Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article
Special Issue

Lie Symmetry with Applications to Problems in Non-Newtonian Fluid Mechanics

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 6666863 | https://doi.org/10.1155/2021/6666863

M. Irfan, M. Asif Farooq, A. Aslam, A. Mushtaq, Z. H. Shamsi, "Magnetohydrodynamic Time-Dependent Bio-Nanofluid Flow in a Porous Medium with Variable Thermophysical Properties", Mathematical Problems in Engineering, vol. 2021, Article ID 6666863, 16 pages, 2021. https://doi.org/10.1155/2021/6666863

Magnetohydrodynamic Time-Dependent Bio-Nanofluid Flow in a Porous Medium with Variable Thermophysical Properties

Academic Editor: Hafiz Muhammad Ali
Received29 Oct 2020
Revised12 Mar 2021
Accepted29 Mar 2021
Published14 Apr 2021

Abstract

In this work, a theoretical model with a numerical solution is brought forward for a bio-nanofluid with varying fluid features over a slippery sheet. The partial differential equations (PDEs) involving temperature-dependent quantities have been translated into ordinary differential equations (ODEs) by using similarity variables. Numerical verifications have been done in three different methods: finite difference method, shooting method, and bvp4c. To figure out the influence of parameters on the flows, the graphs are plotted for the velocity, temperature, concentration, and microorganism curves. The boundary layer thickness of the microorganism profile reduces with the Schmidt number and Peclet number. In addition to adding radiative heat flux, we added heat generation, rate of chemical reaction, and first-order slip. Adding these parameters brought new aspects to the underlying profiles. Moreover, the obtained data of the skin friction coefficient, the local Nusselt number, the local Sherwood number, and the local density of motile microorganisms are tabulated against various parameters for the physical parameters. From the results, it is apparent that the local Nusselt number decreases with the Brownian and thermophoretic parameters. The data obtained for physical parameters have a close agreement with the published data. Finally, the graphs for slip conditions are significantly different when the comparison is drawn with no-slip condition.

1. Introduction

Thermal analysis has attracted attention from the scientific community because of its role in our daily lives. The applications of heat transfer range from electrical devices and power plants to the heating and cooling devices, boiler, condenser, and evaporators within houses where efficiency of these devices plays a key role. The efficient devices not only reduce energy consumption but also give additional life to it.

Nanofluid is a mixture of a base fluid with 100 nm-size nanoparticles. Since the work by Choi and Eastman [1] on nanofluid, the research in this direction took a huge stride. The thermal conductivity is significantly higher than that of the traditional fluids as it was reported in Lee et al. [2]. There are many applications in the field of nanofluids including lubricants, automation, electronics, and biomedicine. For the list of references which took multiple paths considering nanofluid in their study, one is referred to in [37].

Bioconvection is another phenomenon which occurs due to the density difference of the fluid. Raees et al. [8] recorded the homotopy analysis method (HAM) solution for an unsteady bioconvection flow in a channel and showed that the velocity component decreases with the increase in time. Uddin et al. [9] discussed bioconvection nanofluid over a wavy surface with slip flow in application to nano-biofuel cells. Khan and Makinde [10] explored bioconvection flow due to gyrotactic microorganisms. They noticed that, with rising the values of the convective variable, the dimensionless temperature on the surface rises. Uddin et al. [11] investigated Stefan blowing with multiple slip effects in bioconvection. For finding similarity transformation, they used Lie group analysis. The resources for further study on this topic can be found in [1216].

One of the ways through which heat transfer occurs is thermal radiation. It has diverse technological applications in combustion, furnace design, turbines, and solar collectors. The thermal radiation with variable fluid properties is reported in [17]. The author found that the skin friction coefficient increases with viscosity parameter. RamReddy and Naveen [18] reported results for activation energy and thermal radiation. Aziz et al. [19] discussed free convection flow in nanofluids with microorganisms. They discovered that the bioconvection parameter affects heat transfer rate. Mutuku and Makinde [20] discussed hydromagnetic fluid flow in microorganisms. Sk et al. [21] presented multiple slip effects in the presence of microorganisms. Anwar et al. [22] discussed MHD flow in a porous channel with generalized conditions. For further study on this topic, one is referred to in [2325].

The magnetic field has many applications including geothermal energy extractions, plasma studies, chemical engineering, and magnetic resonance imaging (MRI) equipment [26]. Ali et al. [27] discussed hybrid nanofluid with slip conditions for Jeffrey fluid. Mburu et al. [28] reported magnetic and thermal radiation effects over an inclined cylinder. Mabood et al. [29] combined electrical and magnetic flows for non-Newtonian nanofluids over a thin needle.

The viscosity of the fluid generally depends on pressure and temperature. However, less effect in fluid flow is observed with pressure. Therefore, viscosity is dependent on the temperature variation. Fatunmbi and Adeniyan [30] reported nonlinear thermal radiation in fluid flow with variable properties. Dandapat et al. [31] discussed thin film unsteady flow with variable fluid properties. Vajravelu et al. [32] discussed unsteady convective flow in a vertical surface with variable fluid properties. Shahsavar et al. [33] investigated the impact of variable fluid properties in hybrid nanofluid. Naganthran et al. [34] found results of the stretching and shrinking sheet with variable fluid properties. They discussed dual solutions in this rotating disk. Salahuddin et al. [35] discussed variable fluid properties for viscoelastic fluid between two rotating plates.

Shafiq et al. [36] presented second-grade bioconvective nanofluid flow and computed the solution from the shooting method. In another study, Rasool and Shafiq [37] discussed Powell–Eyring nanofluid flow in a porous medium over a nonlinear surface. The porosity factor is enhancing the drag force. In another work by Shafiq et al. [38], the numerical solution of the bioconvective tangent hyperbolic nanofluid was found. The effect of temperature-dependent viscosity, thermal radiation, and gyrotactic phenomenon in a convection flow over a cylinder has been discussed in [3941], respectively. In another work by Khan et al. [42], the bioconvection flow was discussed in a truncated cone. For critical review about nanofluid and its effects on viscosity along with thermal conductivity, the reader is referred to in [4346]. For experimental investigation on nanofluids, the reader is referred to in [47].

Most theoretical studies mentioned above are focused on the idea of constant fluid properties in fluid flows. The viscosity of a fluid, however, relies heavily on temperature than on other factors, such as pressure. It comes out that the use of variable properties offers distinct effects on fluid flow motion.

This paper is ordered in the following way: the flow model is presented in Section 2. The numerical procedure for the solution is presented in Section 3. Results and discussion are given in Section 4. Conclusion of the paper is drawn in Section 5.

2. Flow Model

Consider the movement of a nanofluid containing gyrotactic microorganisms past a stretching sheet with variable physical properties. The magnetic field is applied normal to the surface. Due to low magnetic Reynolds number, the induced magnetic field is assumed negligible. The stretching velocity is . The governing model is [48]and the boundary condition corresponding to the considered model is taken aswhere all the variables are defined in the glossary.

The similarity variables are defined as

Inserting equation (7) into equations (1)–(6), we get

Following Amirsom et al. [48], the physical quantities consisting of viscosity, thermal conductivity, nanoparticle, and microorganism diffusivities are written as

Equation (13) when used into equations (8)–(11), one can get

All these parameters are grouped into

The physical quantities of the interest in this study are the local skin friction coefficient , the local Nusselt number , the local Sherwood number , and the local density number of motile microorganisms defined as

Inserting equation (7) into equation (13) yields the following expressions:where the local Reynolds number is defined as .

3. Numerical Process

3.1. Shooting Method

A boundary value problem ((8)–(12)) can be solved with the shooting method. The stable iterative scheme, Newton–Raphson method, has been used in locating the roots followed by obtaining the solution from the fifth-order Runge–Kutta solver. The system of first-order ODEs is

The results’ verification is achieved from the bvp4c solver. For details on bvp4c, the reader is referred to in [49].

3.2. Finite Difference Method

In this section, we present the finite difference method to solve boundary value problem (8)–(12). The spatial discretization is given by first defining in the momentum equation: and the boundary conditions are

4. Results and Discussion

An excellent agreement with published results is obtained for a comparison of the skin friction coefficient which is shown in Tables 13.


Hayat et al. [50]Mabood and Mastroberardino [51]Amirsom et al. [48]Shooting methodbvp4c

01.00001.0000081.00000021.00001.0001
11.414211.41421351.414222111.41421.4142
52.449482.44948972.44949012.44952.4495
103.316623.31662473.31662293.31663.3166
507.141427.14142847.14142797.14147.1414
10010.0498710.04987510.04986810.049910.0499
50022.3830222.38302922.38303122.383022.3830
100031.6385831.63858431.63857831.638631.6386


Hayat et al. [50]Mabood and Mastroberardino [51]Amirsom et al. [48]FDM

01.00001.0000081.00000021.0001
11.414211.41421351.414222111.4142
52.449482.44948972.44949012.4495
103.316623.31662473.31662293.3166
507.141427.14142847.14142797.1414
10010.0498710.04987510.04986810.0499
50022.3830222.38302922.38303122.3830
100031.6385831.63858431.63857831.6386


Andersson [52]Hamad et al. [53]Amirsom et al. [48]Shooting methodbvp4c

01.00001.000000001.000000001.00001.0001
0.10.87210.872082470.872042470.87210.8722
0.20.77640.776377070.775933070.77640.7765
0.50.59120.591195480.591195890.59120.5913
1.00.43020.430159700.430160000.43020.4303
2.00.28400.283979590.283989320.28400.2841
5.00.14480.144840190.144640150.14480.1449
10.00.08120.081241980.081240910.08120.0813
20.00.04380.043788340.043787900.04380.0439
50.00.01860.018596230.018578680.01860.0186
100.00.00950.009549970.009546770.00950.0096

The data in Table 4 show computational results for the local Nusselt number, the local Sherwood number, and the local density number of motile microorganisms obtained with bvp4c. The local Nusselt number is reduced against Brownian motion parameter , thermophoretic parameter , Eckert number , heat source parameter , and thermal conductive parameter .



40.50.10.10.10.250.2210.10.10.10.10.45781.63062.0202
50.51321.61232.0097
6.80.57451.59482.0020
6.810.10.10.10.250.2210.10.10.10.10.72761.60182.0002
1.50.83621.61652.0072
20.90541.63302.0171
6.80.500.10.10.250.2210.10.10.10.10.67711.47371.8969
0.10.57451.59472.0020
0.20.41831.77132.1544
6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020
0.150.44681.75802.1461
0.20.31981.91982.2889
6.80.50.10.10.50.250.2210.10.10.10.10.49741.64522.0171
10.40001.65052.0179
1.50.30401.65142.0170
6.80.50.10.10.1150.2210.10.10.10.10.50801.76462.3215
1.50.46262.15632.7886
20.41412.78113.4623
6.80.50.10.10.10.230.2210.10.10.10.10.57801.10721.6118
50.57451.59472.0020
100.57122.44382.7096
6.80.50.10.10.10.250210.10.10.10.10.57631.14911.6338
0.50.57292.07652.4069
10.57162.65842.9040
6.80.50.10.10.10.250.20.510.10.10.10.10.57451.59471.6129
10.57451.59471.7596
30.57451.59472.2000
6.80.50.10.10.10.250.220.50.10.10.10.10.57451.59471.3577
10.57451.59472.0020
30.57451.59474.6914
6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020
0.50.55731.63362.0454
0.90.53861.66902.0830
6.80.50.10.10.10.25210.10.10.10.10.57451.59472.0020
0.50.54931.61012.0127
0.90.52581.62292.0213
6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020
0.50.57151.27881.7325
0.90.56861.08451.5698
6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020
0.50.57451.59471.5143
0.90.57451.59471.2232

With increasing values of Prandtl number and radiation parameter , the local Nusselt number shows an upward trend.

The physical parameter, the local Sherwood number , depicts an upward trend against Brownian motion parameter , thermophoretic parameter , Schmidt number , and chemical reaction parameter . However, a decreasing trend for the local Sherwood number is observed for rising values of mass diffusivity parameter .

Finally, the values of the local density number of motile microorganisms decline with the increase of mass diffusivity parameter and microorganism diffusivity parameter . However, there is an upsurge for increasing values of the bioconvection Schmidt number and Peclet number .

Figures 1 and 2 illustrate the effects of magnetic parameter and porosity parameter on the velocity profile with and without hydrodynamic slip. The boundary layer thickness reduces with increasing values of and . When fluid flow encounters the Lorentz forces, the velocity of the fluid decelerates which affects the boundary layer thickness. The same argument holds for .

Figure 3 is plotted to perceive the effect of Prandtl number on the temperature profile. It is noted that an enhancement in Prandtl number causes reduction in the temperature distribution. The smaller values of correspond to the increase in thermal conductivities which causes reduction in a thermal boundary layer. For Prandtl number (), the momentum diffusivity is dominant in fluid behavior. Thus, less thermal diffusivity contributes to lowering the thermal boundary layer thickness.

Figure 4 depicts the influence of radiation parameter on the temperature profile. It is seen that an increase in enhances the temperature of the fluid. Larger values of radiation parameter transfer more heat to the fluid which overall increases the temperature and its profile.

Figure 5 reports the influence of Eckert number on the temperature profile. The higher values of Eckert number cause an increase in the thermal boundary layer thickness. The Eckert number enhances kinetic energy, which increases fluid’s temperature.

Figure 6 illustrates the impact of heat source parameter on the temperature distribution. It is observed that temperature of the fluid increases with an increment in the heat generation parameter. The higher values of provide more heat to the fluid resulting in the rise of the temperature of the fluid.

Figure 7 examines the effect of temperature-dependent thermal conductivity parameter on temperature. It is noted that the thermal boundary layer thickness increases by increasing parameter .

Figures 8 and 9 are drawn to perceive the effect of Brownian motion parameter on the temperature and concentration profiles. It is revealed in the figure that, by increasing Brownian motion parameter , thermal boundary layer thickness rises, while concentration boundary layer thickness declines. The Brownian parameter appears due to the presence of nanoparticles’ concentration.

Figures 10 and 11 convey the impacts of thermophoresis parameter on temperature and concentration distributions. The temperature and concentration profile rise for rising values of . The thermophoresis term appears due to the temperature gradient in particulate flows. Larger values of transmit more temperature to the fluid along with the concentration profile.

Figure 12 portrays the influence of chemical reaction parameter on the concentration profile. The rising values of suppress diffusion which lowers the concentration boundary layer.

Figure 13 depicts the effects of Schmidt number on the concentration distribution. The rise in causes reduction in the concentration profile. The higher the Schmidt number, the lower the mass diffusivity which is the reason for reduction in the concentration boundary layer thickness.

Figure 14 presents the influence of mass diffusivity parameter on the concentration profile. One can observe that rise in mass diffusivity parameter results in an increase of the concentration profile.

Figure 15 describes the influence of Peclet number on the density of motile microorganism profile. The incremental values of Peclet number cause reduction in motile microorganisms’ boundary layer thickness. The Peclet number appears in the study of transport processes. It measures the importance of convection over diffusion. For larger values of the Peclet number, the convection is dominant and diffusion is negligible which is happening here in the motile microorganisms’ boundary layer thickness.

Figure 16 investigates the impact of bioconvection Schmidt number on the density of motile microorganism profile. It is shown that rising values of bioconvection Schmidt number lower the boundary layer thickness of the motile microorganism profile. In high values of , the particles are giant which means these diffuse slowly.

Figures 17 and 18 are drawn to perceive the effect of mass diffusivity parameter and microorganism diffusivity parameter . Increasing the values of mass diffusivity parameter and microorganism diffusivity parameter elevates the boundary layer thickness of the motile microorganism profile.

5. Conclusion

The focus of the paper involves unsteady MHD flow of bio-nanofluid in a permeable medium taking thermal radiation and chemical reaction into account over a stretching sheet with variable thermophysical properties. The notable findings of the problem are outlined in the following [51]:(i)The incremental values of Brownian motion parameter , thermophoresis parameter , thermal radiation parameter , Eckert number , and heat source parameter magnify the thermal boundary layer thickness, while an increase in Prandtl number causes reduction in the thermal boundary layer thickness.(ii)The concentration boundary layer thickness rises for thermophoresis parameter and mass diffusivity parameter , whereas it declines for higher values of the Brownian motion parameter , Schmidt number , and chemical reaction parameter .(iii)The increment in bioconvection Schmidt number and Peclet number reduces the boundary layer thickness of motile microorganisms, while the motile microorganisms’ boundary layer shows inverse behavior for mass diffusivity parameter and microorganism parameter .(iv)Graphs have been drawn with and without slip conditions. Difference can be clearly seen through graphs as the boundary layer thickness of the slip condition is different when compared without the slip flow case.

Notations

:A constant ()
:Fluid velocities along and normal to the flow ()
:Orthogonal Cartesian coordinates (m)
:Unsteadiness parameter
:Dimensionless parameter
:Applied magnetic field ()
:Dynamic viscosity ()
:Fluid’s viscosity ()
:Fluid’s electrical conductivity () (S is siemens)
:Dimensionless magnetic variable
:Porosity variable
:Fluid’s temperature (K)
:Constant wall temperature (K)
:Free-stream temperature (K)
:Variable thermal conductivity ()
:Thermal diffusivity ()
:Slip parameter
:Heat capacity at constant pressure ()
:Flux due to radiation ()
:Internal heat generation/absorption
:Skin friction coefficient
:Local Nusselt parameter
MHD:Magnetohydrodynamics
PDEs:Partial differential equations
:Heat source/sink parameter
:Thermal radiation parameter
:Free-stream Prandtl number
:Brownian coefficient ()
:Thermophoretic coefficient ()
:Diffusivity of microorganisms ()
:Diffusivity coefficient ()
:Ratio of effective heat capacitance of the nanoparticle to the base fluid
:Nanoparticle heat capacity
:Brownian motion parameter
:Thermophoresis parameter
:Concentration
:Concentration at the wall
:Ambient fluid concentration
:Concentration of microorganisms
:Microorganisms at the wall
:Microorganisms far from the wall
:Lewis number
:Bioconvection Schmidt number
:Chemotaxis constant (m)
:Maximum cell swimming speed ()
:Peclet number
:Local Sherwood parameter
:Local density parameter of the motile microorganisms
ODEs:Ordinary differential equations
FDM:Finite difference method.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. S. U. Choi and J. A. Eastman, Enhancing Thermal Conductivity of Fluids with Nanoparticles (No. ANL/MSD/CP-84938; CONF-951135-29), Argonne National Lab., Chicago, IL, USA, 1995.
  2. S. Lee, S. U.-S. Choi, S. Li, and J. A. Eastman, “Measuring thermal conductivity of fluids containing oxide nanoparticles,” Journal of Heat Transfer, vol. 121, no. 2, p. 280, 1999. View at: Publisher Site | Google Scholar
  3. S. Rashidi, O. Mahian, and E. M. Languri, “Applications of nanofluids in condensing and evaporating systems,” Journal of Thermal Analysis and Calorimetry, vol. 131, no. 3, pp. 2027–2039, 2018. View at: Publisher Site | Google Scholar
  4. Z. Zhang, J. Cai, F. Chen, H. Li, W. Zhang, and W. Qi, “Progress in enhancement of CO2 absorption by nanofluids: a mini review of mechanisms and current status,” Renewable Energy, vol. 118, pp. 527–535, 2018. View at: Publisher Site | Google Scholar
  5. C. RamReddy, P. V. S. N. Murthy, A. M. Rashad, and A. J. Chamkha, “Numerical study of thermally stratified nanofluid flow in a saturated non-Darcy porous medium,” The European Physical Journal Plus, vol. 129, no. 2, p. 25, 2014. View at: Publisher Site | Google Scholar
  6. M. N. Khan, S. Nadeem, N. Ullah, and A. Saleem, “Theoretical treatment of radiative oldroyd-B nanofluid with microorganism pass an exponentially stretching sheet,” Surfaces and Interfaces, vol. 21, Article ID 100686, 2020. View at: Publisher Site | Google Scholar
  7. W. A. Khan, M. Khan, and R. Malik, “Three-dimensional flow of an Oldroyd-B nanofluid towards stretching surface with heat generation/absorption,” PLoS One, vol. 9, no. 8, Article ID e105107, 2014. View at: Publisher Site | Google Scholar
  8. A. Raees, H. Xu, and S.-J. Liao, “Unsteady mixed nano-bioconvection flow in a horizontal channel with its upper plate expanding or contracting,” International Journal of Heat and Mass Transfer, vol. 86, pp. 174–182, 2015. View at: Publisher Site | Google Scholar
  9. M. J. Uddin, W. A. Khan, S. R. Qureshi, and O. Anwar Bég, “Bioconvection nanofluid slip flow past a wavy surface with applications in nano-biofuel cells,” Chinese Journal of Physics, vol. 55, no. 5, pp. 2048–2063, 2017. View at: Publisher Site | Google Scholar
  10. W. A. Khan and O. D. Makinde, “MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet,” International Journal of Thermal Sciences, vol. 81, pp. 118–124, 2014. View at: Publisher Site | Google Scholar
  11. M. J. Uddin, M. N. Kabir, and O. A. Bég, “Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms,” International Journal of Heat and Mass Transfer, vol. 95, pp. 116–130, 2016. View at: Publisher Site | Google Scholar
  12. Z. Mehmood and Z. Iqbal, “Interaction of induced magnetic field and stagnation point flow on bioconvection nanofluid submerged in gyrotactic microorganisms,” Journal of Molecular Liquids, vol. 224, pp. 1083–1091, 2016. View at: Publisher Site | Google Scholar
  13. S. Siddiqa, M. Sulaiman, M. A. Hossain, S. Islam, and R. S. R. Gorla, “Gyrotactic bioconvection flow of a nanofluid past a vertical wavy surface,” International Journal of Thermal Sciences, vol. 108, pp. 244–250, 2016. View at: Publisher Site | Google Scholar
  14. Z. Iqbal, Z. Mehmood, and E. N. Maraj, “Oblique transport of gyrotactic microorganisms and bioconvection nanoparticles with convective mass flux,” Physica E: Low-Dimensional Systems and Nanostructures, vol. 88, pp. 265–271, 2017. View at: Publisher Site | Google Scholar
  15. M. Waqas, T. Hayat, S. A. Shehzad, and A. Alsaedi, “Transport of magnetohydrodynamic nanomaterial in a stratified medium considering gyrotactic microorganisms,” Physica B: Condensed Matter, vol. 529, pp. 33–40, 2018. View at: Publisher Site | Google Scholar
  16. M. Ramzan, J. D. Chung, and N. Ullah, “Radiative magnetohydrodynamic nanofluid flow due to gyrotactic microorganisms with chemical reaction and non-linear thermal radiation,” International Journal of Mechanical Sciences, vol. 130, pp. 31–40, 2017. View at: Publisher Site | Google Scholar
  17. K. Das, “Impact of thermal radiation on MHD slip flow over a flat plate with variable fluid properties,” Heat and Mass Transfer, vol. 48, no. 5, pp. 767–778, 2012. View at: Publisher Site | Google Scholar
  18. C. RamReddy and P. Naveen, “Analysis of activation energy and thermal radiation on convective flow of a power-law fluid under convective heating and chemical reaction,” Heat Transfer-Asian Research, vol. 48, no. 6, pp. 2122–2154, 2019. View at: Publisher Site | Google Scholar
  19. A. Aziz, W. A. Khan, and I. Pop, “Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms,” International Journal of Thermal Sciences, vol. 56, pp. 48–57, 2012. View at: Publisher Site | Google Scholar
  20. W. N. Mutuku and O. D. Makinde, “Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms,” Computers & Fluids, vol. 95, pp. 88–97, 2014. View at: Publisher Site | Google Scholar
  21. M. T. Sk, K. Das, and P. K. Kundu, “Multiple slip effects on bioconvection of nanofluid flow containing gyrotactic microorganisms and nanoparticles,” Journal of Molecular Liquids, vol. 220, pp. 518–526, 2016. View at: Publisher Site | Google Scholar
  22. T. Anwar, P. Kumam, D. Baleanu, I. Khan, and P. Thounthong, “Radiative heat transfer enhancement in MHD porous channel flow of an Oldroyd-B fluid under generalized boundary conditions,” Physica Scripta, vol. 95, no. 11, Article ID 115211, 2020. View at: Publisher Site | Google Scholar
  23. O. D. Makinde and I. L. Animasaun, “Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution,” International Journal of Thermal Sciences, vol. 109, pp. 159–171, 2016. View at: Publisher Site | Google Scholar
  24. N. Begum, S. Siddiqa, and M. A. Hossain, “Nanofluid bioconvection with variable thermophysical properties,” Journal of Molecular Liquids, vol. 231, pp. 325–332, 2017. View at: Publisher Site | Google Scholar
  25. F. T. Tuz Zohra, M. Jashim Uddin, A. I. Md. Ismail, and O. Anwar Bég, “Bioconvective electromagnetic nanofluid transport from a wedge geometry:Simulation of smart electro-conductive bio-nanopolymer processing,” Heat Transfer-Asian Research, vol. 47, no. 1, pp. 231–250, 2018. View at: Publisher Site | Google Scholar
  26. J. B. Mathieu and S. Martel, “Aggregation of magnetic microparticles in the context of targeted therapies actuated by a magnetic resonance imaging system,” Journal of Applied Physics, vol. 106, no. 4, Article ID 044904, 2009. View at: Publisher Site | Google Scholar
  27. A. Ali, S. Saleem, S. Mumraiz, A. Saleem, M. Awais, and D. N. Khan Marwat, “Investigation on TiO2–Cu/H2O hybrid nanofluid with slip conditions in MHD peristaltic flow of Jeffrey material,” Journal of Thermal Analysis and Calorimetry, vol. 143, no. 3, pp. 1985–1996, 2021. View at: Publisher Site | Google Scholar
  28. Z. M. Mburu, S. Mondal, and P. Sibanda, “Numerical study on combined thermal radiation and magnetic field effects on entropy generation in unsteady fluid flow past an inclined cylinder,” Journal of Computational Design and Engineering, vol. 8, no. 1, pp. 149–169, 2020. View at: Publisher Site | Google Scholar
  29. F. Mabood, T. Muhammad, M. K. Nayak, H. Waqas, and O. D. Makinde, “EMHD flow of non-Newtonian nanofluids over thin needle with Robinson’s condition and Arrhenius pre-exponential factor law,” Physica Scripta, vol. 95, no. 11, Article ID 115219, 2020. View at: Publisher Site | Google Scholar
  30. E. O. Fatunmbi and A. Adeniyan, “Nonlinear thermal radiation and entropy generation on steady flow of magneto-micropolar fluid passing a stretchable sheet with variable properties,” Results in Engineering, vol. 6, Article ID 100142, 2020. View at: Publisher Site | Google Scholar
  31. B. S. Dandapat, B. Santra, and K. Vajravelu, “The effects of variable fluid properties and thermo-capillarity on the flow of a thin film on an unsteady stretching sheet,” International Journal of Heat and Mass Transfer, vol. 50, no. 5-6, pp. 991–996, 2007. View at: Publisher Site | Google Scholar
  32. K. Vajravelu, K. V. Prasad, and C.-O. Ng, “Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 455–464, 2013. View at: Publisher Site | Google Scholar
  33. A. Shahsavar, A. Godini, P. T. Sardari, D. Toghraie, and H. Salehipour, “Impact of variable fluid properties on forced convection of Fe3O4/CNT/water hybrid nanofluid in a double-pipe mini-channel heat exchanger,” Journal of Thermal Analysis and Calorimetry, vol. 137, no. 3, pp. 1031–1043, 2019. View at: Publisher Site | Google Scholar
  34. K. Naganthran, M. Mustafa, A. Mushtaq, and R. Nazar, “Dual solutions for fluid flow over a stretching/shrinking rotating disk subject to variable fluid properties,” Physica A: Statistical Mechanics and Its Applications, vol. 556, Article ID 124773, 2020. View at: Publisher Site | Google Scholar
  35. T. Salahuddin, M. Arshad, N. Siddique, and I. Tlili, “Change in internal energy of viscoelastic fluid flow between two rotating parallel plates having variable fluid properties,” Indian Journal of Physics, pp. 1–11, 2020. View at: Publisher Site | Google Scholar
  36. A. Shafiq, G. Rasool, C. M. Khalique, and S. Aslam, “Second grade bioconvective nanofluid flow with buoyancy effect and chemical reaction,” Symmetry, vol. 12, no. 4, p. 621, 2020. View at: Publisher Site | Google Scholar
  37. G. Rasool and A. Shafiq, “Numerical exploration of the features of thermally enhanced chemically reactive radiative Powell-Eyring nanofluid flow via Darcy medium over non-linearly stretching surface affected by a transverse magnetic field and convective boundary conditions,” Applied Nanoscience, pp. 1–18, 2020. View at: Publisher Site | Google Scholar
  38. A. Shafiq, T. N. Sindhu, and C. M. Khalique, “Numerical investigation and sensitivity analysis on bioconvective tangent hyperbolic nanofluid flow towards stretching surface by response surface methodology,” Alexandria Engineering Journal, vol. 59, no. 6, pp. 4533–4548, 2020. View at: Publisher Site | Google Scholar
  39. M. A. El-Hakiem and A. M. Rashad, “Effect of radiation on non-Darcy free convection from a vertical cylinder embedded in a fluid-saturated porous medium with a temperature-dependent viscosity,” Journal of Porous Media, vol. 10, no. 2, 2007. View at: Google Scholar
  40. S. M. M. El-Kabeir, A. J. Chamkha, and A. M. Rashad, “Effect of thermal radiation on non-Darcy natural convection from a vertical cylinder embedded in a nanofluid porous media,” Journal of Porous Media, vol. 17, no. 3, 2014. View at: Publisher Site | Google Scholar
  41. A. M. Rashad and H. A. Nabwey, “Gyrotactic mixed bioconvection flow of a nanofluid past a circular cylinder with convective boundary condition,” Journal of the Taiwan Institute of Chemical Engineers, vol. 99, pp. 9–17, 2019. View at: Publisher Site | Google Scholar
  42. W. A. Khan, A. M. Rashad, M. M. M. Abdou, and I. Tlili, “Natural bioconvection flow of a nanofluid containing gyrotactic microorganisms about a truncated cone,” European Journal of Mechanics—B/Fluids, vol. 75, pp. 133–142, 2019. View at: Publisher Site | Google Scholar
  43. H. Babar, M. Sajid, and H. Ali, “Viscosity of hybrid nanofluids: a critical review,” Thermal Science, vol. 23, no. 3, pp. 1713–1754, 2019. View at: Publisher Site | Google Scholar
  44. M. U. Sajid and H. M. Ali, “Thermal conductivity of hybrid nanofluids: a critical review,” International Journal of Heat and Mass Transfer, vol. 126, pp. 211–234, 2018. View at: Publisher Site | Google Scholar
  45. A. Wahab, A. Hassan, M. A. Qasim, H. M. Ali, H. Babar, and M. U. Sajid, “Solar energy systems—potential of nanofluids,” Journal of Molecular Liquids, vol. 289, Article ID 111049, 2019. View at: Publisher Site | Google Scholar
  46. M. U. Sajid and Y. Bicer, “Nanofluids as solar spectrum splitters: a critical review,” Solar Energy, vol. 207, pp. 974–1001, 2020. View at: Publisher Site | Google Scholar
  47. M. U. Sajid, H. M. Ali, A. Sufyan, D. Rashid, S. U. Zahid, and W. U. Rehman, “Experimental investigation of TiO2-water nanofluid flow and heat transfer inside wavy mini-channel heat sinks,” Journal of Thermal Analysis and Calorimetry, vol. 137, no. 4, pp. 1279–1294, 2019. View at: Publisher Site | Google Scholar
  48. N. A. Amirsom, M. J. Uddin, M. F. Md Basir, A. Kadir, O. A. Bég, and A. I. Md. Ismail, “Computation of melting dissipative magnetohydrodynamic nanofluid bioconvection with second-order slip and variable thermophysical properties,” Applied Sciences, vol. 9, no. 12, p. 2493, 2019. View at: Publisher Site | Google Scholar
  49. J. Kierzenka and L. F. Shampine, “A BVP solver based on residual control and the Maltab PSE,” ACM Transactions on Mathematical Software, vol. 27, no. 3, pp. 299–316, 2001. View at: Publisher Site | Google Scholar
  50. T. Hayat, Q. Hussain, and T. Javed, “The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 966–973, 2009. View at: Publisher Site | Google Scholar
  51. F. Mabood and A. Mastroberardino, “Melting heat transfer on MHD convective flow of a nanofluid over a stretching sheet with viscous dissipation and second order slip,” Journal of the Taiwan Institute of Chemical Engineers, vol. 57, pp. 62–68, 2015. View at: Publisher Site | Google Scholar
  52. H. I. Andersson, “Slip flow past a stretching surface,” Acta Mechanica, vol. 158, no. 1-2, pp. 121–125, 2002. View at: Publisher Site | Google Scholar
  53. M. A. A. Hamad, M. J. Uddin, and A. I. M. Ismail, “Investigation of combined heat and mass transfer by lie group analysis with variable diffusivity taking into account hydrodynamic slip and thermal convective boundary conditions,” International Journal of Heat and Mass Transfer, vol. 55, no. 4, pp. 1355–1362, 2012. View at: Publisher Site | Google Scholar

Copyright © 2021 M. Irfan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views442
Downloads358
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.