Radiation Effect on Natural Convection Near a Vertical Plate Embedded in Porous Medium with Ramped Wall Temperature

ABSTRACT

Radiation effect on the natural convection flow of an optically thin viscous incompressible fluid near a vertical plate with ramped wall temperature in a porous medium has been studied. The exact solution of momentum and energy equations is obtained by the use of Laplace transform technique. The variations in fluid velocity and temperature are shown graphically whereas the numerical values of shear stress and the rate of heat transfer at the wall are presented in tabular form for various values of flow parameters. The results show that the fluid velocity increases with increase in Grashof number, Darcy number and time parameters whereas the fluid velocity decreases with increase in the radiation parameter and Prandtl number for ramped temperature as well as isothermal wall temperature. It is found that an increase in radiation parameter leads to rise the temperature for both ramped wall temperature as well as isothermal wall temperature. Further, it is found that an increase in Prandtl number leads to fall the temperature for both ramped wall temperature as well as isothermal wall temperature. The shear stress at the wall decreases with increases in either Prandtl number or porosity parameter while the result shows reverse in the case of radiation parameter. Finally, the rate of heat transfer is increased with increases in the radiation parameter for both ramped wall temperature as well as isothermal wall temperature.

Radiation effect on the natural convection flow of an optically thin viscous incompressible fluid near a vertical plate with ramped wall temperature in a porous medium has been studied. The exact solution of momentum and energy equations is obtained by the use of Laplace transform technique. The variations in fluid velocity and temperature are shown graphically whereas the numerical values of shear stress and the rate of heat transfer at the wall are presented in tabular form for various values of flow parameters. The results show that the fluid velocity increases with increase in Grashof number, Darcy number and time parameters whereas the fluid velocity decreases with increase in the radiation parameter and Prandtl number for ramped temperature as well as isothermal wall temperature. It is found that an increase in radiation parameter leads to rise the temperature for both ramped wall temperature as well as isothermal wall temperature. Further, it is found that an increase in Prandtl number leads to fall the temperature for both ramped wall temperature as well as isothermal wall temperature. The shear stress at the wall decreases with increases in either Prandtl number or porosity parameter while the result shows reverse in the case of radiation parameter. Finally, the rate of heat transfer is increased with increases in the radiation parameter for both ramped wall temperature as well as isothermal wall temperature.

KEYWORDS

Natural Convection, Darcy Number, Radiation Parameter, Prandtl Number, Porous Medium, Ramped Wall Temperature and Isothermal Wall Temperature

Natural Convection, Darcy Number, Radiation Parameter, Prandtl Number, Porous Medium, Ramped Wall Temperature and Isothermal Wall Temperature

Cite this paper

nullS. Das, M. Jana and R. Jana, "Radiation Effect on Natural Convection Near a Vertical Plate Embedded in Porous Medium with Ramped Wall Temperature,"*Open Journal of Fluid Dynamics*, Vol. 1 No. 1, 2011, pp. 1-11. doi: 10.4236/ojfd.2011.11001.

nullS. Das, M. Jana and R. Jana, "Radiation Effect on Natural Convection Near a Vertical Plate Embedded in Porous Medium with Ramped Wall Temperature,"

References

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[2] D. A. Nield and A. Bejan, “Convection in Porous Media,” Springer, New York, 2006, pp. 94-97.

[3] M. A. Mansour, “Radiation and Free Convection Effects on the Oscillating Flow past a Vertical Plate,” Astrophysics and Space Science, Vol. 166, No. 2, 1990, pp. 269- 275. doi:10.1007/BF01094898

[4] A. Raptis and C. Perdikis, “Radiation and Free Convection Flow past a Moving Plate,” Applied Mechanics and Engineering, Vol. 4, No. 4, 1999, pp. 817-821.

[5] U. N. Das, R. Deka and V. M. Soundalgekar, “Radiation Effect on Flow past an Impulsively Started Vertical Plate: An Exact Solutions,” Journal of Applied Mathematics and Fluid Mechanics, Vol. 1, No. 2, 1966, pp. 111-115.

[6] R. Grief, I. S. Habib and J. C. Lin, “Laminar Convection of Radi-ating Gas in a Vertical Channel,” Journal of Fluid Mechanics, Vol. 46, 1991, pp. 513-520. doi:10.1017/S0022112071000673

[7] P. Ganesan and P. Loganathan, “Radiation and Mass Transfer Effects on Flow of an Incompressible Viscous Fluid past a Moving Vertical Cylinder,” International Journal of Heat and Mass Transfer, Vol. 45, No. 21, 2002, pp. 4281-4288. doi:10.1016/S0017-9310(02)00140-0

[8] I. U. Mbeledogu, A. R. C. Amakiri and A. Ogulu, “Unsteady MHD Free Convection Flow of a Compressible Fluid past a Moving Vertical Plate in the Presence of Radiative Heat Transfer,” International Journal of Heat and Mass Transfer, Vol. 50, No. 9-10, 2007, pp. 326-331. doi:10.1016/j.ijheatmasstransfer.2006.10.032

[9] O. D. Mak-inde, “Free-Convection Flow with Thermal Radiation and Mass Transfer past a Moving Vertical Porous Plate,” International Communications in Heat and Mass Transfer, Vol. 32, No. 10, 2005, pp. 1411-1419. doi:10.1016/j.icheatmasstransfer.2005.07.005

[10] M. D. Ab-dus-Sattar and M. D. Hamid Kalim, “Unsteady Free-Convection Intraction with Thermal Radiation in a Boundary Layer Flow past a Vertical Porous Plate,” Journal of Mathematical Physics, Vol. 30, 1996, pp. 25-37.

[11] C. Is-rael-Cookey, A. Ogulu and V. B. Omubo-Pepple, “Influence of Viscous Dissipation and Radiation on Unsteady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction,” Interna-tional Journal of Heat and Mass Transfer, Vol. 46, No. 13, 2003, pp. 2305-2311. doi:10.1016/S0017-9310(02)00544-6

[12] K. D. Alagoa, G. Tay and T. M. Abbey, “Radiative and Free Convective Effects of a MHD Flow through a Porous Medium between Infinite Parallel Plates with Time-Dependent Suction,” Astrophysics and Space Science, Vol. 260, 1999, pp. 455-468. doi:10.1016/S0017-9310(02)00544-6

[13] C. Israel-Cookey, E. Amos and C. Nwaigwe, “MHD Oscillatory Couette Flow of a Radiating Viscous Fluid in a Porous Medium with Periodic Wall Temperature,” American Journal of Scientific and Industrial Research, Vol. 1, No. 2, 2010, pp. 326-331. doi:10.5251/ajsir.2010.1.2.326.331

[14] M. A. Sattar and M. A. Maleque, “Unsteady MHD Natural Convection Flow along an Accelerated Porous Plate with Hall Current and Mass Transfer in a Rotating Porous Medium,” Journal of Energy, Heat and Mass Transfer, Vol. 22, 2000, pp. 67-72.

[15] M. A. Sattar, M. M. Rahman and M. M. Alam, “Free Convection Flow and Heat Transfer through a Porous Vertical Flat Plate Immersed in a Porous Medium,” Journal of Energy Research, Vol. 22, No. 1, 2000, pp. 17-21.

[16] M. D. Abdus Samad and M. Mansur Rahman, “Thermal Radiation Interaction with Unsteady MHD Flow past a Vertical Porous Plate Immersed in a Porous Medium,” Journal of Naval Architecture and Marine Engineering, Vol. 3, No. 1, 2006, pp. 7-14.

[17] N. C. Mahanti and P. Gaur, “The Effects 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,” Jour-nal of Applied Fluid Mechanics, Vol. 2, No. 1, 2009, pp. 23-28.

[18] H. S. Takhar, P. Ganesan, K. Ekambavanan and V. M. Soundalgekar, “Transient Free Convection past a Semi Infinite Vertical Plate with Variable Surface Temperature,” International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 7, No. 4, 1997, pp. 280-296. doi:10.1108/09615539710165804

[19] A. C. L. Cogley, W. G. Vincenti and E. S. Gilles, “Differential Approximation for Radiative Heat Transfer in a Non Grey Gas near Equilibrium,” American Institute of Aeronautics and Astronautics, Vol. 6, No. 3, 1968, pp. 551-553.

[1] P. S. Ghoshdastidar, “Heat Transfer,” Oxford University Press, Oxford, 2004.

[2] D. A. Nield and A. Bejan, “Convection in Porous Media,” Springer, New York, 2006, pp. 94-97.

[3] M. A. Mansour, “Radiation and Free Convection Effects on the Oscillating Flow past a Vertical Plate,” Astrophysics and Space Science, Vol. 166, No. 2, 1990, pp. 269- 275. doi:10.1007/BF01094898

[4] A. Raptis and C. Perdikis, “Radiation and Free Convection Flow past a Moving Plate,” Applied Mechanics and Engineering, Vol. 4, No. 4, 1999, pp. 817-821.

[5] U. N. Das, R. Deka and V. M. Soundalgekar, “Radiation Effect on Flow past an Impulsively Started Vertical Plate: An Exact Solutions,” Journal of Applied Mathematics and Fluid Mechanics, Vol. 1, No. 2, 1966, pp. 111-115.

[6] R. Grief, I. S. Habib and J. C. Lin, “Laminar Convection of Radi-ating Gas in a Vertical Channel,” Journal of Fluid Mechanics, Vol. 46, 1991, pp. 513-520. doi:10.1017/S0022112071000673

[7] P. Ganesan and P. Loganathan, “Radiation and Mass Transfer Effects on Flow of an Incompressible Viscous Fluid past a Moving Vertical Cylinder,” International Journal of Heat and Mass Transfer, Vol. 45, No. 21, 2002, pp. 4281-4288. doi:10.1016/S0017-9310(02)00140-0

[8] I. U. Mbeledogu, A. R. C. Amakiri and A. Ogulu, “Unsteady MHD Free Convection Flow of a Compressible Fluid past a Moving Vertical Plate in the Presence of Radiative Heat Transfer,” International Journal of Heat and Mass Transfer, Vol. 50, No. 9-10, 2007, pp. 326-331. doi:10.1016/j.ijheatmasstransfer.2006.10.032

[9] O. D. Mak-inde, “Free-Convection Flow with Thermal Radiation and Mass Transfer past a Moving Vertical Porous Plate,” International Communications in Heat and Mass Transfer, Vol. 32, No. 10, 2005, pp. 1411-1419. doi:10.1016/j.icheatmasstransfer.2005.07.005

[10] M. D. Ab-dus-Sattar and M. D. Hamid Kalim, “Unsteady Free-Convection Intraction with Thermal Radiation in a Boundary Layer Flow past a Vertical Porous Plate,” Journal of Mathematical Physics, Vol. 30, 1996, pp. 25-37.

[11] C. Is-rael-Cookey, A. Ogulu and V. B. Omubo-Pepple, “Influence of Viscous Dissipation and Radiation on Unsteady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction,” Interna-tional Journal of Heat and Mass Transfer, Vol. 46, No. 13, 2003, pp. 2305-2311. doi:10.1016/S0017-9310(02)00544-6

[12] K. D. Alagoa, G. Tay and T. M. Abbey, “Radiative and Free Convective Effects of a MHD Flow through a Porous Medium between Infinite Parallel Plates with Time-Dependent Suction,” Astrophysics and Space Science, Vol. 260, 1999, pp. 455-468. doi:10.1016/S0017-9310(02)00544-6

[13] C. Israel-Cookey, E. Amos and C. Nwaigwe, “MHD Oscillatory Couette Flow of a Radiating Viscous Fluid in a Porous Medium with Periodic Wall Temperature,” American Journal of Scientific and Industrial Research, Vol. 1, No. 2, 2010, pp. 326-331. doi:10.5251/ajsir.2010.1.2.326.331

[14] M. A. Sattar and M. A. Maleque, “Unsteady MHD Natural Convection Flow along an Accelerated Porous Plate with Hall Current and Mass Transfer in a Rotating Porous Medium,” Journal of Energy, Heat and Mass Transfer, Vol. 22, 2000, pp. 67-72.

[15] M. A. Sattar, M. M. Rahman and M. M. Alam, “Free Convection Flow and Heat Transfer through a Porous Vertical Flat Plate Immersed in a Porous Medium,” Journal of Energy Research, Vol. 22, No. 1, 2000, pp. 17-21.

[16] M. D. Abdus Samad and M. Mansur Rahman, “Thermal Radiation Interaction with Unsteady MHD Flow past a Vertical Porous Plate Immersed in a Porous Medium,” Journal of Naval Architecture and Marine Engineering, Vol. 3, No. 1, 2006, pp. 7-14.

[17] N. C. Mahanti and P. Gaur, “The Effects 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,” Jour-nal of Applied Fluid Mechanics, Vol. 2, No. 1, 2009, pp. 23-28.

[18] H. S. Takhar, P. Ganesan, K. Ekambavanan and V. M. Soundalgekar, “Transient Free Convection past a Semi Infinite Vertical Plate with Variable Surface Temperature,” International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 7, No. 4, 1997, pp. 280-296. doi:10.1108/09615539710165804

[19] A. C. L. Cogley, W. G. Vincenti and E. S. Gilles, “Differential Approximation for Radiative Heat Transfer in a Non Grey Gas near Equilibrium,” American Institute of Aeronautics and Astronautics, Vol. 6, No. 3, 1968, pp. 551-553.