ORIGINAL RESEARCH article

Front. Mater., 13 March 2023
Sec. Colloidal Materials and Interfaces
Volume 10 - 2023 | https://doi.org/10.3389/fmats.2023.1139284

Numerical computation for dual stratification of slip flow of sutterby nanofluids with heat generation features

Ubaid Ullah1 Syed Inayat Ali Shah1 Kottakkaran Sooppy Nisar2,3* Hamid Khan1 Naeem Ullah1* Muhammad Yousaf4
• 1Department of Mathematics, Faculty of Technologies and Engineering Sciences, Islamia College University Peshawar, Peshawar, KP, Pakistan
• 2Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Abdulaziz, Saudi Arabia
• 3School of Technology, Woxsen University, Hyderabad, Telangana, India
• 4Department of Mathematics, University of Malakand, Chakdara, KP, Pakistan

The current communication, manifest mathematical modelling and numerical computations of Sutterby nanofluids with radiant heat assessment subject to heat generation/absorption. The thermophoresis and Brownian motion effects are incorporated via the Buongiorno model in flow governing equations. Moreover, the present analysis reveals the impacts of thermal stratification, velocity slip, and a magnetic field on flow phenomena. The non-Newtonian nature is modelled using Sutterby fluid. The proposed model is formulated mathematically through basic partial differential equations relating mass, momentum, energy, and nanoparticle concentration conservations using boundary layer theory. We adapted the generated governed equations to ordinary differential equations utilizing similarity variables mechanism. Numerical treatment for the reduced system of ordinary differential equations is performed using the built-in MATLAB code bvp4c. The impacts of distinct characterizing parameters on velocity, temperature, and concentration profiles are determined and analyzed via graphs. The existence of velocity slip parameter, fluid flow is significantly dwindle, while the surface friction growth is sophisticated. Brownian and thermophoresis mechanisms degrade the heat transmission rate and escalate the mass flux. The thermal and solutal stratification exhibits opposite conduct for thermal and concentration of the nanoparticles.

1 Introduction

In the era of ever-increasing high demand for the improved thermal capabilities of ordinary fluids, like water, glycol, ethylene, etc., nanotechnology is an opening gate for a revolutionary modernized world. Nanofluids are a new generation of heat enhancement fluids with tiny metallic particles (1–100 nm) mixed in the ordinary fluids, thereby enhancing their thermal characteristics significantly. These particles consist of different metals (copper, gold, silver, titanium, etc.) or their oxides. They have novel implications in biomedical and engineering sciences, along with other industrial processes where heat transfer enhancement is prominent. The nanofluids have a wide range of uses in transportation, nuclear reactors, vehicle thermal applications, solar-based science, imaging and sensing applications, food packaging, etc. Choi (Choi, 1995) was the first to describe this type of fluid, which he dubbed “nanofluids”. Xuan and Li (Xuan and Li, 2000) developed preparatory techniques for multiple sampling nanoparticles, and their thermophysical properties, such as, shape, amount, attributes, and dimensions were investigated. Buongiorno (Buongiorno, 2006) discussed the significance of a heterogeneous scientific mixture of nanomaterial processing stimulus and heat transport limits. Later, Kuznetsov an d Nield (Kuznetsov and Nield, 2010) conducted an analytic study for natural convective flow of viscous nanofluids across a vertical flat surface. Makinde and Aziz (Makinde and Aziz, 2011) discussed the convective heat transport for boundary layer flow of nanofluid generated by a linearly stretched surface. In another study, Mustafa et al. (Mostafa et al., 2011) reported an analytic investigation for stagnation-point flow of nanofluid using Buongiorno model. After that, Ibrahim et al. (Ibrahim et al., 2013) gave numerical solutions for MHD stagnation point flow generated by stretching surface in the presence of nanoparticles. The comprehensive analysis of literature in the framework of nanofluids flow and heat transport characteristics can be found in the works of (Hamad and Ferdows (Hamad and Ferdows, 2012), Kalidas Das (Das, 2012), Bachok et al. (Bachok et al., 2012), Turkyilamzoglu (Turkyilmazoglu, 2012), and Alsaedi et also (Effect of heat generation, 2012), etc.).

The fundamental manifestation of Navier-Stoke’s theory of fluid dynamics is the no-slip boundary condition. However, there seem to be cases when this scenario is inappropriate. Specifically, for several non-Newtonian fluids and nanofluids, the no-slip boundary condition is inadequate, as certain polymer melts frequently exhibit microscopic wall slip, which is governed in general by a non-linear and monotone relationship between slip velocity and traction. Micro electromechanical systems (MEMS) have developed numerous microfluidic devices in the medical, physical, biological, chemical, engineering, and energy domains in recent years. The physical aspects of microscale flow and heat transmission, which may differ from macroscale flow, must be thoroughly understood to meet the technical needs. In an earlier studies, Andersson (Andersson, 2002) conducted the investigations that considered the slip boundary condition during the flow over a stretching sheet. He gave the exact solutions for modelled flow equations. Wang (Wang, 2002) gave the exact similarity solution for flow equations driven by a stretching geometry with partial slip condition. Again, Wang (Wang, 2006) researched at stagnation slip flow and heat transfer on a moving plate. Later, Wu (Wu, 2008) suggested a new second-order slip model based on the linear Boltzmann equation’s. Similarly, Fang et al. (Fang et al., 2009) used mathematical methods to study slip magnetohydrodynamics viscous flow over a stretching sheet. Some notable research works showcasing the innovative properties of velocity slip during fluid flows can found in the references (Sutterby, 1965; Sutterby, 1966; Aziz, 2010; Fang et al., 2010; Hayat et al., 2011; Mahantesh et al., 2012).

The stratification phenomenon in fluids has recently attracted a lot of attention in heat and mass transport assessment, and it is a mechanism that occurs as a result of temperature, concentration, and density fluctuations in various fluids. In fact, due to its prevalence in geophysical flows such as oceans, rivers, ponds, solar ponds, and thermal energy storage technologies, evaluating the flow through a double stratified media is an essential fluid topic. Hayat et al. (Hayat et al., 2015) evaluate a dual stratified radiative flow of reactive species Oldroyd-B fluid in the presence of mixed convection. Hayat et al. (Hayat et al., 2017) report an assessment of chemical reactions in mixed convective squeezing fluid flows with thermal radiation. Muhammad et al. (Rehman et al., 2016) demonstrate a dual stratified flow of squeezed viscous fluid utilizing modified Fick’s and Fourier’s theories. The stagnation point Magnetohydrodynamic flowing of reactive chemical Powell-Eyring nano-fluid through dual stratified substrate with thermal radiation was established by Ramzan et al. (Ramzan et al., 2017). Some applications of thermal radiation and chemical reaction of various stratified flows in physiology and industry can be seen in literature (Mahanthesh et al., 2016; Rehman et al., 2016; Daniel et al., 2017; Muhammad et al., 2017; Bég et al., 2020; Lin and Ghaffari, 2021; Rehman et al., 2021; Unyong et al., 2021; Nandi et al., 2022). However Rashid et al. studied the unsteady slip flow amicroplor nanofluid over an impulsively stretched vertical surface see (El-Hakiem and Rashad, 2007; El-Kabeir et al., 2007; El-Kabeir et al., 2010; Chamkha et al., 2011; Tlili et al., 2019; Reddy et al., 2020; Nabwey et al., 2022). Furthermore, Tripathi et al. investigated the peristaltic pumping of hybrid nanofluids throught an asymmetric microchannel in the presence of electromagnetic field see (Akram et al., 2020a; Akram et al., 2020b; Prakash et al., 2020; Tripathi et al., 2020; Akram et al., 2021; Tripathi et al., 2021; Akram et al., 2022; Prakash et al., 2022; Saleem et al., 2022).

In any of the aforementioned analyses, the Sutterby fluid model was never used in conjunction with thermal radiation and a heat generating or absorbing source. The constitutive equations of Sutterby fluid are used in mathematical modelling to achieve this goal. The Sutterby fluid framework illustrates diluted polymer solutions and is one of the non-Newtonian fluid models used to study the rheological properties of various materials. As a result, the current effort is to investigate the heat-producing/absorbing Sutterby fluid flow across horizontal geometry with thermal radiation, as well as double stratification near thermal radiation. As an outcome, the influence of different flow, heat, and mass transport attributes is reported. Nusselt and Sherwood values, as well as drag force (skin friction coefficient), are visually assessed versus numerous emerging parameters.

2 Model development

As shown in Figure 1, an incompressible two-dimensional flow of non-Newtonian Sutterby fluid with heat and mass transport past an infinite flat surface is deliberated. The stretching surface has the linear velocity uw(x)=ax, where a is constant. Non-linearly varying thermal radiation, velocity slip, heat generation/absorption are also incorporated in this analysis. The thermal stratification mechanism is studied by considering the surface temperature Tw(x)=T0+m1x and ambient temperature T(x)=T0+m2x, respectively. Similarly, the surface concentration Cw (x)=C0+m1x and free stream concentration C(x)=C0+m2x, is taken to discuss the solutal stratification. A magnetic force of intensity B0 is applied parallel to the plate's y-axis, which is perpendicular to it. The induced magnetic field created by the velocity of an electrically conducting fluid is supposed to be insignificant in this case. The fluid is supposed to be grayish, generating, and absorbent but quasi medium, and the optically thick radiating limit is taken into account in this study, where the radiant heat flux term can be reduced using the Rosselant estimation. The use of Cartesian coordinates (x,y) to study the flow under discussion is acceptable because the x-axis runs parallel to the stretched sheet and the y-axis runs perpendicular to it. The following are the rheological equations that characterize the incompressible flow of Sutterby fluid:

2.1 Sutterby fluid model

The Cauchy stress tensor for Sutterby fluid (Sutterby, 1965; Sutterby, 1966) is expressed as

$τ=−pI+μS(1)$

where extra stress tensor has the form

$S=μ0sinh−1βγ.βγ.A1(2)$

The shear rate can be defined as:

$γ.=12trA12(3)$

where μ0 is dynamic viscosity, B is the fluid parameter, n denoted the power-law index and “tr” means trace. For Sinh−1 function, the second order approximation is considered as:

$sinh−1βγ.≈βγ.−βγ.36(4)$

As a result, the stress tensor takes the form:

$S=μ01−βγ.26nA1(5)$

For two-dimensional steady flow, we take the velocity field of the form

$V=ux,y,vx,y,0(6)$

In view of Eq. 4, the shear rate takes the form.

The extra stress components are expressed as below

$γ.=2∂u∂x2+∂u∂y+∂v∂x2+2∂u∂x212(7)$

The extra stress components are expressed as below

$Sxx=−P+2μ01−βγ.26n∂u∂y(8)$
$Sxy=μ01−βγ.26n∂u∂y+∂v∂x(9)$
$Syy=−P+2μ01−βγ.26n∂v∂y(10)$

Based on the above restrictions and involving boundary layer approximations, the prevailing equations for Sutterby nanofluids using the Buongiorno model are given as

$∂u∂x+∂v∂y=0(11)$
$u∂u∂x+v∂u∂y=v1−β26∂u∂y2n∂2⁡u∂y2−nvβ261−β26×∂u∂yn−1∂u∂y2∂2u∂y2−σB02uρ,(12)$
$u∂T∂x+v∂T∂y=α∂2T∂y2+τnpDB∂C∂y∂T∂y+DTT∞∂T∂y2−1ρCp16σ*T∞3kk*∂2T∂y2+Q1T−T∞,(13)$
$u∂T∂x+v∂C∂y=DB∂2c∂y2+DTT∞∂2T∂y2,(14)$

The associated boundary conditions

$u=uw+L1−B26∂u∂y22∂u∂y,v=0,T=Tw,C=Cwaty=0(15)$
$u→U∞,T→T∞,C→∞asy→∞(16)$

2.2 Transformations

The dimensionless form of the modelled problem is obtained by utilizing the following dimensionless variables:

$ψ=avxfη,η=yav,θη=T−T∞Tw−TT∞,ϕη=C−C∞Cw−C∞.(17)$

Here, f′(η), θ(η),and ϕ(η) represents the dimensionless stream function, dimensionless temperature and concentration. Making use of non-dimensional transformation, the leading Eqs. (4.12)–(4.14) reduced to

$1−16DeRef′′2nf′′′−nDeRe31−16DeRef′′2n−1f′′′f′′2+ff′−f′2−Mf′=0(18)$
$1+43Rθ″+Prfθ′−f′θ+PrNbθ′ϕ′+Ntθ′2−Prstf′+PrQθ=0,(19)$
$h″+Lefϕ′−Leϕf′+NbNtθ″−LeScf′=0(20)$
$f0=1,f′0=1+Af′01−16DeRef′′20n,θ0=1−St,ϕ0=1−Sc,atη=0,(21)$
$f′∞=0,θ∞=0,ϕ∞=0asη→∞(22)$

where η is the similarity variables and the prime denotes differentiation with respect toη. The dimensionless parameters in the above equations are the Deborah number De, heat generation/absorption parameter Q, thermal stratification parameter st and solutal stratification parameter sc, given as $De=a2B2,Q=Q1ρCpa,St=m2m1,St=m4m3$

$CfRex=1−16DeRef″02f″0,NuxRex=11−s1Q′1,ShxRex=−11−S2ϕ′1.(23)$
$f=Z1,f′=Z2,f″=Z3,θ=Z4,θ′=Z5,ϕ′=Z7.(24)$
$Z1′=Z2,Z2′=Z3,Z3′=Z22−Z1z2+μZ2A1−nDeRe3A2,Z5′=PrNbZ5Z7−NtZ52+Z1Z5−Z2Z4−Z2ZStQZ41+43R,Z7′=−LeZ1z7+LeZ2Z6+LeZ2Sc−NbNtZ5′(25)$

along with boundary conditions

$Z10=0,Z20=1+AZ301−16DeReZ320n,Z40=1−St,Z60=1−Sc(26)$
$Z2∞→0,Z4∞→0,Z6∞→0.(27)$

The above set of seven first-order ODEs Eq. 25 with boundary conditions Eqs 26 and 27 is numerically integrated by exercising the MATLAB routine bvp4c which is a finite difference code that uses the collocation method to execute the three - stage Labatto IIIa formula. The numerical analysis is performed by selecting a suitable finite value (η = 10) to fulfill the for field boundary conditions. The error tolerance of 10−6is set for computational purpose. The RFK45 procedure is adaptive because it modifies the number and location of grid points throughout each iteration, keeping the local error within acceptable ranges. The asymptotic boundary conditions in Eq. 27 are supplemented in the existing situation by a predefined limit in the range 10–15, based on the parameter values. To assure that all numerical results approach the asymptotic values accurately, infinity is desired. The choice of an appropriately large number for infinity is essential for maintaining desirable reliability in boundary layer flows, and it is a common blunder seen in many investigations. The stepped equations used to estimate Eq. 25, employing fifth–fourth-order Runge-Kutta-Fehlberg procedures under conditions Eqs 26 and 27 are listed below (Bég et al., 2020; Bég et al., 2022).K0 = f (x0, y0).$K1=fx0+14h,y0+14K0h$$K2=fx0+38h,y0+332K0+932k1$$K3=fx0+1213h,y0+19322197k0−72002197K1+72962197k2h$$K4=fx0+h,y0+439216K0−8K1+3860513K2−8454104K3h$$K5=fx0+12h,y0+827k0+2K1−35442565K2−18594104K3−1140K4h$$yi+1+25216K0+14082565K2+21974104K3−115K4h$$Zi+1=Zi+16135K0+665612825K2+2856156430K3−940K4+255K5h$The fourth-order Runge-Kutta component is denoted by y, and the fifth-order Runge-Kutta stage is denoted by Z. By subtracting the two values obtained, an estimate of the error can be obtained. The findings can be redone with a reduced step size if the deviation reaches a certain threshold. The following is an example of how to determine the new step size:

$hnew=holdϵhold2Zi+1−yi+114(28)$

2.3 Results and discussion

FIGURE 1

FIGURE 1. The physical model and coordinate systems.

FIGURE 2

FIGURE 2. f′(η), alteration via M.

FIGURE 3

FIGURE 3. f′(η), alteration via De.

FIGURE 4

FIGURE 4. f′(η), alteration via Re.

FIGURE 5

FIGURE 5. f′(η), alteration via A.

FIGURE 6

FIGURE 6. θ(η), alteration via Pr.

FIGURE 7

FIGURE 7. θ(η), alteration via R.

FIGURE 8

FIGURE 8. θ(η), alteration via Nt.

FIGURE 9

FIGURE 9. θ(η), alteration via St.

FIGURE 10

FIGURE 10. θ(η), alteration via Q.

FIGURE 11

FIGURE 11. ϕ(η), alteration via R.

FIGURE 12

FIGURE 12. ϕ(η), alteration via Q.

FIGURE 13

FIGURE 13. ϕ(η)), alteration via Sc.

FIGURE 14

FIGURE 14. $skinCfRex$, sketch for frequent values M and A.

FIGURE 15

FIGURE 15. $skinCfRex$, sketch for frequent values of A and M.

FIGURE 16

FIGURE 16. $NussltNuxRex$, sketch for frequent values of Nb and Nt.

FIGURE 17

FIGURE 17. $ShaerwoodShxRex$, sketch for frequent values of Nb and Nt.

3 Conclusion

The investigation presented in this paper helps us to understand, physically as well as numerically, the aspects of activation energy on magnetized Sutterby nanoliquid subjected to heat generation/absorption through a Darcy porous medium. The well-established Buongiorno model is employed to examine the features of Brownian and thermophoresis diffusion of the nanofluid. The most significant outcomes noted from present research are described as follows.

g(ζ) nanoliquid temperature deteriorates for larger S1while it intensifies against Q.

• Positive values (heat generation) of Q, improve the temperature distribution while reverse trend is seen in case of negative values (heat absorption).

• Velocityfζ), is increasing function of n, and Da and dwindle for large M.

• Opposite trend of h(ζ) is detected against Nb (Brownian moment parameter) and Nb (thermophoresis parameter).

• The solutal stratification parameter S1, Le and γ (Chemical reaction rate) exhibitsdiminishing behavior for concentration of the nanofluid.

• Heat transfer rate is significantly rises with heat generating parameter.

• Large magnetic number reduces the skin friction of the nanofluid.

Data availability statement

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

Author contributions

There are six authors in this manuscript and each one has contributed properly. The mathematical model has been proposed by SA, all the numerical computations and their graphs have been carried out by UU. The discussion of graphs and their physical interpretation has been given by HK and KSN. The literature review and comparison of the present simulations with the classical data has been established by MY. The final review and amendments in the manuscript has been carried out by NU.

Funding

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

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.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Glossary

x, y Cartesian Coordinates

Da Darcy

u, v velocity components in x(y) directions

cp specific heat (JKg−1k−1)

D mass diffusivity

Db Brownian diffusion coefficient

Dt Thermophoretic diffusion coefficient

T0 Reference temperature(K)

Tw wall temperature(K)

C0 Concentration

E Activation energy J)

Q Heat generating absorbing J)

s1 $bd$ Thermal stratification

s2 $ce$ Solutal stratification

τnp $(ρcp)p(ρcp)f$ Ratio of the nano particle to fluid particle

K Thermal conductivity (Wm−1k−1)

tw sheer stress on the wall

Dimensionless functions

η similarity variables

h(η) represent the velocity component u

g(η) represent the velocity component v

τ shear stress

Re Reynolds number

Pr Prandtl number

Le Lewis number

Sc Schmidth number

Sh Sherwood number

Nu Nusselt number

Nt Thermophoresis parameter

Nb Brownian motion parameter

Grx Grashof number along xaxis

Greek Letters

α Thermal diffusivity

β Sutterby fluid coefficient

γ Chemical reaction

ϵ Sutterby fluid

δ Temperature ratio

μ Viscosity (Nsm−2)

ρ Density (Kgm−1)

ν Kinematic viscosity (m−2s−1)

Keywords: Double stratification, activation energy, heat generating, Sutterby fluid, Darcy porous medium, dual stratification, slip flow, MHD

Citation: Ullah U, Shah SIA, Nisar KS, Khan H, Ullah N and Yousaf M (2023) Numerical computation for dual stratification of slip flow of sutterby nanofluids with heat generation features. Front. Mater. 10:1139284. doi: 10.3389/fmats.2023.1139284

Received: 06 January 2023; Accepted: 20 February 2023;
Published: 13 March 2023.

Edited by:

Hammad Khalil, University of Education Lahore, Pakistan

Reviewed by:

A. M. Rashad, Aswan University, Egypt
Dharmendra Tripathi, National Institute of Technology Uttarakhand, India

Copyright © 2023 Ullah, Shah, Nisar, Khan, Ullah and Yousaf. 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.

*Correspondence: Kottakkaran Sooppy Nisar, n.sooppy@psau.edu.sa; Naeem Ullah, naeemullah1989@gmail.com