Mixed Convection MHD Flow of Viscoelastic Fluid in a Porous Medium past a Hot Vertical Plate

Affiliation(s)

Department of Mathematics, Garbeta College, Paschim Medinipur, India.

Department of Mathematics, Jadavpur University, Kolkata, India.

Department of Mathematics, Garbeta College, Paschim Medinipur, India.

Department of Mathematics, Jadavpur University, Kolkata, India.

ABSTRACT

The boundary layer flow of a steady incompressible and visco-elastic fluid with short memory (obeying Walters’ B fluid model) passing over a hot vertical porous plate has been investigated in the presence of transverse magnetic field. The momentum and energy equations are reduced to couple non-linear partial differential equations along with the boundary conditions by using a suitable similarity transformation. These partial differential equations are transformed to a system of coupled non-linear ordinary differential equations by employing a perturbation technique. The system is solved by developing a suitable numerical procedure such as implicit finite difference scheme along with Newton’s linearization method. The computational results for the flow quantities have presented graphically for the effects of thermal radiation, viscous dissipation, heat generation/absorption, visco-elasticity, Hartmann number and the permeability parameter. Results demonstrated that Prandtl number has more pronouncing effect on the temperature distribution rather than the viscosity parameter as well as the thermal radiation parameter. Further the velocity gradient changes significantly due to the presence of temperature dependent variable viscosity.

The boundary layer flow of a steady incompressible and visco-elastic fluid with short memory (obeying Walters’ B fluid model) passing over a hot vertical porous plate has been investigated in the presence of transverse magnetic field. The momentum and energy equations are reduced to couple non-linear partial differential equations along with the boundary conditions by using a suitable similarity transformation. These partial differential equations are transformed to a system of coupled non-linear ordinary differential equations by employing a perturbation technique. The system is solved by developing a suitable numerical procedure such as implicit finite difference scheme along with Newton’s linearization method. The computational results for the flow quantities have presented graphically for the effects of thermal radiation, viscous dissipation, heat generation/absorption, visco-elasticity, Hartmann number and the permeability parameter. Results demonstrated that Prandtl number has more pronouncing effect on the temperature distribution rather than the viscosity parameter as well as the thermal radiation parameter. Further the velocity gradient changes significantly due to the presence of temperature dependent variable viscosity.

Cite this paper

nullS. Ghosh and G. Shit, "Mixed Convection MHD Flow of Viscoelastic Fluid in a Porous Medium past a Hot Vertical Plate,"*World Journal of Mechanics*, Vol. 2 No. 5, 2012, pp. 262-271. doi: 10.4236/wjm.2012.25032.

nullS. Ghosh and G. Shit, "Mixed Convection MHD Flow of Viscoelastic Fluid in a Porous Medium past a Hot Vertical Plate,"

References

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[2] A. G. Raptis, G. Tzivanidis and N. Kafousias, “Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction”, Letters in Heat and Mass Transfer, Vol. 8, 1981, pp. 417-424. doi:10.1016/0094-4548(81)90029-1

[3] A. G. Raptis, N. Kafousias and C. Massalas, “Free convection and mass transfer flow through a porous me dium bounded by an infinite vertical limiting surface with constant heat flux”, Journal of Applied Mathematics and Mechanics (ZAMM), Vol. 62, 1982, pp. 489-491.

[4] F. C. Lai and F. A. Fulacki, “The effect of variable vis-cosity on convective heat transfer along a vertical surface in a saturated porous media”, International Journal of Heat and Mass Transfer, Vol. 33, 1990, 1028-1031. doi:10.1016/0017-9310(90)90084-8

[5] H. P. Rani and Chang Nyung Kim, “Transient convection on vertical cylinder with variable viscosity and thermal conductivity”, Journal of Thermo Physics and Heat Transfer, Vol. 22, No. 2, 2008, pp. 254—261. doi:10.2514/1.32501

[6] P. K. Sharma, “Simultaneous thermal and mass diffusion in three dimensional mixed convection flow through a po-rous media”, Journal of Porous Media, Vol. 8, 2005, pp. 409-417. doi:10.1615/JPorMedia.v8.i4.70

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[9] P. Laganathan, P. Ganesan and D. Iranian, “Effects of thermal conductivity on unsteady MHD free convective flow over a semi-infinite vertical plate”, International Journal of Engineering Sciences, Vol. 52, No. 11, 2010, pp. 6257—6268.

[10] G. C. Shit and R. Haldar, “Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet”, Applied Mathematics and Mechanics, Vol. 32, No. 6, 2011, pp. 677-688. doi:10.1007/s10483-011-1448-6

[11] J. Y. Jang and J. S. Leu, “ Variable viscosity effects on the vortex instability of free convection boundary layer over a horizontal surface, Numerical Heat Transfer, Vol. 25, 1994, pp. 495-500. doi:10.1080/10407789408955962

[12] M. Misra, N. Ahmad and Z. U. Siddiqui, “Unsteady boundary layer flow past a stretching plate and heat transfer with variable thermal conductivity”, World Journal of Mechanics, Vol. 2, No. 1, 2012, pp. 35-41. doi:10.4236/wjm.2012.21005

[13] B. K. Sharma,, R. C. Chaudhury and M. Agarwal, “Radiation effect of steady free convective flow along a uniform moving porous vertical plate in presence of heat source/sink and transverse magnetic field”, Bulletin of Calcutta Mathematical Society, Vol. 100, 2008, pp. 529-538.

[14] M. Massoudi and M. Ramezan, “ Effect of injection or suction on the Falkner-Skan Flows of second grade flu-ids”, International Journal of non-Linear Mechanics, Vol. 24, 1989, pp. 221-227. doi:10.1016/0020-7462(89)90041-3

[15] A.K. Singh and R. S. R. Gorla, “Free convective heat and mass transfer with Hall current,Joule heating and thermal diffusion”, Heat and Mass Transfer, Vol. 45, 2009, pp. 1341—1349. doi:10.1007/s00231-009-0506-9

[16] D. Pal and H Mondal “Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium\with non-uniform heat source/sink” Communications in Non-Linear Science and Numerical Simulation, Vol. 15, No. 6, 2010, pp. 1553—1564. doi:10.1016/j.cnsns.2009.07.002

[17] S. Shateyi and S. S. Motsa, “Variable Viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with Hall effect”, Boundary Value Problems, Vol. 1, 2010, pp. 1-20.

[18] N. C. Mahanti and P. Gaur,”The effect of varying viscosity and thermal conductivity on steady free convective flow and heat transfer along an isothermal vertical plate in the presence of heat sink, Journal of Applied Fluid Mechanics, Vol. 2, No. 1, 2009, pp. 23—28.

[19] R. Cortell, “Similarity solutions for flow and heat transfer of a viscoelastic fluid over a stretching sheet, Interna-tional Journal of non-Linear Mechanics, Vol. 29, 1994, pp. 155-161.

[20] R. Cortell, “Numerical solutions for flow and heat transfer of voscoelastic fluid over a stretching sheet”, International Journal of non-Linear Mechanics, Vol. 28, 1993, pp. 623-626. doi:10.1016/0020-7462(93)90023-E

[21] K. R. Rajagopal, A. Z. Szeri and W. Troy, An existence theorem for the flow of Newtonian fluid past an infinite porous plate”, International Journal of non-Linear Me chanics, Vol.21, 1986, pp. 279-289. doi:10.1016/0020-7462(86)90035-1

[22] R. Cortell, “Visco-elastic fluid flow and heat transfer over a stretching sheet under the effects of a non-uniform heat source, viscous dissipation and thermal radiation”, International Journal of Heat and Mass Transfer, Vol. 50, 2007, pp. 3152-3162. doi:10.1016/j.ijheatmasstransfer.2007.01.003

[23] H. S. Takhar, V. M. Soundalgekar and A. S. Gupta, “Mixed convction of an incompressible fluid in a porous media past a hot vertical plate, International Journal of non-Linear Mechanics, Vol. 25, 1990, pp. 723-728. doi:10.1016/0020-7462(90)90010-7

[24] T. R. Mahapatra and A. S. Gupta, “Stagnation point flow of a viscoelastic fluid towards a stretching surface, International Journal of non-Linear Mechanics, Vol. 39, 2004, pp. 811-820. doi:10.1016/S0020-7462(03)00044-1

[25] Misra J, C. and Shit G. C., “Flow of a Biomagnetic visco-elastic fluid in a channel with stretching walls”, ASME Journal of Applied Mechanics, Vol. 76, 2009, pp. 1-9. doi:10.1115/1.3130448

[26] Cebeci, T., Cousteix, J., Modeling and computation of boundary-layer flows”, Springer-Verlag, 1999

[27] Misra, J. C., Shit, G. C., Biomagnetic viscoelastic fluid flow over a stretching sheet”, Applied Mathematics and Computation, Vol. 210, 2009, pp. 350-361. doi:10.1016/j.amc.2008.12.088

[28] C. H. Chen, “Combined effects of joule heating and viscous dissipation on MHD flow past a permeable, stretching surface with free convection and radiative heat transfer”, ASME Journal of Heat Transfer, Vol. 132, 2010, pp. 1-3. doi:10.1115/1.4000946

[1] A. G. Raptis, Perdikis and G. Tzivanidis, “Free convection flow through a porous medium bounded by a vertical surface”, Journal of Physics D: Applied Physics, Vol. 14, 1981, pp. 99-102. doi:10.1088/0022-3727/14/7/001

[2] A. G. Raptis, G. Tzivanidis and N. Kafousias, “Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction”, Letters in Heat and Mass Transfer, Vol. 8, 1981, pp. 417-424. doi:10.1016/0094-4548(81)90029-1

[3] A. G. Raptis, N. Kafousias and C. Massalas, “Free convection and mass transfer flow through a porous me dium bounded by an infinite vertical limiting surface with constant heat flux”, Journal of Applied Mathematics and Mechanics (ZAMM), Vol. 62, 1982, pp. 489-491.

[4] F. C. Lai and F. A. Fulacki, “The effect of variable vis-cosity on convective heat transfer along a vertical surface in a saturated porous media”, International Journal of Heat and Mass Transfer, Vol. 33, 1990, 1028-1031. doi:10.1016/0017-9310(90)90084-8

[5] H. P. Rani and Chang Nyung Kim, “Transient convection on vertical cylinder with variable viscosity and thermal conductivity”, Journal of Thermo Physics and Heat Transfer, Vol. 22, No. 2, 2008, pp. 254—261. doi:10.2514/1.32501

[6] P. K. Sharma, “Simultaneous thermal and mass diffusion in three dimensional mixed convection flow through a po-rous media”, Journal of Porous Media, Vol. 8, 2005, pp. 409-417. doi:10.1615/JPorMedia.v8.i4.70

[7] A. Raptis and C. Perdikis,”Viscoelastic flow by the presence of rotation” Journal of Applied Mathematics and Mechanics (ZAMM), Vol. 78, No. 4, 1998, pp. 277-279.

[8] M. M. Rahman, A. A. Mamun, M. A. Azim, “Effects of temperature dependent thermal conductivity on MHD free convective flow along a vertical flat plate with heat conduction”, Non-linear analysis Modelling and Con-troll, Vol. 13, No. 4, 2008, pp. 513—524.

[9] P. Laganathan, P. Ganesan and D. Iranian, “Effects of thermal conductivity on unsteady MHD free convective flow over a semi-infinite vertical plate”, International Journal of Engineering Sciences, Vol. 52, No. 11, 2010, pp. 6257—6268.

[10] G. C. Shit and R. Haldar, “Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet”, Applied Mathematics and Mechanics, Vol. 32, No. 6, 2011, pp. 677-688. doi:10.1007/s10483-011-1448-6

[11] J. Y. Jang and J. S. Leu, “ Variable viscosity effects on the vortex instability of free convection boundary layer over a horizontal surface, Numerical Heat Transfer, Vol. 25, 1994, pp. 495-500. doi:10.1080/10407789408955962

[12] M. Misra, N. Ahmad and Z. U. Siddiqui, “Unsteady boundary layer flow past a stretching plate and heat transfer with variable thermal conductivity”, World Journal of Mechanics, Vol. 2, No. 1, 2012, pp. 35-41. doi:10.4236/wjm.2012.21005

[13] B. K. Sharma,, R. C. Chaudhury and M. Agarwal, “Radiation effect of steady free convective flow along a uniform moving porous vertical plate in presence of heat source/sink and transverse magnetic field”, Bulletin of Calcutta Mathematical Society, Vol. 100, 2008, pp. 529-538.

[14] M. Massoudi and M. Ramezan, “ Effect of injection or suction on the Falkner-Skan Flows of second grade flu-ids”, International Journal of non-Linear Mechanics, Vol. 24, 1989, pp. 221-227. doi:10.1016/0020-7462(89)90041-3

[15] A.K. Singh and R. S. R. Gorla, “Free convective heat and mass transfer with Hall current,Joule heating and thermal diffusion”, Heat and Mass Transfer, Vol. 45, 2009, pp. 1341—1349. doi:10.1007/s00231-009-0506-9

[16] D. Pal and H Mondal “Effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium\with non-uniform heat source/sink” Communications in Non-Linear Science and Numerical Simulation, Vol. 15, No. 6, 2010, pp. 1553—1564. doi:10.1016/j.cnsns.2009.07.002

[17] S. Shateyi and S. S. Motsa, “Variable Viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with Hall effect”, Boundary Value Problems, Vol. 1, 2010, pp. 1-20.

[18] N. C. Mahanti and P. Gaur,”The effect of varying viscosity and thermal conductivity on steady free convective flow and heat transfer along an isothermal vertical plate in the presence of heat sink, Journal of Applied Fluid Mechanics, Vol. 2, No. 1, 2009, pp. 23—28.

[19] R. Cortell, “Similarity solutions for flow and heat transfer of a viscoelastic fluid over a stretching sheet, Interna-tional Journal of non-Linear Mechanics, Vol. 29, 1994, pp. 155-161.

[20] R. Cortell, “Numerical solutions for flow and heat transfer of voscoelastic fluid over a stretching sheet”, International Journal of non-Linear Mechanics, Vol. 28, 1993, pp. 623-626. doi:10.1016/0020-7462(93)90023-E

[21] K. R. Rajagopal, A. Z. Szeri and W. Troy, An existence theorem for the flow of Newtonian fluid past an infinite porous plate”, International Journal of non-Linear Me chanics, Vol.21, 1986, pp. 279-289. doi:10.1016/0020-7462(86)90035-1

[22] R. Cortell, “Visco-elastic fluid flow and heat transfer over a stretching sheet under the effects of a non-uniform heat source, viscous dissipation and thermal radiation”, International Journal of Heat and Mass Transfer, Vol. 50, 2007, pp. 3152-3162. doi:10.1016/j.ijheatmasstransfer.2007.01.003

[23] H. S. Takhar, V. M. Soundalgekar and A. S. Gupta, “Mixed convction of an incompressible fluid in a porous media past a hot vertical plate, International Journal of non-Linear Mechanics, Vol. 25, 1990, pp. 723-728. doi:10.1016/0020-7462(90)90010-7

[24] T. R. Mahapatra and A. S. Gupta, “Stagnation point flow of a viscoelastic fluid towards a stretching surface, International Journal of non-Linear Mechanics, Vol. 39, 2004, pp. 811-820. doi:10.1016/S0020-7462(03)00044-1

[25] Misra J, C. and Shit G. C., “Flow of a Biomagnetic visco-elastic fluid in a channel with stretching walls”, ASME Journal of Applied Mechanics, Vol. 76, 2009, pp. 1-9. doi:10.1115/1.3130448

[26] Cebeci, T., Cousteix, J., Modeling and computation of boundary-layer flows”, Springer-Verlag, 1999

[27] Misra, J. C., Shit, G. C., Biomagnetic viscoelastic fluid flow over a stretching sheet”, Applied Mathematics and Computation, Vol. 210, 2009, pp. 350-361. doi:10.1016/j.amc.2008.12.088

[28] C. H. Chen, “Combined effects of joule heating and viscous dissipation on MHD flow past a permeable, stretching surface with free convection and radiative heat transfer”, ASME Journal of Heat Transfer, Vol. 132, 2010, pp. 1-3. doi:10.1115/1.4000946