Radiative MHD Walter’s Liquid-B Flow Past a Semi-Infinite Vertical Plate in the Presence of Viscous Dissipation with a Heat Source
Abstract
The free convective magnetohydrodynamics (MHD) flow of a non-Newtonian fluid due to a semi-infinite vertical plate under the influence of radiation and viscous dissipation is investigated. The system of partial differential equations is derived and solved for the solutions of velocity and temperature profiles along with the Nusselt number and skin friction by using the perturbation technique. The related important dimensionless parameters of Eckert, Grashof, and Prandtl numbers, magnetic field, radiation and heat source are discussed and shown in graphs. Also, the Nusselt number and skin friction at the plate are obtained and presented in the tabular forms. Finally, the corresponding result of Newtonian fluid is obtained by setting viscoelastic parameter k1 = 0. It is worth mentioning that the obtained results coincide with the previously published results.Keywords:
radiation, magnetohydrodynamics (MHD), viscous dissipation, porous medium, heat source and viscoelastic fluid, vertical plateReferences
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4. Sattar Md.A., Alam Md.M., MHD free convective heat and mass transfer flow with Hall current and constant heat flux through a porous medium, Indian Journal of Pure and Applied Mathematics, 26(2): 157–167, 1995.
5. Sahoo P.K., Datta N., Biswal S., Magnetohydrodynamic unsteady free convective flow past an infinite vertical plate with constant suction and heat sink, Indian Journal of Pure and Applied Mathematics, 34(1): 145–155, 2003.
6. Maqbool K., Mann A.B., Tiwana M.H., Unsteady MHD convective flow of a Jeffery fluid embedded in a porous medium with ramped wall velocity and temperature, Alexandria Engineering Journal, 57(2): 1071–1078, 2018, https://doi.org/10.1016/j.aej.2017.02.012
7. Hamza M.M., Free convection slip flow of an exothermic fluid in a convectively heated vertical channel, Ain Shams Engineering Journal, 9(4): 1313–1323, 2018, https://doi.org/10.1016/j.asej.2016.08.011
8. Un Nisa Z., Hajizadeh A., Nazar M., Free convection flow of nanofluid over infinite vertical plate with damped thermal flux, Chinese Journal of Physics, 59: 175–188, 2019, https://doi.org/10.1016/j.cjph.2019.02.029
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10. Shah N., Zafar A.A., Fetecau C., Free convection flows over a vertical plate that applies shear stress to a fractional viscous fluid, Alexandria Engineering Journal, 57(4): 2529–2540, 2018, https://doi.org/10.1016/j.aej.2017.08.023
11. Gholinia M., Hoseini M.E., Gholinia S., A numerical investigation of free convection MHD flow of Walters-B nanofluid over an inclined stretching sheet under the impact of Joule heating, Thermal Science and Engineering Progress, 11: 272–282, 2019, https://doi.org/10.1016/j.tsep.2019.04.006
12. Wang Z.H., Zhou Z.K., Eternal natural convection heat transfer of liquid metal under the influence of the magnetic field, International Journal of Heat and Mass Transfer, 134: 175–184, 2019, https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.173
13. Patel H.R., Effects of cross diffusion and heat generation on mixed convective MHD flow of Casson fluid through porous medium with non-linear thermal radiation, Heliyon, 5(4): 1–26, 2019, https://doi.org/10.1016/j.heliyon.2019.e01555
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15. Satya Narayana P.V., Venkateswarlu B., Venkataramana S., Thermal radiation and heat source effects on a MHD nanofluid past a vertical plate in a rotating system with porous medium, Heat Transfer Asian Research, 44(1): 1–19, 2015, https://doi.org/10.1002/htj.21101
16. Cortell R., MHD (magneto-hydrodynamic) flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, Energy, 74: 896–905, 2014, https://doi.org/10.1016/j.energy.2014.07.069
17. Amir Hamzah N.S., Kandasamy R., Muhammad R., Thermal radiation energy on squeezed MHD flow of Cu, Al2O3 and CNTs-nanofluid over a sensor surface, Alexandria Engineering Journal, 55(3): 2405–2421, 2016, https://doi.org/10.1016/j.aej.2016.04.019
18. Fagbade A.I., Falodun B.O., Omowaye A.J., MHD natural convection flow of viscoelastic fluid over an accelerating permeable surface with thermal radiation and heat source or sink: spectral homotopy analysis approach, Ain Shams Engineering Journal, 9(4): 1029–1041, 2018, https://doi.org/10.1016/j.asej.2016.04.021
19. Satya Narayana P.V., Akshit S.M., Ghori J.P., Venkateswarlu B., Thermal radiation effects on an unsteady MHD nanofluid flow over a stretching sheet with non-uniform heat source/sink, Journal of Nanofluids, 6(5): 899–907, 2017, https://doi.org/10.1166/jon.2017.1374
20. Harish Babu D., Ajmath K.A., Venkateswarlu B., Satya Narayana P.V., Thermal radiation and heat source effects on MHD non-Newtonian nanofluid flow over a stretching sheet, Journal of Nanofluids, 8(5): 1085–1092, 2018, https://doi.org/10.1166/jon.2019.1666
21. Makanda G., Makinde O.D., Sibanda P., Natural convection of viscoelastic fluid from a cone embedded in a porous medium with viscous dissipation, Mathematical Problems in Engineering, 2013: Article ID 934712, 11 pages, 2013, https://doi.org/10.1155/2013/934712
22. Venkateswarlu B., Satya Narayana P.V., Influence of variable thermal conductivity on MHD Casson fluid flow over a stretching sheet with viscous dissipation, Soret and Dufour effects, Frontiers in Heat and Mass Transfer, 7(1): 1–9, 2016, https://doi.org/10.5098/hmt.7.16
23. Hayat T., Ijaz Khan M., Waqas M., Yasmeen T., Alsaedi A., Viscous dissipation effect in flow of magnetonanofluid with variable properties, Journal of Molecular Liquids, 222: 47–54, 2016, https://doi.org/10.1016/j.molliq.2016.06.096
24. Nayak M.K., MHD 3D flow and heat transfer analysis of nanofluid by shrinking surface inspired by thermal radiation and viscous dissipation, International Journal of Mechanical Sciences, 124–125: 185–193, 2017, https://doi.org/10.1016/j.ijmecsci.2017.03.014
25. Khan M.I., Hayat T., Khan M.I., Alsaedi A., A modified homogeneous-heterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating, International Journal of Heat and Mass Transfer, 113: 310–317, 2017, https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.082
26. Hayat T., Khan M.I., Alsaedi A., Khan M.I., Joule heating and viscous dissipation in flow of nanomaterial by a rotating disk, International Communications in Heat and Mass Transfer, 89: 190–197, 2017, https://doi.org/10.1016/j.icheatmasstransfer.2017.10.017
27. Ramesh K., Effects of viscous dissipation and Joule heating on the Couette and Poiseuille flows of a Jeffery fluid with slip boundary conditions, Propulsion and Power Research, 7(4): 329–341, 2018, https://doi.org/10.1016/j.jppr.2018.11.008
28. Muhammad T., Hayat T., Shehzad S.A., Alsaedi A., Viscous dissipation and Joule heating effects in MHD 3D flow with heat and mass fluxes, Results in Physics, 8: 365–371, 2018, https://doi.org/10.1016/j.rinp.2017.12.047
29. Venkateswarlu B., Satya Narayana P.V., Tarakaramu N., Melting and viscous dissipation effects on MHD flow over a moving surface with constant heat source, Transactions of A. Razmadze Mathematical Institute, 172(3B): 619–630, 2018, https://doi.org/10.1016/j.trmi.2018.03.007
30. Ezzat M.A., Abd-Elaal M.Z., Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, Journal of the Franklin Institute, 334(4): 685–706, 1997, https://doi.org/10.1016/S0016-0032%2896%2900095-6
31. Prasad K.V., Pal D., Umesh V., Prasanna Rao N.S., The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, Communication in Non-Linear Science and Numerical Simulation, 15(2): 331–344, 2010, https://doi.org/10.1016/j.cnsns.2009.04.003
32. Goyal M., Bhargava R., Numerical solution of MHD viscoelastic nanofluid flow over a stretching sheet with partial slip and heat source/sink, ISRN Nanotechnology, 2013: Article ID 931021, 11 pages, 2013, https://doi.org/10.1155/2013/931021
33. Rashidi M.M., Ali M., Freidoonimehr N., Rostami B., Hossain M.A., Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation, Advances in Mechanical Engineering, 6, 2014, https://doi.org/10.1155/2014/735939
34. Venkateswarlu B., Satya Narayana P.V., MHD visco-elastic fluid flow over a continuously moving vertical surface with chemical reaction, Walailak Journal of Science and Technology, 12(9): 775–783, 2015.
35. Li J., Zheng L., Liu L., MHD viscoelastic flow and heat transfer over a vertical stretching sheet with Cattaneo-Christov heat flux effects, Journal of Molecular Liquids, 221: 19–25, 2016, https://doi.org/10.1016/j.molliq.2016.05.051
36. Bilal S., Malik M.Y., Awais M., Khalil-ur-Rehman, Hussain A., Khan I., Numerical investigation on 2D viscoelastic fluid due to exponentially stretching surface with magnetic effects: an application of non-Fourier flux theory, Natural Computing and Applications, 30(9): 2749–2758, 2018, https://doi.org/10.1007/s00521-016-2832-4
37. Satya Narayana P.V., Tarakaramu N., Makinde O.D., Venkateswarlu B., Sarojamma G., MHD stagnation point flow of viscoelastic nanofluid past a convectively heated stretching surface, Defect and Diffusion Forum, 387:106–120, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.106
38. Choudhury R., Dey D., Free convective MHD flow of a non-Newtonian fluid past an infinite vertical plate with constant suction and heat sink, International Journal of Dynamics of Fluids, 8(2): 83–94, 2012.
39. Dessie H., Kishan N., MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/sink, Engineering Physics and Mathematics. Ain Shams Engineering Journal, 5(3): 967–977, 2014, https://doi.org/10.1016/j.asej.2014.03.008
40. Besthapu P., Ul Haq R., Bandari S., Al-Mdallal Q.M., Mixed convection flow of thermally stratified MHD nanofluid over an exponentially stretching surface with viscous dissipation effect, Journal of the Taiwan Institute of Chemical Engineers, 71: 307–314, 2017, https://doi.org/10.1016/j.jtice.2016.12.034
2. Singh K.R., Cowling T.G., Thermal convection in magnetohydrodynamics: I. Boundary layer flow up a hot vertical plate, The Quarterly Journal of Mechanics and Applied Mathematics, 16(1): 1–15, 1963, https://doi.org/10.1093/qjmam/16.1.1
3. Sacheti N.C., Chandran P., Singh A.K., An exact solution for unsteady magnetohydrodynamic free convective flow with constant heat flux, International Communications in Heat and Mass Transfer, 21(1): 131–142, 1994, https://doi.org/10.1016/0735-1933%2894%2990090-6
4. Sattar Md.A., Alam Md.M., MHD free convective heat and mass transfer flow with Hall current and constant heat flux through a porous medium, Indian Journal of Pure and Applied Mathematics, 26(2): 157–167, 1995.
5. Sahoo P.K., Datta N., Biswal S., Magnetohydrodynamic unsteady free convective flow past an infinite vertical plate with constant suction and heat sink, Indian Journal of Pure and Applied Mathematics, 34(1): 145–155, 2003.
6. Maqbool K., Mann A.B., Tiwana M.H., Unsteady MHD convective flow of a Jeffery fluid embedded in a porous medium with ramped wall velocity and temperature, Alexandria Engineering Journal, 57(2): 1071–1078, 2018, https://doi.org/10.1016/j.aej.2017.02.012
7. Hamza M.M., Free convection slip flow of an exothermic fluid in a convectively heated vertical channel, Ain Shams Engineering Journal, 9(4): 1313–1323, 2018, https://doi.org/10.1016/j.asej.2016.08.011
8. Un Nisa Z., Hajizadeh A., Nazar M., Free convection flow of nanofluid over infinite vertical plate with damped thermal flux, Chinese Journal of Physics, 59: 175–188, 2019, https://doi.org/10.1016/j.cjph.2019.02.029
9. Hajizadeh A., Shah N.A., Shah S.I.S., Animasaun I.L., Gorji M.-R., Alarifi I.M., Free convection flow of nanofluids between two vertical plates with damped thermal flux, Journal of Molecular Liquids, 289: Article ID 110964, 2019, https://doi.org/10.1016/j.molliq.2019.110964
10. Shah N., Zafar A.A., Fetecau C., Free convection flows over a vertical plate that applies shear stress to a fractional viscous fluid, Alexandria Engineering Journal, 57(4): 2529–2540, 2018, https://doi.org/10.1016/j.aej.2017.08.023
11. Gholinia M., Hoseini M.E., Gholinia S., A numerical investigation of free convection MHD flow of Walters-B nanofluid over an inclined stretching sheet under the impact of Joule heating, Thermal Science and Engineering Progress, 11: 272–282, 2019, https://doi.org/10.1016/j.tsep.2019.04.006
12. Wang Z.H., Zhou Z.K., Eternal natural convection heat transfer of liquid metal under the influence of the magnetic field, International Journal of Heat and Mass Transfer, 134: 175–184, 2019, https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.173
13. Patel H.R., Effects of cross diffusion and heat generation on mixed convective MHD flow of Casson fluid through porous medium with non-linear thermal radiation, Heliyon, 5(4): 1–26, 2019, https://doi.org/10.1016/j.heliyon.2019.e01555
14. Chamkha A.J., Thermal radiation and buoyancy effects on hydromagnetic flow over an accelerating permeable surface with heat source or sink, International Journal of Engineering Science, 38(15): 1699–1712, 2000, https://doi.org/10.1016/S0020-7225%2899%2900134-2
15. Satya Narayana P.V., Venkateswarlu B., Venkataramana S., Thermal radiation and heat source effects on a MHD nanofluid past a vertical plate in a rotating system with porous medium, Heat Transfer Asian Research, 44(1): 1–19, 2015, https://doi.org/10.1002/htj.21101
16. Cortell R., MHD (magneto-hydrodynamic) flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, Energy, 74: 896–905, 2014, https://doi.org/10.1016/j.energy.2014.07.069
17. Amir Hamzah N.S., Kandasamy R., Muhammad R., Thermal radiation energy on squeezed MHD flow of Cu, Al2O3 and CNTs-nanofluid over a sensor surface, Alexandria Engineering Journal, 55(3): 2405–2421, 2016, https://doi.org/10.1016/j.aej.2016.04.019
18. Fagbade A.I., Falodun B.O., Omowaye A.J., MHD natural convection flow of viscoelastic fluid over an accelerating permeable surface with thermal radiation and heat source or sink: spectral homotopy analysis approach, Ain Shams Engineering Journal, 9(4): 1029–1041, 2018, https://doi.org/10.1016/j.asej.2016.04.021
19. Satya Narayana P.V., Akshit S.M., Ghori J.P., Venkateswarlu B., Thermal radiation effects on an unsteady MHD nanofluid flow over a stretching sheet with non-uniform heat source/sink, Journal of Nanofluids, 6(5): 899–907, 2017, https://doi.org/10.1166/jon.2017.1374
20. Harish Babu D., Ajmath K.A., Venkateswarlu B., Satya Narayana P.V., Thermal radiation and heat source effects on MHD non-Newtonian nanofluid flow over a stretching sheet, Journal of Nanofluids, 8(5): 1085–1092, 2018, https://doi.org/10.1166/jon.2019.1666
21. Makanda G., Makinde O.D., Sibanda P., Natural convection of viscoelastic fluid from a cone embedded in a porous medium with viscous dissipation, Mathematical Problems in Engineering, 2013: Article ID 934712, 11 pages, 2013, https://doi.org/10.1155/2013/934712
22. Venkateswarlu B., Satya Narayana P.V., Influence of variable thermal conductivity on MHD Casson fluid flow over a stretching sheet with viscous dissipation, Soret and Dufour effects, Frontiers in Heat and Mass Transfer, 7(1): 1–9, 2016, https://doi.org/10.5098/hmt.7.16
23. Hayat T., Ijaz Khan M., Waqas M., Yasmeen T., Alsaedi A., Viscous dissipation effect in flow of magnetonanofluid with variable properties, Journal of Molecular Liquids, 222: 47–54, 2016, https://doi.org/10.1016/j.molliq.2016.06.096
24. Nayak M.K., MHD 3D flow and heat transfer analysis of nanofluid by shrinking surface inspired by thermal radiation and viscous dissipation, International Journal of Mechanical Sciences, 124–125: 185–193, 2017, https://doi.org/10.1016/j.ijmecsci.2017.03.014
25. Khan M.I., Hayat T., Khan M.I., Alsaedi A., A modified homogeneous-heterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating, International Journal of Heat and Mass Transfer, 113: 310–317, 2017, https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.082
26. Hayat T., Khan M.I., Alsaedi A., Khan M.I., Joule heating and viscous dissipation in flow of nanomaterial by a rotating disk, International Communications in Heat and Mass Transfer, 89: 190–197, 2017, https://doi.org/10.1016/j.icheatmasstransfer.2017.10.017
27. Ramesh K., Effects of viscous dissipation and Joule heating on the Couette and Poiseuille flows of a Jeffery fluid with slip boundary conditions, Propulsion and Power Research, 7(4): 329–341, 2018, https://doi.org/10.1016/j.jppr.2018.11.008
28. Muhammad T., Hayat T., Shehzad S.A., Alsaedi A., Viscous dissipation and Joule heating effects in MHD 3D flow with heat and mass fluxes, Results in Physics, 8: 365–371, 2018, https://doi.org/10.1016/j.rinp.2017.12.047
29. Venkateswarlu B., Satya Narayana P.V., Tarakaramu N., Melting and viscous dissipation effects on MHD flow over a moving surface with constant heat source, Transactions of A. Razmadze Mathematical Institute, 172(3B): 619–630, 2018, https://doi.org/10.1016/j.trmi.2018.03.007
30. Ezzat M.A., Abd-Elaal M.Z., Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, Journal of the Franklin Institute, 334(4): 685–706, 1997, https://doi.org/10.1016/S0016-0032%2896%2900095-6
31. Prasad K.V., Pal D., Umesh V., Prasanna Rao N.S., The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, Communication in Non-Linear Science and Numerical Simulation, 15(2): 331–344, 2010, https://doi.org/10.1016/j.cnsns.2009.04.003
32. Goyal M., Bhargava R., Numerical solution of MHD viscoelastic nanofluid flow over a stretching sheet with partial slip and heat source/sink, ISRN Nanotechnology, 2013: Article ID 931021, 11 pages, 2013, https://doi.org/10.1155/2013/931021
33. Rashidi M.M., Ali M., Freidoonimehr N., Rostami B., Hossain M.A., Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation, Advances in Mechanical Engineering, 6, 2014, https://doi.org/10.1155/2014/735939
34. Venkateswarlu B., Satya Narayana P.V., MHD visco-elastic fluid flow over a continuously moving vertical surface with chemical reaction, Walailak Journal of Science and Technology, 12(9): 775–783, 2015.
35. Li J., Zheng L., Liu L., MHD viscoelastic flow and heat transfer over a vertical stretching sheet with Cattaneo-Christov heat flux effects, Journal of Molecular Liquids, 221: 19–25, 2016, https://doi.org/10.1016/j.molliq.2016.05.051
36. Bilal S., Malik M.Y., Awais M., Khalil-ur-Rehman, Hussain A., Khan I., Numerical investigation on 2D viscoelastic fluid due to exponentially stretching surface with magnetic effects: an application of non-Fourier flux theory, Natural Computing and Applications, 30(9): 2749–2758, 2018, https://doi.org/10.1007/s00521-016-2832-4
37. Satya Narayana P.V., Tarakaramu N., Makinde O.D., Venkateswarlu B., Sarojamma G., MHD stagnation point flow of viscoelastic nanofluid past a convectively heated stretching surface, Defect and Diffusion Forum, 387:106–120, 2018, https://doi.org/10.4028/www.scientific.net/DDF.387.106
38. Choudhury R., Dey D., Free convective MHD flow of a non-Newtonian fluid past an infinite vertical plate with constant suction and heat sink, International Journal of Dynamics of Fluids, 8(2): 83–94, 2012.
39. Dessie H., Kishan N., MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/sink, Engineering Physics and Mathematics. Ain Shams Engineering Journal, 5(3): 967–977, 2014, https://doi.org/10.1016/j.asej.2014.03.008
40. Besthapu P., Ul Haq R., Bandari S., Al-Mdallal Q.M., Mixed convection flow of thermally stratified MHD nanofluid over an exponentially stretching surface with viscous dissipation effect, Journal of the Taiwan Institute of Chemical Engineers, 71: 307–314, 2017, https://doi.org/10.1016/j.jtice.2016.12.034
