}\frac{\partial\psi^{(m+1)}}{\partial\eta} \\ &- \mathcal{N}\frac{\partial^{2}\eta^{(m+1)}}{\partial\eta^{2}} - \frac{\mathcal{N}\_{1}}{\left(\eta+\boldsymbol{\gamma}\right)}\frac{\partial\mathbf{w}^{(m+1)}}{\partial\eta} - \frac{1}{\left(\eta+\boldsymbol{\gamma}\right)^{3}} - \frac{\mathcal{N}^{2}H\mathbf{d}^{2}\left(1-\mathbf{N}\_{1}\right)}{\left(\eta+\boldsymbol{\gamma}\right)^{2}} = 0. \tag{32} \\ &\frac{\partial^{2}\theta^{(m+1)}}{\partial\eta^{2}} + \frac{1}{\left(\eta+\boldsymbol{\gamma}\right)}\frac{\partial\theta^{(m+1)}}{\partial\eta} + \mathcal{R}\left(\frac{1}{\eta+\boldsymbol{\gamma}}\left(1-\frac{\partial\boldsymbol{\gamma}^{(m)}}{\partial\eta}\right) + \nu^{(m)}\right)\left(\frac{\partial^{2}\psi^{(m)}}{\partial\eta^{2}} + \frac{1}{\eta+\boldsymbol{\gamma}}\left(1-\frac{\partial\boldsymbol{\gamma}^{(m)}}{\partial\eta}\right)(1+N\_{1}) + N\_{1}\nu^{(m)}\right) + \end{split} \tag{32}$$

$$\begin{split} \mathrm{Br}\left(\frac{\partial^{2}\psi^{(m)}}{\partial\eta^{2}} - w^{(m)}\right) & \left(\frac{\partial^{2}\psi^{(m)}}{\partial\eta^{2}} + \frac{1}{\eta+\chi}\left(1 - \frac{\partial\psi^{(m)}}{\partial\eta}\right) + N\_{1}\left(\frac{\partial^{2}\psi^{(m)}}{\partial\eta^{2}} - w^{(m)}\right)\right) + N\_{2}\mathrm{Br}\left(\frac{\partial w^{(m)}}{\partial\eta}\right)^{2} = 0, \end{split} \tag{33}$$

$$\frac{\partial^2 \phi^{(m+1)} }{\partial \eta^2} + \frac{1}{\eta + \chi} \frac{\partial \phi^{(m+1)}}{\partial \eta} - R\_c \phi^{(m+1)} = -\text{SrSc} \left( \frac{\partial^2 \theta^{(m)}}{\partial \eta^2} + \frac{1}{(\eta + \chi)} \frac{\partial \theta^{(m)}}{\partial \eta} \right) + R\_c,\tag{34}$$

$$\psi^{m+1} = -\frac{q}{2}, \frac{\partial \psi^{m+1}}{\partial \eta} = 1, \; w^{m+1} = 0, \; \theta^{(m+1)} = 0, \; \phi^{(m+1)} = 0, \; at \eta \tag{35}$$

$$\psi^{(m+1)} = \frac{q}{2}, \frac{\partial \psi^{(m+1)}}{\partial \eta} = 1, \; \mathbf{w}^{n+1} = 0, \theta^{(m+1)} = 1, \; \phi^{(m+1)} = 1, \; at \eta \; = h\_2. \tag{36}$$

In the above expression, m is the iterative step index. It is emphasized that the above transmuted set of equations is linear in ψ (m+1) .


performed up to a desirable accuracy of solution. The fast convergence solution has been obtained by employing a successive under-relaxation technique. The values at of ψ˜ (m+1) , w˜ (m+1) ,θ˜(m+1) and φ˜(m+1) at the (m+1)th iterative step are determined as

$$\begin{aligned} \psi^{(m+1)} &= \psi^{(m)} + \mathfrak{r} (\tilde{\psi}^{(m+1)} - \psi^{(m)}), \\ \psi^{(m+1)} &= \psi^{(m)} + \mathfrak{r} (\tilde{\psi}^{(m+1)} - \psi^{(m)}), \\ \theta^{(m+1)} &= \theta^{(m)} + \mathfrak{r} (\tilde{\theta}^{(m+1)} - \theta^{(m)}), \\ \phi^{(m+1)} &= \phi^{(m)} + \mathfrak{r} (\tilde{\phi}^{(m+1)} - \phi^{(m)}), \end{aligned}$$

where τ denotes the under relaxation parameter. For excellent accuracy of the solution, the values of τ should be taken small. In

the current situation, the convincing accuracy of 10−<sup>8</sup> has been achieved for ψ,w, θ, and φ.

### RESULTS AND DISCUSSION

To understand some momentous consequences of peristaltic aspects of flow features, pumping phenomenon, temperature distribution and trapping phenomenon for various values of coupling number (N1), micropolar parameter (N2), Brinkman number (Br), Hartmann number (Ha), and curvature parameter (γ ), various graphs are provided in **Figures 2**–**5** with relevant consequences. The heat transfer characteristics at both wall surfaces are also visualized.

The effects of Hartmann number (Ha) on axial velocity u (η) and micro rotation w (η) are shown in **Figures 2A,B**. **Figure 2A** shows that u (η) reached at peak level with a larger variation of Ha at the upper channel level in contrast to the lower wall channel. **Figure 2B** exhibits the effect of Hartmann number (Ha) on micro-rotation w (η). In the lower channel region, w (η) boosted up with the increment of Ha while its behavior is reversed in the upper part. The decrease in velocity with increasing Ha in the lower part of the channel is attributed to the resistive nature of the Lorentz force due to the applied magnetic field. In order to maintain the prescribed flux, the velocity attained a peak variation in the upper channel portion due to Ha. **Figure 3A** displays the effects of N<sup>1</sup> on axial velocity. The parameter N<sup>1</sup> reflects the vortex to the dynamic viscosity ratio of the fluid. In fact, it is a measure of which viscosity dominates the flow under consideration. Larger values of N<sup>1</sup> correspond to the situation in which vortex viscosity due to spinning motion of fluid particles dominates the flow, and as a result, axial velocity u (η) decreases in the upper channel region. In order to preserve the prescribed flow rate, the axial velocity u (η) increases in the lower part of the channel with increasing N1. **Figure 3B** shows an enhancement in the magnitude of micro-rotation component w (η) with increasing N<sup>1</sup> in both parts of the channel. **Figure 4A** shows the impact of micropolar parameter (N2) on u (η). It is observed that u (η) increases with increasing N<sup>2</sup> in the lower part of the channel. In contrast, u (η) decreases with increasing (N2) in the upper part of the channel. **Figure 4B** shows the effect of N<sup>2</sup> on w (η). It is observed that w (η) decreases in the lower portion of the channel while it increases in the upper portion with increasing N2.

**Figures 5A–C** exhibit the effect of Hartmann number (Ha), coupling number (N1), micropolar parameter (N2), and curvature parameter (γ ) on pressure rise per wavelength (1p). The profiles of the pressure rise per wavelength for different values of Ha (Hartmann number) and coupling number (N1) are shown in **Figures 5A,B**. It is observed that in pumping region 2 > 0, 1p > 0 , the pressure rise per wavelength increases with increasing Ha and N1. The situation is different in the free pumping (1p = 0) and co-pumping regions (2 > 0, 1p < 0). Here, 1p decreases by increasing Ha and N1. **Figure 5C** shows the effects of micropolar parameter (N2) on 1p. In the case of the micropolar parameter, an opposite trend is observed as seen in the figure.

The profiles of the temperature field for different values of the Brinkman number (Br), Hartmann number (Ha), coupling number (N1), and micropolar parameter (N2) are shown in **Figures 6A–D**. It is noted that θ increases over the entire cross section with each increase in Br,N1, and N2. The increase in θ with increasing N<sup>1</sup> and N<sup>2</sup> is due to the retarding effect of these parameters on velocity u (η). The Brinkman number is a parameter which is the ratio of viscous heat to the heat transported by conduction. Larger values of Brinkman correspond to the scenario when heat generated due to viscous dissipation is dominant. In such situation, an enhanced temperature distribution in the channel has been justified. **Figure 6D** shows that θ decreases with increasing Ha. In order to determine how heat transfer coefficient z is altered for diverse values of Ha, N<sup>1</sup> and N<sup>2</sup> are displayed in **Figures 7A–C**. The behavior of z is clearly oscillating which is attributed to the oscillatory nature of the channel walls. A damping in amplitude of oscillations is observed with increasing Ha. The effects of B r (Brinkman number), R<sup>c</sup> (rate of chemical reaction), Ha (Hartmann number), Sc (Schmidt number), and Sr (Soret number) on mass concentration (φ(η)) can be observed in **Figures 8A–E**. It is observed that φ(η) is enhanced with increasing Br, R<sup>c</sup> , Sc, and Sr. On the contrary, φ(η) turns down by varying Ha. The streamlines of flow inside the channel for different values of curvature parameter (γ ) and coupling number (N1) are shown in **Figures 9**, **10**. The objective is to investigate the trapping phenomenon. **Figure 9** shows the effect of the curvature parameter on streamlines. We noticed that for minimum values of γ , the fluid bolus has been concentrated in the upper channel region which is divided into symmetric parts due to increment in γ . The physical consequences of coupling number N<sup>1</sup> on streamlines are investigated by preparing **Figure 10**. Similar to earlier observations, the fluid bolus concentrated in the upper channel portion exists for lesser variation of coupling number (N<sup>1</sup> = 0.8, 1). The bolus has been ripped into two shapes as N<sup>1</sup> gets maximum values. However, the upper part is relatively bigger as compared to the lower part. The lower part of the bolus increases in size with increasing N<sup>1</sup> to 1.4. It is strongly anticipated that the upper part of the bolus vanishes with further increasing N<sup>1</sup> and the channel is only filled with a single bolus concentrated in the lower part.

#### REFERENCES


### CONCLUDING REMARKS

We have reported the transportation of the heat/mass phenomenon in the peristaltic study of micropolar liquid in a curved channel. The flow model is constructed via relevant equations which are treated numerically by employing the finite difference procedure. We summarized important observations from the current analysis in the following points:


#### DATA AVAILABILITY STATEMENT

The datasets generated for this study are available on request to the corresponding author.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Ahmed, Ali, Khan, Rashad, Nabwey and Tlili. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

## NOMENCLATURE


# Triple Diffusive Unsteady Flow of Eyring–Powell Nanofluid Over a Periodically Accelerated Surface With Variable Thermal Features

Sami Ullah Khan<sup>1</sup> , Hanumesh Vaidya<sup>2</sup> , Wathek Chammam<sup>3</sup> , Sa'ed A. Musmar <sup>4</sup> , K. V. Prasad<sup>2</sup> and Iskander Tlili 5,6 \*

*<sup>1</sup> Department of Mathematics, COMSATS University Islamabad, Sahiwal, Pakistan, <sup>2</sup> Department of Mathematics, Vijayanagara Sri Krishnadevaraya University, Karnataka, India, <sup>3</sup> Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al Majma'ah, Saudi Arabia, <sup>4</sup> Industrial Engineering Department, The University of Jordan, Amman, Jordan, <sup>5</sup> Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam, <sup>6</sup> Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam*

This research communicates the triple diffusion perspective of Eyring–Powell nano-materials configured by a periodically moving configuration. The thermal consequences of variable natures are utilized as a novelty. Combined magnetic and porous medium effects are also involved, which result in a magneto-porosity parameter. The thermophoretic and Brownian motion aspects are reported by using Buongiorno's nanofluid theory. The formulated flow equations in non-dimensional forms are tackled with the implementation of a homotopy analysis algorithm. A detailed physical investigation against derived parameters is presented graphically. Due to periodically accelerated surface, the oscillations in velocity and wall shear stress have been examined.

Keywords: eyring–powell nanofluid, triple diffusion, variable thermal conductivity, oscillatory stretching sheet, homotopy analysis method

### INTRODUCTION

Recent advances in nanotechnology have discovered an advanced source of energy based on utilization nanoparticles. Nanofluids have been interacted for the impressive thermal properties that turn into enhancement of energy transportation. The enhancement of thermo-physical features of conventional base liquids with the addition of micro-sized metallic particles is a relatively new and interesting development in nanotechnology. Nanoparticles attain microscopic size, having a range between 1 and 100 nm. Recently, the investigations on nano-materials become a new class of intense engineering research due to inherent significances in biomedical, chemical, and mechanical industries, electronic field, nuclear reactors, power plants, cooling systems, diagnoses, diseases, etc. The primary investigation on this topic was reported by Choi [1], which was further worked out by several scientists, especially in the current century. The convective features for nano-materials based on thermophoresis and Brownian movement phenomenon were notified by Buongiorno [2]. This investigation revealed that the role of thermophoresis and Brownian motion factors was quite essential for convective slip mechanism. Khan and Pop [3] discussed the feature of nanofluid immersed in base material confined by moving configuration. Sheikholeslami et al. [4] reported the features of thermal radiation in magneto-nanoparticle flow between circular cylinders. The slip flow in nano-material due to porous surface has been reported by

#### Edited by:

*Muhammad Mubashir Bhatti, Shanghai University, China*

#### Reviewed by:

*Kh S. Mekheimer, Al-Azhar University, Egypt Mohammad Rahimi Gorji, Ghent University, Belgium*

#### \*Correspondence:

*Iskander Tlili iskander.tlili@tdtu.edu.vn*

#### Specialty section:

*This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics*

> Received: *28 March 2020* Accepted: *04 June 2020* Published: *24 July 2020*

#### Citation:

*Khan SU, Vaidya H, Chammam W, Musmar SA, Prasad KV and Tlili I (2020) Triple Diffusive Unsteady Flow of Eyring–Powell Nanofluid Over a Periodically Accelerated Surface With Variable Thermal Features. Front. Phys. 8:246. doi: 10.3389/fphy.2020.00246*

**191**

Khan et al. Triple Diffusive Unsteady Flow

Shahzadi et al. [5]. Khan et al. [6] directed their investigation regarding stability prospective of nanoliquids in a curved geometry and successfully estimated a dual solution for the formulated problem. Turkyilmazoglu [7] imposed zero mass flux constraints regarding asymmetric channels filled by nanoparticles. Vaidya et al. [8] enrolled the fundamental thermal characteristics in the three-dimensional (3D) flow of Maxwell nanofluid where analytical expressions were developed by using optimal homotopic procedure. Hayat et al. [9] focused on the thermal properties and developed the 3D flow of Oldroyd-B fluid featuring mixed convection effects. Some valuable closedform expressions for a nanofluid flow problem in porous space have been computed by Turkyilmazoglu [10]. Krishna and Chamkha [11] investigated the ion and hall slip effects in the rotating flow of nanofluid configured by a vertical porous plate. The enhancement of heat transfer by using hybrid nanofluids having variable thermal viscosity was reported by Manjunatha et al. [12]. Sardar et al. [13] used non-Fourier's expressions for Carreau nanofluid and suggested some useful multiple numerical solutions successfully. Alwatban et al. [14] performed a numerical analysis to examine the rheological consequences in Eyring– Powell fluid subjected to the second-order slip along with activation energy. The stability analysis for bioconvection flow of nanofluid was reported by Zhao et al. [15]. Alkanhal et al. [16] involved thermal radiation and external heat source for nanofluid enclosed by a wavy shaped cavity. Kumar et al. [17] discussed the thermo-physical properties of hybrid ferrofluid in thin-film flow impacted by uniform magnetic field. Bhattacharyya et al. [18] evaluated the characteristics of different carbon nanotubes for coaxial movement of disks. Mekheimer and Ramdan [19] investigated the flow of Prandtl nanofluid in the presence of gyrotactic microorganisms over a stretching/shrinking surface.

Recently, researchers specified their attention toward the complex and interesting properties of non-Newtonian meterials due to their miscellaneous application in many industries and technologies. The non-Newtonian materials due to convoluted features attracted special attention especially in the current century. The novel physical importance of such non-Newtonian liquids in various engineering and physical processes, biological sciences, physiology, and manufacturing industries is associated due to complex rheological features. Some useful applications associated with the non-Newtonian fluids include polymer solutions, certain oil, petroleum industries, blood, honey, lubricants, and many more. It is commonly observed that distinctive features of such non-Newtonian fluid cannot be pointed out via single relation. Therefore, different non-Newtonian fluid models are suggested by investigators according to their rheology. Among these, Eyring–Powell fluid is inferred from kinetic laws of gases instead of any empirical formulas. This model reduces the viscous fluid at both low and high shear rates (see Powell and Eyring [19]). Gholinia et al. [20] carried out the homogenous and heterogeneous impact in flow of Powell–Eyring liquid due to rotating. Khan et al. [21] focused on viscositydependent mixed convection flow of Eyring–Powell nanofluid encountered by inclined surface. Salawu and Ogunseye [22] reported the entropy generation prospective in Eyring–Powell nanofluid featuring variable thermal consequences and electric field. Another useful continuation performed by Abegunrin et al. [23] examined the change in the boundary layer for the flow of Eyring–Powell fluid subjected by the catalytic surface reaction. Rahimi et al. [24] adopted a numerical technique to compute the numerical solution of a boundary value problem modeled due to the flow of Eyring–Powell fluid. Reddy et al. [25] involved some interesting thermal features, like activation energy, chemical reaction, and non-linear thermal radiation in the 3D flow of Eyring–Powell nanofluid induced via slandering surface. Hayat and Nadeem [26] examined the flow of Eyring–Powell fluid and suggested modification in energy and concentration expressions by using generalized Fourier's law. Ghadikolaei et al. [27] reported the Joule heating and thermal radiation features inflow of Eyring–Powell non-Newtonian fluid in a stretching walls channel.

The double diffusion convection is a natural phenomenon that encountered multiple novel applications in area soil sciences, groundwater, oceanography, petroleum engineering, food processing, etc. The double-diffusive convection refers to the intermixing of components of two fluid having different diffuse rates. However, the situation becomes quite interesting when double-diffusive convection depends upon more than two components of fluids. Examples of such multiple diffusive phenomenons include seawater, molten alloy solidification, and geothermally heated lakes. The triple diffusion flow appears in diverse engineering and scientific fields like geology, astrophysics, disposals of nuclear waste, deoxyribonucleic acid (DNA), chemical engineering, etc. [28–30].

After careful observation of the previously cited work, it is claimed that no efforts have been made to report the triple diffusion flow of Eyring–Powell nanofluid induced by an oscillatory stretching surface with variable thermal features. Although some investigations on flow that is due to periodical acceleration have been available in the literature, thermodiffusion features for Eyring–Powell nanofluid are not studied yet. Therefore, our prime objective of this contribution is to report the triple diffusion aspects of Eyring–Powell nanofluid flow by using variable thermal properties. The most interesting convergent technique homotopy analysis procedure is followed to simulate the solution [31–35]. The graphs are prepared to see the impact of different flow parameters with relevant physical consequence.

### FLOW PROBLEM

To develop governing equations for unsteady flow of Eyring– Powell nanofluid, we have considered a periodically stretching surface where x-axis is assumed along with the stretched configuration, whereas y-axis is taken normally. The source of induced flow is based on the periodically moving surface where amplitude of oscillations are assumed to be small. Let velocity of the moving surface as u = u<sup>ω</sup> = bx sin ωt, b as stretching rate, ω being angular frequency, whereas t represent time. The uniform features of the magnetic field are reported by implementing it vertically. Let T represent the temperature, C solutal concentration, whereas 8 report the nanoparticle volume fraction. Furthermore, T∞, C∞, and 8∞ denote free stream nanoparticle temperature, free stream solutal concentration, and volume fraction of nanofluid, respectively. After using such assumptions, the flow problem is modeled through the following equations [22, 33]:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial \nu}{\partial y} = 0,\tag{1}
$$

$$\begin{split} u\left(\frac{\partial u}{\partial \mathbf{x}}\right) &+ \nu \left(\frac{\partial u}{\partial \mathbf{y}}\right) + \frac{\partial u}{\partial t} = \left(\nu + \frac{1}{\rho\_f \beta^\* \mathbf{C}}\right) \left(\frac{\partial^2 u}{\partial \mathbf{y}^2}\right) \\ &- \frac{1}{2\rho \beta^\* \mathbf{C}^3} \left[ \left(\frac{\partial u}{\partial \mathbf{y}}\right)^2 \frac{\partial^2 u}{\partial \mathbf{y}^2} \right] - \left(\frac{\sigma B\_0^2}{\rho\_f} + \frac{\nu \vartheta}{k'}\right) u, \end{split} \tag{21}$$

$$\begin{split} \mu \left( \frac{\partial T}{\partial x} \right) &+ \left( \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right) + \frac{\partial T}{\partial t} = \frac{1}{\left( \rho \boldsymbol{c} \right)\_p} \frac{\partial}{\partial \boldsymbol{\mathcal{Y}}} \left( K \left( T \right) \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right) \\ &+ \tau\_T \left[ D\_T \frac{\partial \boldsymbol{\phi}}{\partial \boldsymbol{\mathcal{Y}}} \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right)^2 \right] + DK\_{TC} \left( \frac{\partial^2 C}{\partial \boldsymbol{\mathcal{Y}}^2} \right), \end{split} \tag{3}$$

$$u\left(\frac{\partial C}{\partial \mathbf{x}}\right) + \nu \left(\frac{\partial C}{\partial \mathbf{y}}\right) + \frac{\partial C}{\partial t} = D\_s \left(\frac{\partial^2 C}{\partial \boldsymbol{\eta}^2}\right) + D \mathcal{K}\_{\text{CT}} \left(\frac{\partial^2 T}{\partial \boldsymbol{\eta}^2}\right), \text{(4)}$$

$$u\left(\frac{\partial \Phi}{\partial \mathbf{x}}\right) + \nu \left(\frac{\partial \Phi}{\partial \boldsymbol{\eta}}\right) + \frac{\partial \Phi}{\partial t} = D\_B \left(\frac{\partial^2 \Phi}{\partial \boldsymbol{\eta}^2}\right) + \frac{D\_T}{T\_{\infty}} \left(\frac{\partial^2 T}{\partial \boldsymbol{\eta}^2}\right), \text{(5)}$$

where ν is viscosity, ρ<sup>f</sup> fluid density, (β ∗ , C) fluid parameters, ϑ permeability of porous medium, σ<sup>e</sup> electrical conductivity, α<sup>1</sup> thermal diffusivity, DKTC Dufour diffusivity, τ<sup>T</sup> = (ρc)p/(ρc)<sup>f</sup> ratio of heat capacity of nanoparticles to heat capacity of fluid, D<sup>B</sup> Brownian diffusion coefficients, D<sup>s</sup> solutal diffusivity, D<sup>T</sup> thermophoretic diffusion coefficient, whereas DKCT Soret diffusivity.

Following boundary assumptions are articulated for current flow problem

$$\begin{aligned} \mu &= \mu(\mathbf{x}, t) = u\_{\text{w}} \sin \omega t = b\mathbf{x} \sin \omega t, & \nu &= 0, \\ T &= T\_{\text{w}}, \ C = C\_{\text{w}}, \Phi = \Phi\_{\text{w}} \text{ at } \ \mathcal{Y} = 0, & \text{(6)} \\ \mu &\to 0, \ \nu &\to 0, \ T \to T\_{\infty}, \ C \to C\_{\infty}, \Phi \to \Phi\_{\infty} \text{ at } \ \mathcal{Y} \to \infty. \end{aligned} \tag{7}$$

In order to suggest modification in energy equation (3), we used the following relations for variable thermal conductivity [33, 34]

$$K(T) = K\_{\infty} \left[ 1 + \varepsilon \frac{(T - T\_{\infty})}{\Delta T} \right],\tag{8}$$

where K∞ ambient fluid conductivity and ε thermal dependence conductivity constant. Now, before perfume analytical simulations, first, we reduce the number of independent variables in the governing equations by using the following variables:

$$\begin{aligned} \xi &= \left(\frac{b}{\upsilon}\right)^{1/2} \wp, \ \mathfrak{r} = t\wp, \ u = u\_{\mathfrak{w}} f\_{\mathfrak{f}}\left(\xi, \ \mathfrak{r}\right), \\\ \upsilon &= -\sqrt{\upsilon b f}\left(\xi, \ \mathfrak{r}\right), \\\ \theta\left(\xi, \mathfrak{r}\right) &= \frac{\left(T - T\_{\infty}\right)}{\left(T\_{\mathfrak{w}} - T\_{\infty}\right)}, \varphi\left(\xi, \mathfrak{r}\right) = \left(\frac{\left(C - C\_{\infty}\right)}{C\_{\mathfrak{w}} - C\_{\infty}}\right), \phi\left(\xi, \mathfrak{r}\right) \end{aligned} \tag{9}$$

$$=\frac{(\Phi-\Phi\_{\infty})}{(\Phi\_{\text{w}}-\Phi\_{\infty})},\tag{10}$$

The dimensionless set of equations in view of the previously mentioned transformations is

$$(1+K)f\_{\xi\xi\xi} - \S f\_{\xi\tau} - f\_{\xi}^2 + f f\_{\xi\xi} - \Omega f\_{\xi} - \Gamma K f\_{\xi\xi}^2 f\_{\xi\xi\xi} = 0,\tag{11}$$

$$\begin{aligned} \left( \left( 1 + \delta \theta \right) \theta\_{\xi} \xi \right) + \delta \left( \theta\_{\xi} \right)^{2} + \Pr \left[ f \phi\_{\xi} - \mathbb{S} \phi\_{\xi} + \mathrm{Nb} \theta\_{\xi} \phi\_{\xi} + \mathrm{Nt} \left( \theta\_{\xi} \right)^{2} \right] \\ + \left( \mathrm{Nd} \right) \varphi\_{\xi} \xi \right] = 0, \end{aligned} \tag{12}$$

$$\begin{aligned} \text{2. } \begin{pmatrix} \text{Nd} \end{pmatrix} \begin{aligned} \varphi\_{\xi\xi} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \xi \end{aligned} \Big| \begin{aligned} \text{2. } \xi \end{aligned}$$

$$
\varphi\_{\xi\xi} - \mathcal{S}\varphi\_{\mathfrak{t}} + L e\left(f\varphi\_{\xi}\right) + L d\theta\_{\xi\xi} = 0,\tag{13}
$$

$$
\phi\_{\xi}\xi \, - \, \mathcal{S}\phi\_{\mathsf{T}} + \ln \left( f \phi\_{\xi} \right) + \frac{Nt}{Nb} \theta\_{\xi}\xi \, = 0,\tag{14}
$$

The boundary constraints in the non-dimensional form are

$$\begin{aligned} f\_{\xi} \left( 0, \tau \right) &= \sin \tau, \ f \left( 0, \tau \right) = 0, \\ \theta \left( 0, \tau \right) &= 1, \; \phi(0, \tau) = 1, \phi(0, \tau) = 1, \\ \left. f\_{\mathcal{Y}} \left( \infty, \tau \right) &\to 0, \theta \left( \infty, \tau \right) \to 0, \phi \left( \infty, \tau \right) \to 0, \phi \left( \infty, \tau \right) \to 0, \end{aligned} \tag{15}$$

where K = 1/µβ∗C and Ŵ = u<sup>w</sup> <sup>2</sup>b/2νC <sup>2</sup> denote the material parameters, = σB 2 0 /ρ<sup>f</sup> b + νϑ/k ′b is magneto-porosity constant, S = ω/b oscillating frequencyto-stretching rate ratio, N<sup>t</sup> = (ρc)pD<sup>T</sup> (T<sup>w</sup> − T∞) /(ρc)<sup>f</sup> T∞ν thermophoresis parameter, Pr = ν/α<sup>m</sup> is Prandtl number, N<sup>b</sup> = (ρc)pD<sup>B</sup> (C<sup>w</sup> − C∞) /(ρc)<sup>f</sup> ν Brownian motion constant, Nd = DTC (C<sup>w</sup> − C∞) /α<sup>m</sup> (T<sup>w</sup> − T∞) modified Dufour number, Ld = DCT (T<sup>w</sup> − T∞) /α<sup>m</sup> (C<sup>w</sup> − C∞) Dufour Lewis number, Le = ν/D<sup>s</sup> regular Lewis number, whereas Ln = ν/D<sup>B</sup> nano-Lewis number.

We define the following relations associated with the definitions of wall shear stress, local Nusselt number, Sherwood

number, and nano-Sherwood number:

$$\begin{split} \mathcal{C}\_{f} &= \frac{\mathsf{r}\_{\text{w}}}{\rho u\_{\text{w}}^{2}}, \mathsf{N}u\_{\text{x}} = \frac{\mathsf{x}q\_{\text{s}}}{k \,(T\_{\text{w}} - T\_{\text{\infty}})}, \mathsf{Sh}\_{\text{x}} = \frac{\mathsf{x}j\_{\text{s}}}{D\_{\text{B}} \,(\mathsf{C}\_{\text{w}} - \mathsf{C}\_{\text{\infty}})},\\ \mathcal{S}\mathsf{h}\_{\text{xn}} &= \frac{\mathsf{x}q\_{\text{m}}}{D\_{\text{s}} \,(\varphi\_{\text{w}} - \varphi\_{\text{\infty}})}, \end{split} \tag{17}$$

where q<sup>s</sup> , js , and g<sup>s</sup> stand for surface heat flux, surface mass flux, and motile microorganisms flux, respectively. The dimensionless forms of the previously mentioned physical quantities are

$$\begin{array}{c} \text{Re}\_{\mathbf{x}}^{1/2}C\_{\mathbf{f}} = (1+K)f\_{\xi\xi} - \frac{K}{3}\theta\left(f\_{\xi\xi}\right)\_{\xi=0}, \\\quad \text{Nu}\_{\mathbf{x}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\theta\_{\xi}\left(0,\tau\right), \\\quad \text{Sh}\_{\mathbf{x}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\varphi\_{\xi}\left(0,\tau\right), \\\quad \text{Sh}\_{\mathbf{n}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\phi\_{\xi}\left(0,\tau\right). \end{array} \tag{18}$$

TABLE 1 | Comparison of *f*ξ ξ (0, τ ) for τ with Abbas et al. [35] when *S* = 1, = 12, Ŵ = 0, and *K* = 0.


where Re<sup>x</sup> = uwx¯/ν is mentioned for local Reynolds number.

### SOLUTION METHODOLOGY

The structured set of non-linear partial differential equations (12–16) with boundary conditions (17–18) are simulated analytically via homotopy analysis technique. Due to efficient and convincing accuracy, various physical problems in recent years have been solved by following this procedure. The initial guesses for the present flow problem are

$$f\_0\left(\xi,\tau\right) = \sin\tau \left(1 - e^{-\xi}\right), \ \theta\_0\left(\xi\right) = e^{-\xi}, \ \varphi\_0\left(\xi\right) = e^{-\xi},$$

$$\phi\_0\left(\xi\right) = e^{-\xi},\tag{19}$$

Following auxiliary linear operators that are followed to precede the solution

$$\mathfrak{E}\_f = \frac{\partial^3}{\partial \xi^3} - \frac{\partial}{\partial \xi}, \mathfrak{E}\_\theta = \frac{\partial^2}{\partial \xi^2} - 1, \mathfrak{E}\_\varphi = \frac{\partial^2}{\partial \xi^2} - 1,$$

$$\mathfrak{E}\_\phi = \frac{\partial^2}{\partial \xi^2} - 1,\tag{20}$$

satisfying

$$\left[\mathfrak{E}\_f\left[a\_1 + a\_2e^{\xi} + a\_3e^{-\xi}\right] = 0,\tag{21}$$

FIGURE 2 | Impact of (A) *K* (B) *S* on skin friction coefficient.

$$
\varepsilon\_{\theta} \left[ a\_4 e^{\xi} + a\_5 e^{-\xi} \right] = 0,\tag{22}
$$

$$
\xi\_{\varphi} \left[ a\_6 e^{\xi} + a\_7 e^{-\xi} \right] = 0,\tag{23}
$$

$$
\varepsilon\_{\phi} \left[ a s e^{\xi} + a \wp e^{-\xi} \right] = 0,\tag{24}
$$

where a1, . . . a<sup>9</sup> are arbitrary constants.

### CONVERGENT REGION

The convergence procedure in homotopic solution is regulated with auxiliary parameters h<sup>f</sup> , hθ , hϕ, and hφ. On this end, we prepare h−curves to report the convincible range of such parameters. It is obvious from **Figure 1** that the preferable range of these such parameters can be utilized from −2 ≤ h<sup>f</sup> ≤ 0, −1.2 ≤ h<sup>θ</sup> ≤ −0.2, −1.4 ≤ h<sup>ϕ</sup> ≤ 0 and −1.2 ≤ h<sup>φ</sup> ≤ −0.2.

### VALIDATION OF RESULTS

**Table 1** shows the comparison of present results with Abbas et al. [35] as a limiting case. An excellent accuracy of results is noted with these reported investigations.

### DISCUSSION

This section aims to manifest the features of some interesting flow parameters that appeared in the dimensionless equations, where material parameter K, magneto-porosity constant , oscillating frequency-to-rate of stretching ratio S, Brownian motion Nb, thermophoresis constant Nt, variable thermal conductivity δ, Prandtl number Pr, Dufour Lewis number Ld, modified Dufour constant Nd, regular Lewis number Le, and nano-Lewis number Ln. During variation of each flow parameter, we fixed some numerical values to remaining parameters, like K=0.2, = 0.4, S = 0.2, Nt = 0.3, Pr = 0.7, Nb = 0.4, Nd, = 0.5, Ld = 0.3, Le = 0.2, and Ln = 0.3.

#### Skin Friction Coefficient

The impact of the skin friction coefficient against time τ for diverse variation of K and S is evaluated in **Figures 2A,B**. An interesting periodic oscillation in the wall shear stress is evaluated by both figures. Furthermore, the growing values of both parameters increase the amplitude of oscillation sufficiently. Due to no-slip conditions at the surface, the fluid particles accelerated together with surface in same amplitude and phase. However, the occurrence of a phase shift in both curves is almost negligible.

### Velocity Profile

The results reported in **Figures 3A,B** show the change in velocity f<sup>ξ</sup> , verse time τ and leading values of material constant K, and magneto-porosity parameter . **Figure 3A** characterized the influence of K on fξ , which shows that an increment in K leads to higher velocity amplitude. The physical justification of such enhancing velocity distribution is attributed to the lower viscosity of fluid associated with the higher values of K. However, reverse observations are predicated for . In fact, magneto-porosity

constant is the combination of magnetic field and porous space. The existence of a magnetic force encountered the effects of Lorentz force, in which a declining oscillation behavior is noticed. Similarly, the permeability of a porous medium also retards the velocity amplitude due to the loss of fluid. Moreover, the utilization of magnetic force enhances the apparent viscosity of fluid up to a certain point of becoming an elastic solid, and subsequently, the fluid stress can be managed upon changing the magnetic force. These interesting observations can be used in various processes, like magnetohydrodynamic drive ion propulsion, magnetohydrodynamic drive power generators, electromagnetic material casting, etc.

#### Temperature Distribution

To visualize the alter profile of nanoparticle temperature θ due to δ, K, and Nd, **Figures 4A–C** are prepared. **Figure 4A** reveals that temperature distribution θ increases with variable thermal conductivity constant δ. **Figure 4B** is constituted to observe the change in θ due to material parameter K. A fall in θ is associated with leading variation of K. An increment in viscosity would yield for arising values of K that increases the fluid velocity but a decline in the temperature of fluid. The change in θ with effect of modified Dufour number Nd has been reported in **Figure 4C**. A slightly dominant variation in θ is seen with larger values of Nd.

### Solutal Concentration Profile

Now, we observe the variation in solutal concentration profile ϕ by varying regular Lewis number (Le), Dufour Lewis constant Ld , and material constant(K). **Figure 5A** is designed to observe the impact of Le on ϕ. A decreasing solutal concentration profile ϕ is notified due to Le. Physical explanation of such decling variation of ϕ can be justified on the fact that Le captures reverse relation with species diffusion, which means that when Le is maximum, species diffusion is lower, which leads to the decrement of the resulting solutal concentration. From **Figure 5B**, ϕ increases with the growth of Ld. Physically, Ld depends upon the Lewis number due to lower mass diffusivity. **Figure 5C** presents change in ϕ due to material constant K. Again, an enhanced distribution of solutal concentration profile ϕ has resulted for maximum values of K.

### Nanoparticle Concentration

The physical consequences of Ln, Nt, and Nb on concentration distribution φ are deliberated in **Figures 6A–C**. **Figure 6A** specified the input of Ln on φ. A declining concentration distribution φ is examined in the peak values of Ln. This decreasing behavior of φ is attributed to the fact that Ln is associated with the Brownian diffusion coefficient because Ln expresses the thermal diffusivity-to-mass diffusivity ratio. This parameter that is referred to the fluid flow in a

phenomenon of heat and mass transfer occurs due to convection. The consequences for another important parameter, namely thermophoresis constant Nt, are analyzed in **Figure 6B**. As expected, a larger concentration distribution φ is reported due to involvement of Nt. The larger variation of Nt helps to improve the thermal conductivity of fluid. Physically, the thermophoretic process is based on collective migrated heat particles in the region of low temperature and plays a momentous role in many physical



phenomenons. The curve of φ attained maximum level due to Nt. However, a reduced concentration distribution φ is associated with Nb, as shown in **Figure 6C**. The Brownian movement is based on random pattern moving fluid particles in flow surface. It is further justified from Equation (16), which clearly shows that reverse relation is developed between Nb and φ. In fact, the specified numerical values of Nb are associated to the more prominent nanoparticle moments that are being pushed back from accelerated plate to quiescent, which resulted in a retarded concentration distribution.

#### Physical Quantities

To perform the numerical simulations for local Nusselt number −θ<sup>ξ</sup> (0, τ ), local Sherwood Number −ϕ<sup>ξ</sup> (0, τ ) and nanofluid Sherwood number −φ<sup>ξ</sup> (0, τ ), **Table 2** is designed. It is observed that when , ε, and K assigned larger numerical values, a decreasing trend in −θ<sup>ξ</sup> (0, τ ), −ϕ<sup>ξ</sup> (0, τ ), and −φ<sup>ξ</sup> (0, τ ) is reported. However, these physical quantities increase with the variation of Pr.

### CONCLUSIONS

We have focused on periodically accelerated unsteady flow of Eyring–Powell nanofluid with utilization of thermal diffusive features. The variable impact of thermal conductivity, porous medium, and magnetic field consequences are also utilized. The important observations from current flow problem are summarized as:

➢ The magneto-porosity parameter declined the periodic oscillation in the velocity, and subsequently, the magnitude of velocity declined.

### REFERENCES


The observation based on the reported results can be used to improve thermal extrusion processes, heat exchangers, solar technology, energy production, cooling processes, etc.

### DATA AVAILABILITY STATEMENT

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number: RGP-2019-1.


presence of radiation heat source, Int J Heat Mass Transfer. (2019) **139**:87–94. doi: 10.1016/j.ijheatmasstransfer.2019.05.006


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Khan, Vaidya, Chammam, Musmar, Prasad and Tlili. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Entropy Optimization of Third-Grade Nanofluid Slip Flow Embedded in a Porous Sheet With Zero Mass Flux and a Non-Fourier Heat Flux Model

K. Loganathan<sup>1</sup> \*, G. Muhiuddin<sup>2</sup> \*, A. M. Alanazi <sup>2</sup> , Fehaid S. Alshammari <sup>3</sup> , Bader M. Alqurashi <sup>4</sup> and S. Rajan<sup>1</sup>

*<sup>1</sup> Department of Mathematics, Erode Arts & Science College, Erode, India, <sup>2</sup> Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia, <sup>3</sup> Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia, <sup>4</sup> Department of Mathematics, Faculty of Arts and Science in Almandaq, Albaha University, Albaha, Saudi Arabia*

#### Edited by:

*Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan*

#### Reviewed by:

*Ali Chamkha, Prince Mohammad bin Fahd University, Saudi Arabia Precious Sibanda, University of KwaZulu-Natal, South Africa*

#### \*Correspondence:

*G. Muhiuddin chishtygm@gmail.com K. Loganathan loganathankaruppusamy304@ gmail.com*

#### Specialty section:

*This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics*

> Received: *21 April 2020* Accepted: *08 June 2020* Published: *24 November 2020*

#### Citation:

*Loganathan K, Muhiuddin G, Alanazi AM, Alshammari FS, Alqurashi BM and Rajan S (2020) Entropy Optimization of Third-Grade Nanofluid Slip Flow Embedded in a Porous Sheet With Zero Mass Flux and a Non-Fourier Heat Flux Model. Front. Phys. 8:250. doi: 10.3389/fphy.2020.00250*

The prime objective of this article is to explore the entropy analysis of third-order nanofluid fluid slip flow caused by a stretchable sheet implanted in a porous plate along with thermal radiation, convective surface boundary, non-Fourier heat flux applications, and nanoparticle concentration on zero mass flux conditions. The governing physical systems are modified into non-linear ordinary systems with the aid of similarity variables, and the outcomes are solved by a homotopy analysis scheme. The impression of certain governing flow parameters on the nanoparticle concentration, temperature, and velocity is illustrated through graphs, while the alteration of many valuable engineering parameters viz. the Nusselt number and Sherwood number are depicted in graphs. Entropy generation with various parameters is obtained and discussed in detail. The estimation of entropy generation using the Bejan number find robust application in power engineering and aeronautical propulsion to forecast the smartness of entire system.

Keywords: entropy generation, Christov-Cattaneo heat flux, third-grade nanofluid, porous medium, homotopy analytic technique

### INTRODUCTION

Nanoliquids are the type of liquids that have small volumetric quantities of nanoscale 10−<sup>9</sup> − 10−7m metallic Cu, Ti, Hg, Fe, Ag, Au, etc. or non-metallic particles TiO2, SiO2, CuO, Al2O3, etc taken as nano particles. Usually nanoliquids have a colloidal suspension of nanoparticles inside a base liquid for example water, oil, ethylene glycol, etc. Initially, Choi [1] proposed the "nanofluid" term. In general, the effective heat transfer enhancement has the reason of nanoliquids generally restrict up to volume fraction of nanoparticles. Therefore, in the latest technologies and engineering areas, nanoliquids receiving a phenomenal impact. Mushtaq et al. [2] attempted the numerical study of the nanoliquids induced by an exponentially stretchable sheet with rotating flow model. MHD nanoliquid flow toward a porous plate with internal heat generation effects was presented by Reddy and Chamka [3]. Shit et al. [4] studied the convective flow of hydromagnetic nanoliquid with entropy generation mechanism. Recently, Gireesha et al. [5] explained a Hall current effects of two-phase transient nanoliquid flow induced by a stretchable sheet. Reddy et al. [6] performed the combined convection flow of nanoliquid toward a semi-infinite vertical flat sheet with convectively heated boundary and Soret effect. Few more significant studies in this research area are seen in ref's [7–13].

Due to various applications in different technical and industrial areas, the fluid flow problems toward a stretching surface have developed. It finds application in rubber and plastic sheets production, melt-spinning, production of glass-fiber, and metallic plate cooling systems. Sakiadis [14] studied the uniform velocity of a magnetohydrodynamic flow past a solid medium. Magyari et al. [15] examined the first order chemical reaction and heat generation combination on micropolar fluid flow induced by a permeable stretching surface. Gupta [16] analyzed the heat and mass transfer effects of a boundary layer flow induced by a stretchy sheet with suction or blowing impact. Magyari and Keller [17] exposed the fluid flow over exponentially extending sheet with heat and mass transfer impacts. Significances of thermal boundary layer flow for a linearly stretchy sheet with viscous dissipation were examined by Cortell [18]. The cutting-edge research reports on stretching sheet flow are highlighted in these works of literature [19–24].

Fluid flow saturated in the porous surface have numerous applications in various fields like geothermal energy, fuel cell technologies, material processing, etc. Chamkha et al. [25] described the free convective flow past an inclined plate fitted in a porous medium of variable porosity with solar radiation. Khan and Aziz [26] studied the natural convective flow with double diffusion caused by a vertical porous sheet. Oyelakin et al. [27] analyzed the slip flow of unsteady radiative Casson nanofluid toward a stretchy surface. Gorla and Chamkha [28] studied the nanofluid flow toward a non-isothermal vertical plat entrenched in a porous sheet. The same research group extended their work for different models for various applications [29–32].

Heat transport problems in the flow of liquids have been examined by several researchers for the last decade. In 1822, Fourier [33] constructed the heat conduction law. This states that "the heat transfer in a medium with inertial rate." A parabolictype equation was used to state the heat conduction equation. The problem rising at this time is that there exists no such object or material that satisfies Fourier's law, as argued by Cattaneo [34] when using thermal relaxation time to customize Fourier's law. Later, Christov [35] developed and joined the upper convected Maxwell fluid. This developed model is called a Christov-Cattaneo heat flux model. Loganathan et al. [36] studied the thermal relaxation time effects on Oldroyd-B liquid with second-order slip and cross-diffusion impacts. The Christov-Cattaneo heat flux model for a third-grade liquid with chemical reaction effects was examined by Imtiaz et al. [37].

In the last decade, several scientists have researched entropy generation in the flow of fluids and heat transfer over a stretching surface. In various engineering and industrial divisions, the performance of heating and cooling are of massive importance in different electronic and energy issues. Aiboud and Saouli [38] examined the MHD viscoelastic fluid flow with the application of entropy analysis using Kummer's function. Makinde [39] presented the thermal radiation and Newtonian heating impacts of variable viscosity fluid caused by a semi-infinite plate using shooting quadrature and obtained the entropy generation number. Loganathan et al. [40, 41] verified the entropy analysis for the third grade and Williamson nanoliquid flow caused by a stretchable sheet with various effects. They employed HAM to solve the non-linear governing systems.

Based on our research in previously published works, entropy generation of third-grade nanofluid flow caused by a stretching sheet with a modified Fourier law has not been discussed with a high standard of scientific attention. As far as we noticed in literature, the studies taken for entropy generation are limited with some parameters viz the Brinkmann number, Reynolds number, Temperature difference parameter, and Hartmann number. We have extended our investigations to include thermal relaxation time, the Biot number, thermal radiation, slip parameter, porous parameter, etc. The effective collection and analysis of these results will open new gateways for diverse engineering application in various streams.

#### PROBLEM DEVELOPMENT

We were interested in analyzing the entropy and Bejan number of third-grade nanofluid saturated in a porous medium with Christov-Cattaneo heat flux. In **Figure 1**, the stretching parameter is taken along x direction where T<sup>W</sup> and C<sup>W</sup> are represented the wall temperature and concentration, respectively. T∞ and C∞ are used to index the ambient temperature and concentration, respectively. A convective heating temperature Tf is stimulated at the bottom of the sheet surface. The Buongiorno nanofluid [42] is used for the present case.

Based on his consideration, the slip mechanisms, namely, the Brownian diffusion, inertia, Magnus effect, diffusiophoresis, thermophoresis, gravity, and fluid drainage, were analyzed. He recommended that the thermophoresis and Brownian diffusion are essential slip mechanism in low dimensional materials.

The following is an incompressible fluid model containing body forces with the equation of continuity and motion:

$$d\dot{\nu}\,\nu = 0\tag{1}$$

$$
\rho \frac{d\nu}{dt} = di\nu T + \rho b + J + B \tag{2}
$$

Here, ρ is the fluid density, which is taken as a constant, v is the velocity field, b indicates the body forces, J denotes the electric current, and T states the third-grade incompressible fluids Cauchy stress tensor [43]

$$T = -pI + \mu E\_1 + A\_1^\* E\_2 + A\_2^\* E\_1^2 + \beta\_1 E\_3$$

$$+ \beta\_2 \left(E\_1, E\_2 + E\_2 E\_1\right) + \beta\_3 \left(tr E\_1^2\right) E\_1 \tag{3}$$

where µ,(E1, E2, E3) and A ∗ 1 , β<sup>i</sup> indicate the viscosity coefficient, kinematics tensors, and material modulis as in

$$E\_1 = L + \left(L\right)^T \tag{4}$$

$$E\_n = \frac{d}{dt}E\_{n-1} + E\_{n-1}L + (L)^T E\_{n-1}, \qquad n = 2, 3, \text{and} \tag{5}$$

$$L = \nabla \mathbf{v}.\tag{6}$$

d dt is expressed as the material time derivative

$$\frac{d\langle\rangle}{dt} = \frac{\partial\langle\rangle}{\partial t} + \nu.\nabla\langle\rangle. \tag{7}$$

The relationship between Clausius-Duhem inequality and thermodynamically compatible fluid is stated by Fosdick and Rajagopal [44]:

$$
\mu \ge 0, \ A\_1^\* \ge 0, \ \beta\_1 = \beta\_2 = 0, \ \beta\_3 \ge 0 \tag{8}
$$

$$\left|A\_1^\* + A\_2^\*\right| \le 2\sqrt{6\mu\beta\_3} \tag{9}$$

$$T = -pI + \mu E\_1 + A\_1^\* E\_2 + A\_2^\* E\_1^2 + \beta\_3 \left( trE\_1^2 \right) E\_1 \tag{10}$$

Boussinesq and normal boundary layer approximations were considered by Pakdemirli [45]. We made the following assumptions:


$$\frac{\partial \mu}{\partial \mathbf{x}} + \frac{\partial \nu}{\partial y} = \mathbf{0} \tag{11}$$

$$\begin{split} \frac{\partial u}{\partial x} &+ \nu \frac{\partial u}{\partial \boldsymbol{\nu}} \\ &= \nu \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} + \frac{A\_1^\*}{\rho} \left( u \frac{\partial^3 u}{\partial \boldsymbol{\nu}^2 \partial \boldsymbol{x}} + \nu \frac{\partial^3 u}{\partial \boldsymbol{\nu}^3} + \frac{\partial u}{\partial \boldsymbol{x}} \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} + 3 \frac{\partial u}{\partial \boldsymbol{\nu}} \frac{\partial^2 u}{\partial \boldsymbol{x} \partial \boldsymbol{\nu}} \right) \\ &+ 2 \frac{A\_2^\*}{\rho} \frac{\partial u}{\partial \boldsymbol{\nu}} \frac{\partial^2 u}{\partial \boldsymbol{x} \partial \boldsymbol{\nu}} + 6 \frac{\theta\_1^\*}{\rho} \left( \frac{\partial u}{\partial \boldsymbol{\nu}} \right)^2 \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} - \frac{\sigma B\_0^2}{\rho} u - \frac{\nu}{k\_p} u \end{split} \tag{12}$$

u

$$\begin{aligned} \mu \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} &= \frac{k}{\rho c\_p} \frac{\partial^2 T}{\partial \mathbf{y}^2} - \frac{1}{\rho c\_p} \frac{\partial q\_r}{\partial \mathbf{y}} + \frac{Q\_0}{\rho c\_p} \left(T - T\_\infty\right) \\ &+ \pi \left[D\_B \frac{\partial C}{\partial \mathbf{y}} \frac{\partial T}{\partial \mathbf{y}} + \frac{D\_T}{T\_\infty} \left(\frac{\partial T}{\partial \mathbf{y}}\right)^2\right] \end{aligned} \tag{13}$$

$$\begin{aligned} \mu \frac{\partial C}{\partial x} + \nu \frac{\partial C}{\partial \mathcal{Y}} &= D\_B \frac{\partial^2 C}{\partial \mathcal{Y}^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial \mathcal{Y}^2} \\ \mu = u\_w \left( \mathbf{x} \right) &= a\mathbf{x} + L \frac{\partial u}{\partial \mathcal{Y}}, \quad -k \frac{\partial T}{\partial \mathcal{Y}} = h\_f \left( T\_f - T\_{\infty} \right), \\ D\_B \frac{\partial C}{\partial \mathcal{Y}} + \frac{D\_T}{T\_{\infty}} \frac{\partial T}{\partial \mathcal{Y}} &= 0 \text{ at } \mathcal{y} = 0 \\ \mu \to 0, \ T \to T\_{\infty}, \ c \to c\_{\infty} \text{ as } \mathcal{y} \to \infty \end{aligned} \tag{15}$$

The energy equation with a Cattaneo-Christov heat flux model is stated as

$$\begin{split} \mu \frac{\partial T}{\partial \boldsymbol{x}} + \nu \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} &+ \lambda \boldsymbol{\tau} \left( \mu^{2} \frac{\partial^{2} T}{\partial \boldsymbol{x}^{2}} + \nu^{2} \frac{\partial^{2} T}{\partial \boldsymbol{y}^{2}} + \left( \mu \frac{\partial \mu}{\partial \boldsymbol{x}} \frac{\partial T}{\partial \boldsymbol{x}} + \nu \frac{\partial \mu}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{x}} \right) \right) \\ &+ 2\mu \nu \frac{\partial T^{2}}{\partial \boldsymbol{x} \partial \boldsymbol{y}} \bigg) + \left( \mu \frac{\partial \nu}{\partial \boldsymbol{x}} \frac{\partial T}{\partial \boldsymbol{y}} + \nu \frac{\partial \nu}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{y}} \right) = \frac{k}{\rho c\_{\rho}} \frac{\partial^{2} T}{\partial \boldsymbol{y}^{2}} \\ &- \frac{1}{\rho c\_{p}} \frac{\partial q\_{r}}{\partial \boldsymbol{y}} + \frac{Q\_{0}}{\rho c\_{p}} \left( T - T\_{\infty} \right) \\ &+ \pi \left[ D\_{B} \frac{\partial C}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{y}} + \frac{D\_{T}}{T\_{\infty}} \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^{2} \right]. \tag{16} \end{split}$$

Consider the transformation given below:

$$\psi = \sqrt{a\nu} \text{xf}\left(\eta\right), \ u = \frac{\partial \psi}{\partial \boldsymbol{\eta}}, \ \boldsymbol{\nu} = \ -\frac{\partial \boldsymbol{\psi}}{\partial \boldsymbol{x}}, \ \boldsymbol{\eta} = \sqrt{\frac{a}{\nu}} \boldsymbol{\gamma},$$

$$\boldsymbol{\nu} = -\sqrt{a\boldsymbol{\nu}} \boldsymbol{f}\left(\boldsymbol{\eta}\right), \ \boldsymbol{u} = a \boldsymbol{\text{xf}}'\left(\boldsymbol{\eta}\right), \ \boldsymbol{\theta}\left(\boldsymbol{\eta}\right) = \frac{T - T\_{\infty}}{T\_{f} - T\_{\infty}},$$

$$\boldsymbol{\phi}\left(\boldsymbol{\eta}\right) = \frac{C - C\_{\infty}}{C\_{\infty}}.\tag{17}$$

The non-linear governing equations are:

$$\left(f'' + f f'' - f'^2 + \alpha\_1 \left(2f'f'' - ff^{i\nu}\right) + \left(3\alpha\_1 + 2\alpha\_2\right)f'^2\right.$$

$$\left. + 6\beta \text{Ref}^{\prime\prime} f^{\prime\prime2} - \left(Mf' + Kf'\right) = 0\tag{18}$$

$$\left(1+\frac{4}{3}Rd\right)\theta''+Pr\theta'+Pr\Theta-Pr\gamma f^2\theta''-Pr\gamma f f'\theta'$$

$$+Pr\text{N}b\,\,\theta'\phi'+Pr\text{N}t\theta'^2=0\tag{19}$$

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$$
\phi'' + Lef\phi' + \frac{Nt}{Nb}\theta'' = 0\tag{20}
$$

with the end points

$$\begin{aligned} f\ (0) &= f\_{\mathcal{W}} f'\ (0) = 1 + \Gamma \overline{f}'\ (0), \boldsymbol{\theta}'\ (0) = -Bi(1 - \boldsymbol{\theta}\ (0)),\\ \mathcal{N}b\ \boldsymbol{\phi}'\ (0) + \mathcal{N}t\ \boldsymbol{\theta}'\ (0) &= 1\\ f'\ (\infty) &= 0, \ \boldsymbol{\theta}\ (\infty) = 0, \boldsymbol{\phi}\ (\infty) = 0\end{aligned} \tag{21}$$

The non-dimensional variables are

$$\alpha\_1 = \frac{aA\_1^\*}{\upsilon}, \alpha\_2 = \frac{aA\_2^\*}{\upsilon}, \beta = \frac{a\beta\_1^\*}{\upsilon}, Re = \frac{u\_\upsilon \chi}{\upsilon}, Pr = \rho C\_p/k,$$

$$M = \sigma B\_0^2/\rho a, Rd = \left(4\sigma^\* T\_\infty^3\right)/\left(kk^\*\right), S = \frac{Q\_0}{\rho c\_p}, \gamma = \lambda\_T a,$$

$$Bi = \frac{h\_f}{k}\sqrt{\upsilon\_a^\prime}, Nb = \frac{\tau D\_B}{\upsilon}\left(C\_\infty\right), Nt = \frac{\tau D\_T}{\upsilon}\left(T\_f - T\_\infty\right).$$

The application of physical entitles is such that

$$\operatorname{Re}^{\frac{1}{2}}C\_f = f''(0) + \alpha\_1 f'(0)f''''(0) + \beta \operatorname{Re}[f''(0)]^{\frac{3}{2}} \tag{22}$$

$$Re^{-\frac{1}{2}}Nu\_{\mathbf{x}} = -(1 + \frac{4}{3}Rd)\theta'(0) \,. \tag{23}$$

The local mass transfer rate becomes identically zero due to the zero mass flux state [46]

$$Re^{-\frac{1}{2}}Sh = \frac{Nt}{Nb}\theta'(0) \,. \tag{24}$$

#### ENTROPY OPTIMIZATION

The entropy minimization optimization for fluid friction, heat, and the irreversibility of mass transfer are given below:

$$\begin{split} S\_{\text{gen}}^{'''} &= \frac{K\_1}{T\_{\infty}^2} \left[ \left( \frac{\partial T}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^2 + \frac{16 \boldsymbol{\sigma}^\* T\_{\infty}^3}{3kk^\*} \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^2 \right] \\ &+ \frac{\mu}{T\_{\infty}} \Bigg[ 2 \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{\eta}} \right)^2 \Bigg] + \left[ \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\eta}} + \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{x}} \right]^2 \\ &+ \frac{RD}{C\_{\infty}} \Bigg[ \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{\eta}} \right)^2 \Bigg] + \frac{RD}{T\_{\infty}} \Bigg[ \left( \frac{\partial T}{\partial \boldsymbol{x}} \right) \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{x}} \right) \\ &+ \left( \frac{\partial T}{\partial \boldsymbol{\eta}} \right) \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{\eta}} \right) \Bigg] + \frac{\sigma B\_0^2}{T\_{\infty}} u^2 + \frac{\nu}{k\_p} u^2. \tag{25} \end{split} \tag{25}$$

Using Equation (25) modified with the help of Equation (17),

$$\begin{split} S\_{\text{gen}}^{\prime\prime} &= \frac{K\_1}{T\_{\infty}^2} \left[ \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right)^2 + \frac{16 \boldsymbol{\sigma}^\* T\_{\infty}^3}{3 k k^\*} \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right)^2 \right] + \frac{\mu}{T\_{\infty}} \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\chi}} \right)^2 \\ &+ \frac{RD}{C\_{\infty}} \left( \frac{\partial C}{\partial \boldsymbol{\chi}} \right)^2 + \frac{RD}{T\_{\infty}} \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right) \left( \frac{\partial C}{\partial \boldsymbol{\chi}} \right) + \frac{\sigma B\_0^2}{T\_{\infty}} \boldsymbol{u}^2 + \frac{\nu}{k\_{\text{p}}} \boldsymbol{u}^2. \end{split} \tag{26}$$

Dimensionless system of entropy generation is defined as:

$$E\_G = \operatorname{Re}\left(1 + \frac{4}{3}Rd\right)\left{\boldsymbol{\theta}^{\prime 2}} + \operatorname{Re}\frac{\operatorname{Br}}{\Omega}{\Omega}\boldsymbol{\xi}^{\prime 2} + \operatorname{Re}\left(\frac{\boldsymbol{\xi}}{\Omega}\right)^2\lambda\boldsymbol{\phi}^{\prime 2}$$

$$+\operatorname{Re}\frac{\zeta}{\Omega}\lambda\phi^{\prime}\theta^{\prime}+\frac{Br}{\Omega}(\!M+K)\text{f}^{\prime}.\tag{27}$$

The Bejan number states

Be = Entropy genration due to irrevesablity of heat and mass transfer Total entropy generated

Be =

$$\frac{\left(\text{Re}\left(1+\frac{4}{3}\text{Rd}\right)\boldsymbol{\theta}^{\text{2}}+\text{Re}\left(\frac{\zeta}{\Omega}\right)^{2}\lambda\boldsymbol{\phi}'^{2}+\text{Re}\frac{\zeta}{\Omega}\lambda\boldsymbol{\phi}'\boldsymbol{\theta}'\right)}{\text{Re}\left(1+\frac{4}{3}\text{Rd}\right)\boldsymbol{\theta}'^{\text{2}}+\text{Re}\frac{\text{Re}}{\Omega^{2}}\boldsymbol{f}'^{'1}+\text{Re}\left(\frac{\zeta}{\Omega}\right)^{2}\lambda\boldsymbol{\phi}'^{2}+\text{Re}\frac{\zeta}{\Omega}\lambda\boldsymbol{\phi}'\boldsymbol{\theta}'+\frac{\text{Re}}{\Omega^{2}}(M+K)\boldsymbol{f}'^{2}}.\tag{28}$$

#### HOMOTOPY SOLUTIONS

There are several techniques available to solve non-linear problems. The homotopy analysis method (HAM) is initially constructed by Liao [47]. Moreover, he altered a non-zero auxiliary parameter [48]. This parameter shows the way to calculate the convergence rate. It also offers great independence with which to make the initial guesses of the solutions.

The initial guesses for satisfying the boundary conditions

$$\begin{aligned} f\_0 &= f\_\nu + \left(\frac{1}{1+\Gamma}\right) 1 - e^{-\eta} \\ \theta\_0 &= \frac{Bi \ast e^{-\eta}}{1+Bi} \\ \phi\_0 &= -\left(\frac{Nt}{Nb}\right) \ast \frac{Bi \ast e^{-\eta}}{1+Bi} .\end{aligned}$$

Lf , Lθ , and Lφ are the linear operators

$$\begin{aligned} L\_f &= f''' - f' \\ L\_\theta &= \theta'' - \theta \\ L\_\phi &= \phi'' - \phi \end{aligned}$$

while obeying the resulting properties

$$\begin{aligned} L\_f \left[ E\_1 + E\_2 e^{\eta} + E\_3 e^{-\eta} \right] &= 0 \\ L\_\theta \left[ E\_4 e^{\eta} + E\_5 e^{-\eta} \right] &= 0 \\ L\_\phi \left[ E\_6 e^{\eta} + E\_7 e^{-\eta} \right] \end{aligned}$$

The zeroth order deformation is

$$\begin{aligned} \left(1 - p\right) L\_f \left[ f \left( \eta; p \right) - f \left( \eta \right) \right] &= p \, h\_f \mathcal{N}\_f \left[ f \left( \eta; p \right) \right] \\ \left(1 - p\right) L\_\theta \left[ \theta \left( \eta; p \right) - \theta\_0 \left( \eta \right) \right] \\ &= p \, h\_\theta \mathcal{N}\_\theta \left[ \theta \left( \eta; p \right), f \left( \eta; p \right), \phi \left( \eta; p \right) \right] \\ \left(1 - p\right) \mathcal{N}\_\theta \left[ \phi \left( \eta; p \right), f \left( \eta; p \right) \right] \end{aligned}$$

$$\begin{aligned} &\left[ (1-p)L\_{\phi} \left[ \phi \left( \eta; p \right) - \phi\_0 \left( \eta \right) \right] \right] \\ &= p \left. h\_{\phi} \mathcal{N}\_{\phi} \left[ \phi \left( \eta; p \right), \theta \left( \eta; p \right), f \left( \eta; p \right) \right] \right| \end{aligned}$$

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where p ǫ [0, 1]

hf , h<sup>θ</sup> , and h<sup>φ</sup> are the non-zero auxiliary constants, and N<sup>f</sup> , N<sup>θ</sup> , and N<sup>φ</sup> are the non-linear operators given by

$$\begin{split} \mathcal{N}\_{f}\left[f\left(\eta;p\right),\theta\left(\eta;p\right)\right] &= \frac{\partial^{3}f\left(\eta;p\right)}{\partial\eta^{3}} - \left(\frac{\partial f\left(\eta;p\right)}{\partial\eta}\right)^{2} \\ &+ f\left(\eta;p\right)\frac{\partial^{2}f\left(\eta;p\right)}{\partial\eta^{2}} \\ &+ \alpha\_{1}\left(2\frac{\partial f\left(\eta;p\right)}{\partial\eta}\frac{\partial^{3}f\left(\eta;p\right)}{\partial\eta^{3}} - f\left(\eta;p\right)\frac{\partial^{4}f\left(\eta;p\right)}{\partial\eta^{4}}\right) \\ &+ \left(3\alpha\_{1} + 2\alpha\_{2}\right)\left(\frac{\partial f\left(\eta;p\right)}{\partial\eta}\right)^{2} \\ &+ 6\beta Re \frac{\partial^{3}f\left(\eta;p\right)}{\partial\eta^{3}}\left(\frac{\partial f\left(\eta;p\right)}{\partial\eta}\right)^{2} \\ &- \left(M + K\right)\frac{\partial f\left(\eta;p\right)}{\partial\eta} \end{split}$$

$$\begin{split} \mathcal{N}\_{\theta} \left[ f \left( \eta; p \right), \theta \left( \eta; p \right), \phi \left( \eta; p \right) \right] &= \left( 1 + \frac{4}{3} R d \right) \frac{\partial^{2} \theta \left( \eta; p \right)}{\partial \eta^{2}} \\ &+ \text{Prf} \left( \eta; p \right) \frac{\partial \theta \left( \eta; p \right)}{\partial \eta} - \text{Pr\y} \left[ \left[ f \left( \eta; p \right) \right] \right]^{2} \frac{\partial^{2} \theta \left( \eta; p \right)}{\partial \eta^{2}} \\ &- \text{Pr\y} f \left( \eta; p \right) \frac{\partial f \left( \eta; p \right)}{\partial \eta} \frac{\partial \theta \left( \eta; p \right)}{\partial \eta} + \text{PrS} \theta \left( \eta; p \right) \\ &+ \text{PrNb} \frac{\partial \theta \left( \eta; p \right)}{\partial \eta} \frac{\partial \phi \left( \eta; p \right)}{\partial \eta} + \text{PrNt} \left[ \frac{\partial \theta \left( \eta; p \right)}{\partial \eta} \right]^{2} \end{split}$$

$$\begin{aligned} \mathcal{N}\_{\phi} \left[ f \left( \eta; p \right), \theta \left( \eta; p \right), \phi \left( \eta; p \right) \right] &= \frac{\partial^2 \phi \left( \eta; p \right)}{\partial \eta^2} \\ + \operatorname{Lef} \left( \eta; p \right) \frac{\partial \phi \left( \eta; p \right)}{\partial \eta} + \frac{\operatorname{Nt}}{\operatorname{Nb}} \frac{\partial^2 \theta \left( \eta; p \right)}{\partial \eta^2} \end{aligned}$$

$$\begin{aligned} f\left(0;p\right) &= f\_{\text{w}}f'\left(0;p\right) = 1 + \Gamma f''\left(0;p\right), \, f'\left(\infty;p\right) = 0\\ \theta'\left(0;p\right) &= -Bi\left(1 - \theta\left(0;p\right)\right), \theta\left(\infty;p\right) = 0\\ \phi'\left(0;p\right) &= -\frac{Nt}{Nb}\theta'\left(0;p\right), \phi'\left(\infty;p\right) = 0 \end{aligned}$$

The mth order deformation equations are

$$\begin{aligned} L\_f \left[ f\_m \left( \eta \right) - \chi\_m f\_{m-1} \left( \eta \right) \right] &= h\_f R\_{f,m} \left( \eta \right), \\ L\_\theta \left[ \theta\_m \left( \eta \right) - \chi\_m \theta\_{m-1} \left( \eta \right) \right] &= h\_\theta R\_{\theta,m} \left( \eta \right), \\ L\_\phi \left[ \phi\_m \left( \eta \right) - \chi\_m \phi\_{m-1} \left( \eta \right) \right] &= h\_\phi R\_{\phi,m} \left( \eta \right). \end{aligned}$$

where

$$\chi\_m = \begin{cases} 0, & m \le 1 \\ 1, & m > 1 \end{cases}$$

$$\begin{aligned} R\_{f,m}(\eta) &= f\_{m-1}^{\prime\prime} + \sum\_{k=0}^{m-1} \left[ f\_{m-1-k} f\_k^{\prime\prime} - f\_{m-1-k}^{\prime} f\_k^{\prime\prime} \right] \\ &+ \alpha\_1 \left( 2f\_{m-1-k}^{\prime} f\_k^{\prime\prime} - f\_{m-1-k} f\_k^{\prime\prime} \right) \\ &+ \left( 3\alpha\_1 + 2\alpha\_2 \right) f\_{m-1-k}^{\prime\prime} f\_k^{\prime\prime} \\ &+ 6\beta \text{Re} f\_{m-1-l}^{\prime\prime} \sum\_{j=0}^{l} f\_{l-j}^{\prime\prime} f\_j^{\prime\prime} - \left( M + K \right) \alpha f\_{m-1-k}^{\prime\prime} \end{aligned}$$

$$\begin{aligned} R\_{\theta,m}(\eta) &= \left(1 + \frac{4}{3} R d\right) \theta\_{m-1}^{''} + Pr \sum\_{k=0}^{m-1} \left[\theta\_{m-1-k}^{'} f\_k\right] \\ &- Pr \gamma \left[\left(f\_{m-l-1} \sum\_{j=0}^{l} f\_{1-j} ^{'} \theta\_{j}^{'} + f\_{m-l-1} \theta\_{l}^{''}\right)\right] \\ &+ Pr \text{Nb} \sum\_{k=0}^{m-1} \theta\_{m-1-k} ^{'} \phi\_{k} + Pr \text{Nt} \sum\_{k=0}^{m-1} \theta\_{m-1-k} ^{'} \theta\_{k}^{'} \\ &+ Pr \text{S} \theta\_{m-1} \end{aligned}$$

$$\begin{aligned} R\_{\phi,m}(\eta) &= \stackrel{\circ}{\phi\_{m-1}} + \stackrel{\circ}{L} \sum\_{k=0}^{m-1} \, \phi\_{m-1-k}{}^{'} \acute{f}\_k \\ &+ \frac{Nt}{Nb} \sum\_{k=0}^{m-1} \, \theta\_{m-1-k}{}^{'} \partial\_{k}{}^{'} \end{aligned}$$

$$\begin{aligned} f\_m^{\prime}(0) &= 0, \left. f\_m^{\prime} \right| \left( 0 \right) - \left. \Gamma \right|\_m^{\prime\prime} \left( 0 \right) = 0, \left. \theta\_m^{\prime\prime} \left( 0 \right) - Bi \theta\_m \left( 0 \right) = 0, \\ \phi\_m^{\prime\prime}(0) + \frac{Nt}{Nb} \theta\_m^{\prime\prime}(0) &= 0 \\ \left. f\_m^{\prime\prime}(\eta) \right| \to 0, \left. \theta\_m^{\prime\prime} \left( \eta \right) \to 0, \left. \phi\_m^{\prime\prime} \left( \eta \right) \to 0 \text{as } \eta \to \infty. \end{aligned}$$

with boundary conditions

$$\begin{aligned} f\_m' \left( 0 \right) &= f\_m \left( 0 \right) = f\_m' \left( \infty \right) = \theta\_m \left( 0 \right) = \theta\_m \left( \infty \right) = \phi\_m \left( 0 \right) \\ &= \phi\_m \left( \infty \right) = 0. \end{aligned}$$

The appropriate solutions - f ∗ <sup>m</sup>, θ ∗ <sup>m</sup>, φ ∗ m are

$$\begin{aligned} f\_m(\eta) &= f\_m^\*(\eta) + E\_1 + E\_2 e^{\eta} + E\_3 e^{-\eta}, \\ \eta\_m(\eta) &= \eta\_m^\*(\eta) + E\_4 e^{\eta} + E\_5 e^{-\eta}, \\ \eta\_m(\eta) &= \eta\_m^\*(\eta) + E\_6 e^{\eta} + E\_7 e^{-\eta}. \end{aligned}$$

#### CONVERGENCE ANALYSIS

The auxiliary parameters h<sup>f</sup> , hθ , and hφ act as a vital part of convergence series solutions. The h-charts of f ′′ (0), <sup>θ</sup> ′ (0), and φ ′ (0) for Re,γ, and Nb are shown in **Figure 2**. From these curves, the straight line is referred as the h-curve. The convergent approximation is selected from this straight line of the curves. We note that h-curve of f ′′ (0), <sup>θ</sup> ′ (0), and φ ′ (0) shrinks as we enhance the range of Re,γ, and Nb, which shows the larger order approximation will be needed if the larger value of Re,γ, and Nb is employed. Approximations values of HAM with CPU time is denoted in **Table 1**.


### COMPUTATIONAL RESULTS AND DISCUSSION

The numerical calculations of velocity, concentration, temperature, entropy generation, and the Bejan number are discussed in this section. The homotopy technique is used for solving the non-linear governing Equations (18)–(20) with boundary conditions (21). The graphical results of entropy, Bejan number, temperature, Nusselt number, nanoparticle concentration, Sherwood number, and velocity profiles are computed via different flow parameters included in this study with the fixed values of α<sup>1</sup> = α<sup>2</sup> = β = Bi = f<sup>w</sup> = N<sup>t</sup> = 0.2, Re = Pr = Le = Ŵ = 1.0, M = γ = 0.5, Rd = 0.3, Hg = −0.1, N<sup>b</sup> = 0.4, Br = 5.0, and λ = = ζ = 1.0.

#### Impact on Velocity

It is observed from **Figure 3A** that the velocity is reduced to increase the values of the velocity slip parameter due to the ratio of stretching velocity and the viscosity of the fluid. The effect of the slip parameter in velocity has had more impact in the absence of a porous medium. **Figure 3B** indicates that the velocity diminishes when the magnetic field (M) raises. Improved Lorentz force is observed because of the increasing the values of M that opposes the fluid motion. Thus, we conclude that the velocity profile diminishes.

### Impact on Temperature

The impact of the radiation parameter Rd on the temperature profile is examined in **Figure 4A**. The temperature in the radiation parameter is high. Comparing the radiation effects with Christov-Cattaneo and normal heat flux, we observed that the radiation effect is quite low for Christov-Cattaneo heat flux. **Figure 4B** displays the influence of the Biot number on the temperature profile. From this figure, we observed that temperature is a rising function of Bi close to the sheet. Since Bi affects more temperature near the surface. Heat transfer

resistance is higher within the body compared to the surface of the sheet for rising values Bi.

### Impact on Nanoparticle Volume Concentration

In **Figure 5A**, the effect of concentration profile with growing values of the Brownian motion parameter Nb is depicted. In suction cases, higher values of Nb increase the concentration profile close to that of the surface of the sheet, and, suddenly, the concentration begins to fall, stabilizing far away from the surface of the sheet. This is due to the appearance of passive surface conditions for the concentration profile. Moreover, the injection at the sheet shows the concentration is rising near the sheet, and it diminishes far away from the surface of the sheet. The influence of the thermophoresis parameter Nt in the concentration profile φ (η). is highlighted in **Figure 5B**. When

there is suction in the sheet, with a rise in Nt, the concentration of the liquid decreases near to thsheet at certain stage starts to fall and stabilize away from the endpoints of the surface of the sheet. When there is an injection at the sheet, higher values of Nt decreases the concentration profile near the sheet, and it increases far away from the surface of the sheet.

#### Impact on Entropy Generation

The Effects of distinct fluid parameters on the entropy generation profile are highlighted in **Figures 6A–G**. **Figure 6A** depicts the influence of porous parameter in entropy profile. Initially, the entropy rate increases for the porous parameter at a certain stage (η = 0.6) before it becomes to fall. The responses of the slip parameter on entropy generation were succinctly depicted in **Figure 6B**. From this figure, it was obviously noted that entropy generation was inversely proportional to slip parameter. This causes a decrease in large values of slip parameter and temperature gradients in the boundary layer when retaining the fluid friction as we proceed. This occurrence induces a suppression in entropy generation since heat transfer was committed. **Figures 6C–E** displayed the impact of the suction\ injection parameter (fw.), material parameter (β) and Biot number (Bi) on the entropy profile (EG). Our examination obtained that higher range of fw, reduces the entropy generation profile, and the entropy generation rate is enhanced for higher β and Bi.

The effect of thermal relaxation time (γ ) on the entropy genation profile is sketched in **Figure 6F**. It is obvious that thermal relaxation time is small for temperature and heat transfer rates. In addition, domination of the irreversibility in heat transfer affected the heat flux. Thus, we have seen a small increase in the entropy of the system. Performance of Nt and Nb on entropy generation profile (EG) is shown in **Figures 6G,H**, which shows that entropy is increased with an increase of Nt, whereas E<sup>G</sup> is inversely proportional to Nb. The Brownian motion induces the nanoparticles temperature, but it reduces the temperature gradient on wall. As a result, entropy generation parameter reduces where Nt is directly proportional to temperature gradient and creates ambient atmosphere for higher values of EG.

### Impact on the Bejan Number

The Bejan number (Be) is a dimensionless quantity that specifies the ratio of entropy generation between heat transfer and the total entropy generation where Bejan numbers take values from 0 to 1. If Be is nearly equal to 1, the entropy generation will become more due to heat transfer. It is clear from **Figure 7A** that, with an escalation in the applied magnetic field, there is an augmentation in the Bejan number. The consequence of heat transfer entropy develops as we move up from the surface. The entropy effect has full domination because of the heat transfer while it is also outlying from the region. This is the reason behind how the augmenting value of M brings a stronger frictional effect, which leads to an increase in the liquid temperature. There is also a consequent development in the Be, as shown in **Figure 7A**. From **Figure 7B**, it is observed that the Be is increased for the increase in the slip parameter (Ŵ). Physically, larger values of (Ŵ) enhance the temperature gradient inside the regime, which induces the Bejan number and irreversibility of heat transfer.

**Figure 7C** states that, with the rise of the radiation constant Rd, the Bejan number is boosted. This is due to the total entropy generation dominated by thermal irreversibility. **Figure 7D** shows the effect of the thermal relaxation time (γ ) on the Be. At first, the Be is augmented for higher values of γ at a sudden point (η = 2.2). Consequently, the Bejan number profile reduces for the values of γ . **Figure 7E** shows that the Bi displays a trend of raising the Be. The demonstration of such an increasing trend of the Bejan number explains how the entropy production near the surface is large due to the liquid friction—at least relative to that of the heat transfer irreversibility. The variation of the Brinkmann number Br is sketched in **Figure 7F**. This figure shows that the Bejan number is reduced, as we have to enhance the Br. **Figures 7G,H** shows the influence of thermophoresis (Nt) and Brownian motion (Nb) parameters on the Bejan number. From these plots, we note that Ntand Nb have inverse effects on the Bejan number profile.

### Impact on Physical Entities

**Figures 8**–**11** illustrate the effects of different physical parameters on the local Nusselt number and local Sherwood number. The influence of f<sup>w</sup> and γ on Nu<sup>x</sup> is shown in **Figure 8**: heat transfer decays for higher values of f<sup>w</sup> and γ . The same phenomena can be observed for larger values Hg and γ on the Nusselt number profile, as presented in **Figure 9**. The combined effects of Nb and M as well as Nt and M are shown in **Figures 10**, **11**, respectively. From these figures, we conclude that thermophoresis (Nt) and Brownian motion (Nb) parameters have produce the converse trend in the mass transfer rate.

### NUMERICAL CODE VALIDATION

In this segment, we examine the code validation of early published works of literature. **Table 2** validates the results of

TABLE 2 | Validation of −*f* ′′ (0) and −θ ′ (0) for the limiting case *M* = 0, *K* = 0, *Nb* = 0, *Nt* = 0, *Rd* = 0, *S* = 0, and *Bi* → ∞.


−f ′′(0) and <sup>−</sup><sup>θ</sup> ′ (0) for the limiting case M = 0, K = 0, Nb = 0, Nt = 0, Rd = 0, S = 0, and Bi → ∞ with Imtiaz et al. [37]. Moreover, the skin friction rate is also validated by the same literature [37] when M = K = 0 (see **Table 3**). **Table 4** exhibits the matching results of reduced Nusselt number with the references [20, 49–51]. From the above validation, results show that the current simulation is considered an efficient one.

### KEY RESULTS

The present research work examines the entropy generation influence on third-grade nanoliquid flow caused by a stretching sheet in the appearance of Magnetic field, radiation, and convective heating effects. Christov-Cattaneo heat flux replaces

TABLE 3 | Comparison of *Re*0.5*C<sup>f</sup>* for different values when *M* = *K* = 0.


TABLE 4 | Matching results of reduced Nusselt number with the restricting case *Rd* = *Ec* = *M* = *Nt* = *Nb* = γ = *S* = *K* = Ŵ = *f<sup>w</sup>* = 0, and *Bi* → ∞.


ordinary heat flux. HAM is employed to validate the nonlinear governing equations. Results of velocity, temperature, nanoparticle volume concentration, the system of entropy, the Bejan number, mass, and heat transfer rates are presented graphically. We obtained the following main upshots:

### REFERENCES


### DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### FUNDING

This research was supported by the Research Group Project (RGP), University of Tabuk, Tabuk 71491, Saudi Arabia, under Grant No. RGP-0207-14440.

convective boundary condition. Int J Heat Mass Transfer. (2013) 64:384–92. doi: 10.1016/j.ijheatmasstransfer.2013.04.032


embedded in a porous medium. Int Commun Heat Mass. (2006) 33:1096– 102. doi: 10.1016/j.icheatmasstransfer.2006.04.009


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Loganathan, Muhiuddin, Alanazi, Alshammari, Alqurashi and Rajan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

### LIST OF SYMBOLS


*(Continued)*


ζ Dimensionless concentration difference

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