Abstract
An investigation of an axisymmetric mixed convective boundary layer flow of silver-titanium dioxide/water () hybrid nanofluid towards vertically and rotating stretching cylinder with entropy generation is conducted. The Cattaneo-Christov theory and joule heating effect are used to analyze the features of thermal energy. Moreover, the magnetic impact and convective boundary conditions on the vertical surface also considered in the current investigation. The developing equations for momentum, energy and entropy generation are modelled and by the usage of similarity variables to transform into the system of nonlinear ordinary differential equations (ODEs). The solutions of nonlinear ODEs are obtained numerically with the assistance of BVP4C MATLAB built-in scheme. The graphical consequences and relevant physical reasoning regarding the velocity, temperature, and concentration profiles are discussed. It is noteworthy that strong estimation of buoyancy ratio and mixed convection parameter enhances axial velocity, but the swirl velocity is diminished. The fluid temperature and concentration both are diminished due to thermal and solutal stratification effects. It is also seen that thermal Biot and Eckert numbers enhance the temperature distribution. Further, the Reynold number improves entropy generation.
1 Introduction
Nanofluids exhibit superior heat transport features to those of regular fluids. The effectiveness of heat exchange and the compactness of lower thermal conductivity fluid are delayed in the process, but many techniques are used to develop heat transport phenomena. One innovative trick is used to improve the heat transport/thermal conductivity of a fluid through the suspension of nanoparticles in the base liquids. Nanofluid thermal conductivity mainly depends upon the volume fraction of nanoparticles. The thermal conductivity of nanoparticles is expected to be greater than that of usual fluids. These nanoparticles are very small, not more than 100 mm. The term nanofluids was first used to refer to the fluids along suspended nanoparticles by Choi [1]. Some experimental outcomes have been obtained by Eastman et al. [2] with respect to the development of thermal conductivity with the suspension of in base fluid water. The transport of the heat and mass of the 3D stagnation point flow of a water-based nanofluid toward an exponentially stretching surface was identified by Rehman and Nadeem [3]. Sheikholeslami [4] present the numerical examination of the -water based nanofluid affected by a magnetic field about a porous channel. Bilal et al. [5] surveyed the numerical inspection of MHD and the thermal radiative flow of a Williamson nanofluid influenced by variable thermal conductivity across a stretching cylinder. Maskeen et al. [6] used a stretching cylinder to evaluate the heat transfer features of alumina-coper/water hybrid nanofluid. The heat and mass transport analysis of a chemically reactive Eyring-Powell nanofluid influenced by a Cattaneo-Christov heat flux was performed by Reddy et al. [7] through a stretching cylinder. Ullah et al. [8] explored the collective characteristics of heat sources and the zero mass flux of magnetized nanofluid flow with an activation effect across a rotating and stretchable disk. Khan et al. [9] elaborated a comparison between a linear and exponential stretching surface to identify the rotating impact on a nanofluid flow in stratification conditions. Dawar et al. [10] observed the convective flow of a Williamson nanofluid across the two different geometries with convective boundary conditions. Some meaningful research in the direction of nanofluid is presented in Refs. [11, 12]. Scientists have lately discovered novel techniques to boost thermal conductivity and the heat transport rate of a fluid. One of the best ways to improve the heat conductivity of fluid by the addition of more than two nanoparticles in the convectional fluid, which is said a hybrid nanofluid. A hybrid nanofluid is more beneficial than simple nanofluids. Some key developments in the direction of hybrid nanofluid can be found in Refs. [13–20].
In the modern era, entropy generation is the most valuable subject for the researcher, in which irreversible processes of mass and heat transport occur. The Second Law of Thermodynamics entirely revolves around entropy. The design of the system that depends upon heat transport has a valuable application in the real life of the entropy generation. Entropy minimized energy losses in the system. Heat and mass transport processes occur in heat exchange, heat engines, fluid flow, heat pumps, refrigerators, air conditioners, power plants, anelastic deformation, and substance mixing and expansion, among other sites. Significant work has been done on entropy with application to various fields by Adrian Bejan [21–24]. Mahian et al. [25] elaborated convective heat transport augmentation with the use of nanofluid flow and entropy generation impacts to develop high heat flux cooling devices. Siavashi et al. [26] scrutinized the heat transport and entropy generation characteristics of nanofluid flow across an annular pipe in a two-phase combination model. Rashid et al. [27] observed entropy generation on ferromagnetic fluid flows along the slip and nonlinear radiation impact across a stretching sheet. Khan et al. [28] reviewed the modeling and computational study of hybrid nanofluids subject to entropy generation. Muhammad et al. [29] examined the Darcy–Forchheimer boundary layer flow influenced by activation energy and entropy generation across a curved surface. Recent studies related to the investigation of entropy generation are given in Refs. [30–33].
Heat transportation occurs due to temperature gradients. There are three modes of heat transfer conduction, radiation, and convection. Heat transportation phenomena have many important industrial and engineering applications, such as heat exchange, power generation, nuclear power, refrigeration, petroleum production, and so on. Fourier [34] first developed the law of heat conduction to analyze the rate of heat transport in a system. Ellahi et al. [35] scrutinized the heat transfer rate of a mixed convective boundary layer fluid flow across a vertical permeable slender cylinder. The heat and mass transportation of a non-Newtonian fluid flow influenced by transverse magnetic field and suction/injection effect towards permeable stretching sheet was developed by Sandeep et al. [36]. The heat transfer scrutiny of MHD micropolar fluid flows subject to joule heating and chemical reactions across a stretching sheet were explored by Dawar et al. [37]. Ramadan et al. [38] analyzed that flow and heat transport in a microchannel influenced by gas cooling conditions and thermal creep. Finally, other studies of heat transfer are presented in Refs. [39–48].
The aim of this examination is to study the 3D axisymmetric MHD flow of a viscous nanofluid with nanoparticles generated by vertically rotating a stretching cylinder. The main concern of this existing inquiry is to identify entropy generation evaluation of mixed convective hybrid nanofluid flow with modified Fourier’s and Fick’s law across a rotating and stretching surface. Moreover, convective boundary conditions are also considered to identify a flow regime. The formulated fluid model is converted into a pair of ODEs by adopting appropriate similarity variables. The coupled ODEs are numerically manipulated with the aid of BVP4C MATLAB built-in technique [49–53]. A graphical inquiry into the evolving parameters with respect to temperature distribution, concentration distribution, and velocity profile is established and discussed. The comparison of current outcomes with previous investigation is presented, and a good harmony is shown between them.
2 Flow modelling
Here, we observe a steady, incompressible, axisymmetric laminar, and mixed convective boundary layer flow of hybrid nanofluid induced by a vertical and rotating stretching cylinder with a double diffusion Cattaneo-Christov and entropy generation impact. The equation of thermal and solutal energy are through the consideration of heat generation/absorption, chemical reaction, and joule heating impact. Furthermore, thermal and concentration convective conditions are enforced on the boundary of the stretching cylinder. A magnetic field () is applied in the direction of the axis to analyze the effect of the Lorentz force. The stretching cylinder has radius is , is the radial direction, and axis is measured along the cylinder. The cylinder rotates with an angular velocity and the stretching velocity is directly proportional to the axial distance. Moreover, the concentration and temperature of the surface nanoparticles are and , respectively, and away from the surface they are symbolized by and , respectively. The flow geometry and the coordinate axes are shown in Figure 1. Using the above-mentioned supposition with the presence of Boussinesq approximation, the flow equations are followed by [49, 50],
FIGURE 1
Flow diagram of the article.
The related surface and ambient conditions are stated as follows:
In the above Eqs. 1–6, and are the velocity of the fluid in the and directions, respectively. The symbols and indicate kinematic viscosity of hybrid nanofluid, electrical conductivity of hybrid nanofluid, gravitational force, thermal and concentration volumetric expansion coefficients, thermal relaxation time, thermal diffusivity, heat generation/absorption coefficient, specific heat, concentration relaxation time, density of fluid hybrid nanofluid, chemical reaction constant, mass diffusivity, coefficient of convective heat transfer, coefficient of convective mass transfer, and mass diffusivity, respectively.
2.1 Hybrid nanofluid model
The experimental relationship for an Ag-TiO/water hybrid nanofluid is given as follows [14],
2.1.1 Hybrid nanofluid dynamic viscosity
2.1.2 Hybrid nanofluid density
2.1.3 Hybrid nanofluid specific heat capacity
2.1.4 Hybrid nanofluid thermal conductivity
2.1.4 Thermal and solutal volumetric coefficient
Here, s1 and s2 specify the silver (Ag) and titanium dioxide (TiO2) nanoparticles, respectively. Further, the solid volume fraction of Ag is represented by and TiO2 with .
Table 1 presents the thermo-physical features of convectional fluid and hybrid nanofluid, as follows:
TABLE 1
| Physical properties | Base fluid | Nanoparticles | |
|---|---|---|---|
| H2O | Ag | TiO2 | |
| 4,179.0 | 235.0 | 686 | |
| 997.10 | 10,500 | 4,250 | |
| 0.620 | 429.0 | 8.9538 | |
| 21 | 1.89 | 0.9 | |
| 0.05 | 2.6 × 106 | 6.30 × 107 | |
Thermophysical characteristics of base liquid and nanoparticles [14].
2.2 Similarity variables
The applicable similarity variables are signified as in [50]:
Using Eq. 12, Eqs. 2–10 take the following form:
The convenient conditions are as follows:
The parameters involved are given as Reynold number , magnetic parameter , and mixed convection parameter . The Grashof numbers for temperature and concentration are and , respectively, and the parameters of the buoyancy ratio , local Reynold number , thermal relaxation , concentration relaxation , and Prandtl number are also given.
The above equations hold only for the positive Reynold number, and the solution convergence criteria are very slow for the lower Reynolds numbers. To improve solution convergence, we use the transformation , developed by Fang and Yao [31]. The above ODEs are transformed into following form:
The convenient conditions at the boundary are as follows:
2.3 Entropy generation
To include irreversibility sources, for present article, entropy generation consists of the heat transport, mass transport, and joule heating impact. The equation of entropy generation is stated as follows:
Entropy generation is the ratio between the entropy generation rate and the characteristic entropy generation rate , such that:
Now, using the transformation for fast convergence, we get the following form:
In the above equation, the parameter of the temperature difference is , is the Brinkman number, is the concentration difference parameter, and is the dimensionless constant.
2.4 Skin friction
The quantities of interest, such as skin friction, are very precious for the engineering point of view. No transport of heat and mass rate were observed in the current investigation. The mathematical form of skin fraction is as follows:
In Eq. 29, is defined as:
Equation (30) in its dimensionalized form is as follows: is the Reynolds number.
3 Graphical discussion
The numerical algorithm BVP4C in MATLAB is used to solve Eqs. 20–23 along with the boundary conditions (Eqs. 24, 25). The Bvp4c technique is only applicable to first-order ordinary differential equations. Thus, we first transform the third and second-order equations into a first-order differential equation with the use of a new variable. References [32–35] are recommended to readers because this system is well-known. Table 2 displays an assessment of the present outcomes in comparison with previously published data. From the table, it can be concluded that the current outcomes show good harmony with the results produced by Fang and Yao [31]. Table 3 presents the tabulated values for skin friction for different emerging parameters. It can be observed from the table that due to enhancement of the estimation of the Reynold number and magnetic parameters, the numerical value of skin friction is enhanced, while it falls due to the enhancement of the value of mixed convection and buoyancy ratio parameters it is reduced. For several growing parameters, graphical results are achieved covering axial and swirl velocity, temperature, and concentration field. Figures 2A,B illustrate the influence of the magnetic parameters on swirl and axial velocity. The figure shows that both fluid velocities diminish with a higher estimation of the magnetic parameter. When the magnetic parameter value is increased, Lorentz force appears, which enhances resistive force in the fluid flow, and as a result, the fluid velocity is reduced. The upshot of the influence of the buoyancy ratio parameter on swirl and axial velocity is discussed in Figures 2C,D. The figure shows that axial velocity improves while swirl velocity declines with growing estimation of the buoyancy ratio parameter, and the same result is found for the mixed convection parameter portrayed in Figures 2E,F. Physically, both buoyancy movement forces and forced convection progress in the same direction, boosting the values of the buoyancy ratio parameter. Thus, because the buoyancy effect produces resistive force to the rotation of fluid particles, the axial velocity of liquid flow grows, while the swirl velocity decreases. A relationship between the temperature and concentration distribution via thermal and solutal stratification is observed in Figures 3A,B. The fluid temperature and concentration are both diminished with thermal and solutal relaxation parameter effects. Physically, the mass and heat transport in the fluid flow are reduced with the dimensionless thermal and solutal relaxation parameter; as a result, both the profiles diminish. Figures 4A,B explore the upshot of thermal and solutal Biot number on temperature and concentration field. Greater convection is caused by an increment in the thermal Biot number, resulting in higher temperature and correlated thickness of boundary layer. The same behavior is seen for the solutal Biot number that is depicted in Figure 4B. The response of the Eckert number on temperature and Schmidt number on concentration profile is observed in Figures 5A,B, respectively. If the Eckert number is strong, the fluid components are more energetic due to energy storage. As a result, there is an upsurge in fluid temperature (see in Figure 5A). Figure 5B discloses the influence of the Schmidt number on concentration distribution. It is seen that by boosting the values of the mass diffusivity is reduced, so the concentration profile shrinks. The upshot of parameter of temperature variance and Reynold number on entropy generation is demonstrated in Figures 6A,B. In Figure 6A, it can be seen that the entropy age diminishes along the temperature variance parameter, although the Reynold number augments the entropy generation described in Figure 6B.
TABLE 2
| Fang and Yao [49] | Present outcomes | |||
|---|---|---|---|---|
| 0.1 | −0.48180 | −0.51019 | −0.482506 | −0.512193 |
| 0.2 | −0.61748 | −0.52605 | −0.615485 | −0.525057 |
| 0.4 | −0.806609 | −0.594819 | ||
| 0.5 | −0.88220 | −0.58488 | −0.881203 | −0.585882 |
| 1.0 | −1.17775 | −0.68772 | −1.176753 | −0.688721 |
| 2.0 | −1.59389 | −0.87263 | −1.594892 | −0.873635 |
| 4.0 | −2.176135 | −1.173650 | ||
| 5.0 | −2.41743 | −1.29788 | −2.416435 | −1.296879 |
| 10 | −3.34446 | −1.81006 | −3.344465 | −1.811065 |
Assessment of and with previous data, when .
TABLE 3
| 1.0 | 1.0 | 2.0 | 0.1 | 0.53491 |
| 2.0 | 0.75833 | |||
| 3.0 | 0.97647 | |||
| 1.2 | 0.51590 | |||
| 1.5 | 0.63514 | |||
| 2.0 | 0.75556 | |||
| 0.1 | 0.61458 | |||
| 0.2 | 0.61456 | |||
| 0.3 | 0.61454 | |||
| 0.1 | 0.65162 | |||
| 0.5 | 0.65142 | |||
| 0.7 | 0.65132 |
Numerical values of against the various parameters.
FIGURE 2
(A–F) Variation in and against , and .
FIGURE 3
(A,B) Variation in and against and , respectively.
FIGURE 4
(A,B) Variation in and against and , respectively.
FIGURE 5
(A,B) Variation in and against and , respectively.
FIGURE 6
(A,B) Variation in against and , respectively.
4 Concluding remarks
In the present analysis, the mixed convection hybrid nanofluid flow is discussed using a modified Fourier and Ficks law. Entropy generation is calculated. Further, the influences of heat generation/absorption, chemical reaction, Joule heating, and the thermal and solutal Biot number are discussed. This analysis presents the thermal characteristics of hybrid nanofluid, which presents many novel applications in the thermal engineering, such as nuclear reactions, heat exchangers, cooling and heating devices, fission and fusion chemical reactions, coolant in machining and manufacturing, thermal extrusion processes, and much more. The bvp4c approach is utilized to solve the problem numerically. The key findings of the present results are as follows:
➢ Axial and swirl velocities are reduced due to Lorentz forces produced by higher magnetic parameters.
➢ The axial velocity of fluid flow is enhanced and the swirl velocity is decreased with the buoyancy ratio parameter, as the buoyancy effect generates resistive force to the rotation of fluid particles.
➢ The temperature field is a growing function of thermal Biot number and the Eckert number but presents a diminishing function of thermal relaxation parameter.
➢ For larger values of the Schmidt number and solutal relaxation, the parameter concentration field is reduced, but for larger solutal Biot numbers, it is enhanced.
➢ Entropy generation is boosted with increased Reynolds number and reduced with falling parameter of temperature difference.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, and further inquiries can be directed to the corresponding author.
Author contributions
SA: conceptualization, methodology, software, formal analysis, writing-original draft preparation. NA: Software, Resources, Writing - review and editing. MK: writing original draft preparation, data curation, investigation, visualization, validation. EA: Funding acquisition, Writing - review and editing, Investigation. ET-E: Funding acquisition, Writing - review and editing, Visualization. KGe: Writing - review and editing, validation. KGu: Methodology, Writing - review and editing. AG: Methodology, Writing - review and editing, Resources.
Acknowledgments
The authors thank the Taif University Researcher for supporting project number TURSP-2020/16, Taif University, Taif, Saudi Arabia. The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4331317DSR49.
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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Summary
Keywords
Ag-TiO2/H2O hybrid nanofluid, Cattaneo-Christov double diffusion theory, convective boundary conditions, joule heating, vertical rotating cylinder, entropy generation
Citation
Ahmad S, Ahammad NA, Khan MN, Algehyne EA, Tag-Eldin E, Gepreel KA, Guedri K and Galal AM (2022) Thermal and solutal energy transport analysis in entropy generation of hybrid nanofluid flow over a vertically rotating cylinder. Front. Phys. 10:988407. doi: 10.3389/fphy.2022.988407
Received
07 July 2022
Accepted
25 July 2022
Published
26 September 2022
Volume
10 - 2022
Edited by
Arshad Riaz, University of Education Lahore, Pakistan
Reviewed by
Muhammad Sohail, Khwaja Fareed University of Engineering and Information Technology (KFUEIT), Pakistan
Iskander Tlili, National Engineering School of Monastir, Tunisia
Updates
Copyright
© 2022 Ahmad, Ahammad, Khan, Algehyne, Tag-Eldin, Gepreel, Guedri and Galal.
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: Shafiq Ahmad, ashafiq@math.qau.edu.pk
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
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