Cattaneo-Christov Heat Flux on an MHD 3D Free Convection Casson Fluid Flow Over a Stretching Sheet
Abstract
In this investigation, we analyze the magnetohydrodynamic (MHD) three-dimensional (3D) flow of Casson fluid over a stretching sheet using non-Darcy porous medium with heat source/sink. We also consider the Cattaneo-Christov heat flux and Joule effect. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using suitable transformations and solved by using the shooting technique. The effects of the non-dimensional governing parameters on velocity and temperature profiles are discussed with the graphs. Also, the skin friction coefficient and Nusselt number are discussed through tables. We also validate our results with the ones already available in the literature. It is found that the obtained results are in excellent agreement with the existing studies under some special cases. Our analysis reveals that the thermal relaxation parameter reduces the temperature field for the Newtonian and non-Newtonian fluid cases. It is also found that the temperature profile is decreased in the Newtonian fluid case when compared with the non-Newtonian fluid case.Keywords:
MHD, Casson fluid, non-Darcy flow, Cattaneo-Christov heat flux, 3D, stretching sheet, Joule effectReferences
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7. Nadeem S., Haq R.U., Akbar N.S., Khan Z.H., MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Engineering Journal, 52(4): 577–582, 2013, https://doi.org/10.1016/j.aej.2013.08.005
8. Han S., Zheng L., Li C., Zhang X., Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Applied Mathematics Letters, 38: 87–93, 2014, https://doi.org/10.1016/j.aml.2014.07.013
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10. Straughan B., Thermal convection with the Cattaneo-Christov model, International Journal of Heat and Mass Transfer, 53(1–3): 95–98, 2010, https://doi.org/10.1016/j.ijheatmasstransfer.2009.10.001
11. Imtiaz M., Alsaedi A., Shafiq A., Hayat T., Impact of chemical reaction on third grade fluid flow with Cattaneo-Christov heat flux, Journal of Molecular Liquids, 229: 501–507, 2017, https://doi.org/10.1016/j.molliq.2016.12.103
12. Raju C.S.K., Hoque M.M., Sivasankar T., Radiative flow of Casson fluid over a moving wedge filled with gyrotactic microorganisms, Advanced Powder Technology, 28(2): 575–583, 2017, https://doi.org/10.1016/j.apt.2016.10.026
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15. Raju C.S.K., Sandeep N., Unsteady three-dimensional flow of Casson-Carreau fluids past a stretching surface, Alexandria Engineering Journal, 55(2): 1115–1126, 2016, https://doi.org/10.1016/j.aej.2016.03.023
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17. Sivasankaran S., Ho C.J., Effect of temperature dependent properties on MHD convection of water near its density maximum in a square cavity, International Journal of Thermal Sciences, 47(9): 1184–1194, 2008, https://doi.org/10.1016/j.ijthermalsci.2007.10.001
18. Waqas M., Farooq M., Khan M.I., Alsaedi A., Hayat T., Yasmeen T., Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, International Journal of Heat and Mass Transfer, 102: 766–772, 2016, https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.142
19. Hsiao K.L., MHD mixed convection for viscoelastic fluid past a porous wedge, International Journal of Non-Linear Mechanics, 46(1): 1–8, 2011, https://doi.org/10.1016/j.ijnonlinmec.2010.06.005
20. Srinivas S., Kothandapani M., The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Applied Mathematics and Computation, 213(1): 197–208, 2009, https://doi.org/10.1016/j.amc.2009.02.054
21. Mahmoud M.A.A., Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity, Physica A: Statistical Mechanics and its Applications, 375(2): 401–410, 2007, https://doi.org/10.1016/j.physa.2006.09.010
22. Ishak A., Nazar R. , Pop I., Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface, Computers & Mathematics with Applications, 56(12): 3188–3194, 2008, https://doi.org/10.1016/j.camwa.2008.09.013
23. Prasad P.D., Saleem S., Varma S.V.K., Raju C.S.K., Three dimensional slip flow of a chemically reacting Casson fluid flowing over a porous slender sheet with a nonuniform heat source or sink, Journal of the Korean Physical Society, 74: 855–864, 2019, https://doi.org/10.3938/jkps.74.855
24. Tamoor M., Waqas M., Khan M.I., Alsaedi A., Hayat T., Magnetohydrodynamic flow of Casson fluid over a stretching cylinder, Results in Physics, 7: 498–502, 2017, https://doi.org/10.1016/j.rinp.2017.01.005
25. Upadhya S.M., Mahesha, Raju C.S.K., Shehzad S.A., Abbasi F.M., Flow of Eyring-Powell dusty fluid in a deferment of aluminum and ferrous oxide nanoparticles with Cattaneo-Christov heat flux, Powder Technology, 340: 68–76, 2018, https://doi.org/10.1016/j.powtec.2018.09.015
26. Lakshmi K.S., Sarojamma G., Makinde O.D., Dual stratification on the Darcy-Forchheimer flow of a Maxwell nanofluid over a stretching surface, Defect and Diffusion Forum, 387: 207–217, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.207
27. Kumar S.G., Varma S.V.K., Prasad P.D., Raju C.S.K., Makinde O.D., Sharma R., MHD reacting and radiating 3-D flow of Maxwell fluid past a stretching sheet with heat source/sink and Soret effects in a porous medium, Defect and Diffusion Forum, 387: 145–156, 2018, doi: 0.4028/www.scientific.net/DDF.387.145.
28. Ullah I., Alkanhal T.A., Shafie S., Nisar K.S., Khan I., Makinde O.D., MHD slip flow of Casson fluid along a nonlinear permeable stretching cylinder saturated in a porous medium with chemical reaction, viscous dissipation, and heat generation/ absorption, Symmetry, 11(4): 531 (27 pages), 2019, https://doi.org/10.3390/sym11040531
29. Mamatha S.U., Raju C.S.K., Prasad P.D., Ajmath K.A., Makinde O.D., Exponentially decaying heat source on MHD tangent hyperbolic two-phase flows over a flat surface with convective conditions, Defect and Diffusion Forum, 387: 286–295, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.286
30. Sharma P.R., Choudhary S., Makinde O.D., MHD slip flow and heat transfer over an exponentially stretching permeable sheet embedded in a porous medium with heat source, Frontiers in Heat and Mass Transfer, 9: 1–7, 2017, https://doi.org/10.5098/hmt.9.18
31. Makinde O.D., Nagendramma V., Raju C.S.K., Leelarathnam A., Effects of Cattaneo-Christov heat flux on Casson nanofluid flow past a stretching cylinder, Defect and Diffusion Forum, 378: 28–38, 2017, https://doi.org/10.4028/www.scientific.net/DDF.378.28
[2] Ali F., Sheikh N.A., Khan I., Saqib M., Solution with Wright function for time fraction free convective flow of Casson fluid, Arabian Journal for Science and Engineering, 42: 2565–2572, 2017, https://doi.org/10.1007/s13369-017-2521-3
3. Prasad K.M., Sudha T., Phanikumari M.V., The effect of post-stenotic dilatations on the flow of couple stress fluid through stenosed arteries, American Journal of Computational Mathematics, 6(4): 365–376, 2016, https://doi.org/10.4236/ajcm.2016.64036
4. Raju C.S.K., Sandeep N., MHD slip flow of a dissipative Casson fluid over a moving geometry with heat source/sink: a numerical study, Acta Astronautica, 133: 436–443, 2017, https://doi.org/10.1016/j.actaastro.2016.11.004
5. Mahanta G, Shaw S., 3D Casson fluid flow past a porous linearly stretching sheet with convective boundary condition, Alexandria Engineering Journal, 54(3): 653–659, 2015, https://doi.org/10.1016/j.aej.2015.04.014
6. Mernone A.V., Mazumdar J.N., Lucas S.K., A mathematical study of peristaltic transport of a Casson fluid, Mathematical and Computer Modelling, 35(7–8): 895–912, 2002, https://doi.org/10.1016/S0895-7177%2802%2900058-4
7. Nadeem S., Haq R.U., Akbar N.S., Khan Z.H., MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Engineering Journal, 52(4): 577–582, 2013, https://doi.org/10.1016/j.aej.2013.08.005
8. Han S., Zheng L., Li C., Zhang X., Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Applied Mathematics Letters, 38: 87–93, 2014, https://doi.org/10.1016/j.aml.2014.07.013
9. Hayat T., Khan M.I., Farooq M., Alsaedi A., Waqas M., Yasmeen T., Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, International Journal of Heat and Mass Transfer, 99: 702–710, 2016, https://doi.org/10.1016/j.ijheatmasstransfer.2016.04.016
10. Straughan B., Thermal convection with the Cattaneo-Christov model, International Journal of Heat and Mass Transfer, 53(1–3): 95–98, 2010, https://doi.org/10.1016/j.ijheatmasstransfer.2009.10.001
11. Imtiaz M., Alsaedi A., Shafiq A., Hayat T., Impact of chemical reaction on third grade fluid flow with Cattaneo-Christov heat flux, Journal of Molecular Liquids, 229: 501–507, 2017, https://doi.org/10.1016/j.molliq.2016.12.103
12. Raju C.S.K., Hoque M.M., Sivasankar T., Radiative flow of Casson fluid over a moving wedge filled with gyrotactic microorganisms, Advanced Powder Technology, 28(2): 575–583, 2017, https://doi.org/10.1016/j.apt.2016.10.026
13. Ciarletta M., Straughan B., Uniqueness and structural stability for the Cattaneo-Christov equations, Mechanics Research Communications, 37(5): 445–447, 2010, https://doi.org/10.1016/j.mechrescom.2010.06.002
14. Tibullo V., Zampoli V., A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mechanics Research Communications, 38(1): 77–79, 2011, https://doi.org/10.1016/j.mechrescom.2010.10.008
15. Raju C.S.K., Sandeep N., Unsteady three-dimensional flow of Casson-Carreau fluids past a stretching surface, Alexandria Engineering Journal, 55(2): 1115–1126, 2016, https://doi.org/10.1016/j.aej.2016.03.023
16. Srinivas S., Gayathri R., Kothandapani M., The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport, Computer Physics Communications, 180(11): 2115–2122, 2009, https://doi.org/10.1016/j.cpc.2009.06.015
17. Sivasankaran S., Ho C.J., Effect of temperature dependent properties on MHD convection of water near its density maximum in a square cavity, International Journal of Thermal Sciences, 47(9): 1184–1194, 2008, https://doi.org/10.1016/j.ijthermalsci.2007.10.001
18. Waqas M., Farooq M., Khan M.I., Alsaedi A., Hayat T., Yasmeen T., Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, International Journal of Heat and Mass Transfer, 102: 766–772, 2016, https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.142
19. Hsiao K.L., MHD mixed convection for viscoelastic fluid past a porous wedge, International Journal of Non-Linear Mechanics, 46(1): 1–8, 2011, https://doi.org/10.1016/j.ijnonlinmec.2010.06.005
20. Srinivas S., Kothandapani M., The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Applied Mathematics and Computation, 213(1): 197–208, 2009, https://doi.org/10.1016/j.amc.2009.02.054
21. Mahmoud M.A.A., Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity, Physica A: Statistical Mechanics and its Applications, 375(2): 401–410, 2007, https://doi.org/10.1016/j.physa.2006.09.010
22. Ishak A., Nazar R. , Pop I., Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface, Computers & Mathematics with Applications, 56(12): 3188–3194, 2008, https://doi.org/10.1016/j.camwa.2008.09.013
23. Prasad P.D., Saleem S., Varma S.V.K., Raju C.S.K., Three dimensional slip flow of a chemically reacting Casson fluid flowing over a porous slender sheet with a nonuniform heat source or sink, Journal of the Korean Physical Society, 74: 855–864, 2019, https://doi.org/10.3938/jkps.74.855
24. Tamoor M., Waqas M., Khan M.I., Alsaedi A., Hayat T., Magnetohydrodynamic flow of Casson fluid over a stretching cylinder, Results in Physics, 7: 498–502, 2017, https://doi.org/10.1016/j.rinp.2017.01.005
25. Upadhya S.M., Mahesha, Raju C.S.K., Shehzad S.A., Abbasi F.M., Flow of Eyring-Powell dusty fluid in a deferment of aluminum and ferrous oxide nanoparticles with Cattaneo-Christov heat flux, Powder Technology, 340: 68–76, 2018, https://doi.org/10.1016/j.powtec.2018.09.015
26. Lakshmi K.S., Sarojamma G., Makinde O.D., Dual stratification on the Darcy-Forchheimer flow of a Maxwell nanofluid over a stretching surface, Defect and Diffusion Forum, 387: 207–217, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.207
27. Kumar S.G., Varma S.V.K., Prasad P.D., Raju C.S.K., Makinde O.D., Sharma R., MHD reacting and radiating 3-D flow of Maxwell fluid past a stretching sheet with heat source/sink and Soret effects in a porous medium, Defect and Diffusion Forum, 387: 145–156, 2018, doi: 0.4028/www.scientific.net/DDF.387.145.
28. Ullah I., Alkanhal T.A., Shafie S., Nisar K.S., Khan I., Makinde O.D., MHD slip flow of Casson fluid along a nonlinear permeable stretching cylinder saturated in a porous medium with chemical reaction, viscous dissipation, and heat generation/ absorption, Symmetry, 11(4): 531 (27 pages), 2019, https://doi.org/10.3390/sym11040531
29. Mamatha S.U., Raju C.S.K., Prasad P.D., Ajmath K.A., Makinde O.D., Exponentially decaying heat source on MHD tangent hyperbolic two-phase flows over a flat surface with convective conditions, Defect and Diffusion Forum, 387: 286–295, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.286
30. Sharma P.R., Choudhary S., Makinde O.D., MHD slip flow and heat transfer over an exponentially stretching permeable sheet embedded in a porous medium with heat source, Frontiers in Heat and Mass Transfer, 9: 1–7, 2017, https://doi.org/10.5098/hmt.9.18
31. Makinde O.D., Nagendramma V., Raju C.S.K., Leelarathnam A., Effects of Cattaneo-Christov heat flux on Casson nanofluid flow past a stretching cylinder, Defect and Diffusion Forum, 378: 28–38, 2017, https://doi.org/10.4028/www.scientific.net/DDF.378.28
