Mathematical Modelling of Mass Transfer and Free Convection Current Effects on Unsteady Viscous Flow with Ramped Wall Temperature

ABSTRACT

An exact analysis of the flow of an incompressible viscous fluid past an infinite vertical plate is conducted taking into account the presence of foreign mass or constant mass flux and ramped wall temperature. The dimensionless governing coupled linear partial differential equations are solved using the Laplace transform technique. Two different solutions for the fluid velocity are obtained–one valid for the fluids of Schmidt numbers different from unity, and the other for which the Schmidt number is unity. The effects of Prandtl number (Pr), Schmidt number (Sc), time (t) and mass to thermal buoyancy ratio parameter (N) for both aiding and opposing buoyancy effects on the velocity and skin-friction are studied. Also, the heat and mass transfer effects on the flow near a ramped temperature plate have been compared with the flow near a plate with constant temperature.

An exact analysis of the flow of an incompressible viscous fluid past an infinite vertical plate is conducted taking into account the presence of foreign mass or constant mass flux and ramped wall temperature. The dimensionless governing coupled linear partial differential equations are solved using the Laplace transform technique. Two different solutions for the fluid velocity are obtained–one valid for the fluids of Schmidt numbers different from unity, and the other for which the Schmidt number is unity. The effects of Prandtl number (Pr), Schmidt number (Sc), time (t) and mass to thermal buoyancy ratio parameter (N) for both aiding and opposing buoyancy effects on the velocity and skin-friction are studied. Also, the heat and mass transfer effects on the flow near a ramped temperature plate have been compared with the flow near a plate with constant temperature.

Cite this paper

nullM. Narahari, O. Bég and S. Ghosh, "Mathematical Modelling of Mass Transfer and Free Convection Current Effects on Unsteady Viscous Flow with Ramped Wall Temperature,"*World Journal of Mechanics*, Vol. 1 No. 4, 2011, pp. 176-184. doi: 10.4236/wjm.2011.14023.

nullM. Narahari, O. Bég and S. Ghosh, "Mathematical Modelling of Mass Transfer and Free Convection Current Effects on Unsteady Viscous Flow with Ramped Wall Temperature,"

References

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[2] K. A. Antonopoulos and F. Democritou, “Experimental and Numerical Study of Unsteady Non-periodic Wall Heat Transfer under Step, Ramp and Cosine Temperature Perturbations,” International Journal of Energy Research, Vol. 18, No. 6, 1994, pp. 563-579. HHUUdoi:10.1002/er.4440180602UU

[3] M. A. Kuczmarski and S. A. Gokoglu, “Buoyancy Suppression in Gases at High Temperatures,” International Journal of Heat Fluid Flow, Vol. 28, No. 3, 2007, pp. 496-511.HHUUdoi:10.1016/j.ijheatfluidflow.2006.07.006UU

[4] J. A. Schetz, “On the Approximate Solution of Viscous-Flow Problems,” ASME Journal of Applied Mechanics, Vol. 30, 1963, pp. 263-268.

[5] J. A. Schetz and R. Eichhorn, “Unsteady Natural Convection in the Vicinity of a Doubly Infinite Vertical Plate,” Journal of Heat Transfer, Vol. 84, 1962, pp. 334-338.

[6] A. A. Hayday, D. A. Bowlus and R. A. McGraw, “Free Convection from a Vertical Flat Plate with Step Discontinuities in Surface Temperature,” ASME Journal of Heat Transfer, Vol. 89, 1967, pp. 244-250.

[7] M. Kelleher, “Free Convection from a Vertical Plate with Discontinuous Wall Temperature,” ASME Journal of Heat Transfer, Vol. 93, 1971, pp. 349-356. HHUUdoi:org/10.1115/1.3449830UU

[8] T. T. Kao, “Laminar Free Convective Heat Transfer Response along a Vertical Flat Plate with Step Jump in Surface Temperature,” Letters of Heat Mass Transfer, Vol. 2, No. 5, 1975, pp. 419-428. HHUUdoi:org/10.1016/0094-4548(75)90008-9UU

[9] S. Lee and M. M. Yovanovich, “Laminar Natural Convection from a Vertical Plate with a Step Change in Wall Temperature,” ASME Journal of Heat Transfer, Vol. 113, 1991, pp. 501-504. HHUUdoi:org/10.1115/1.2910591UU

[10] P. Chandran, N. C. Sacheti and A. K. Singh, “Natural con-Vection near a Vertical Plate with Ramped Wall Temperature,” Heat Mass Transfer, Vol. 41, No. 5, 2005, pp. 459-464. HHUUdoi:org/10.1007/s00231-004-0568-7UU

[11] S. C. Saha, C. Lei and J. C. Patterson, “On the Natural Convection Boundary Layer Adjacent to an Inclined Flat Plate Subject to Ramp Heating,” 16th Australian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Australia, 2007, pp. 121-124.

[12] R. C. Chaudhary and P. Jain, “Transient Free Convection Flow along an Infinite Vertical Oscillating Plate with a Step Change in Wall Temperature,” Acta Technica ?SAV, Vol. 52, 2007, pp. 187-204.

[13] B. Gebhart and L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects of Thermal and Mass Diffusion,” International Journal of Heat and Mass Transfer, Vol. 14, No. 12, 1971, pp. 2025-2050. HHUUdoi:org/10.1016/0017-9310(71)90026-3UU

[14] T. S. Chen, C. F. Yuh and A. Moustsoglou, “Combined Heat and Mass Transfer in Mixed Convection along Vertical and Inclined Plates,” International Journal of Heat and Mass Transfer, Vol. 23, No. 4, 1980, pp. 527-537. HHUUdoi:org/10.1016/0017-9310(80)90094-0UU

[15] O. Anwar Bég, Tasveer A. Bég, A. Y. Bakier and V. R. Prasad, “Chemically-Reacting Mixed Convective Heat and Mass Transfer along Inclined and Vertical Plates with Soret and Dufour Effects: Numerical Solutions,” International Journal of Applied Mathematics Mechanics, Vol. 5, No. 2, 2009, pp. 39-57.

[16] V. M. Soundalgekar, “Effects of Mass Transfer and Free Convection on the Flow Past an Impulsively Started Vertical Plate,” ASME Journal of Applied Mechanics, Vol. 46, No. 4, 1979, pp. 757-760. HHUUdoi:org/10.1115/1.3424649UU

[17] U. N. Dass, S. N. Ray and V. M. Soundalgekar, “Mass Transfer Effects on Flow Past an Impulsively Started Infinite Vertical Plate with Constant Mass Flux–an Exact Solution,” Heat and Mass transfer, Vol. 31, No. 3, 1996, pp. 163-167. HHUUdoi: 10.1007/bf02333314UU

[18] R. Muthucumaraswamy, P. Ganesan and V. M. Soundalgekar, “Heat and Mass Transfer Effects on Flow Past an Impulsively Started Vertical Plate,” Acta Mechanica, Vol. 146, No. 1-2, 2001, pp. 1-8. HHUUdoi:org/10.1007/BF01178790UU

[19] S. Ostrach, “An Analysis of Laminar Free Convection Flow and Heat and Mass Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force,” NACA 1111, Technical Report, 1952.

[20] H. S. Takhar and C. P. Perdikis, “Forced and Free Convective Flow of Water at 4?C through a Porous Medium,” International Communications in Heat and Mass Transfer, Vol. 21, 1986, pp. 605-609. HHUUdoi:org/10.1016/0735-1933(94)90005-1UUHHS. Takhar and P. C. Ram, “Magnetohydrodynamic Free Convection Flow of Water at 4?C through a Porous Medium,” International Communications in Heat and Mass Transfer, Vol. 21, No. 3, 1994, pp. 371-376.

[21] M. Ilicak and G. K. Vallis, “Simulations and Scaling of Horizontal Convection,” Tellus, under review, May 25th 2011.

[22] J. H. Siggers, R. R. Kerswell and N. J. Balmforth, “Bounds on Horizontal Convection,” Journal of Fluid Mechanics, Vol. 517, 2004, pp. 55-70. HHUUdoi:org/10.1017/S0022112004000497UU

[23] S. K. Ghosh, O. Anwar Bég and M. Narahari, “Hall Effects on MHD Flow in a Rotating System with Heat Transfer Characteristics,” Meccanica Journal, Vol. 44, No. 6, 2009, pp. 741-765. HHUUdoi:org/10.1007/s11012-009-9210-6UU

[24] S. K. Ghosh, O. Anwar Bég and J. Zueco, “Hydromagnetic Free Convection Rayleigh Flow with Induced Magnetic Field Effects,” Meccanica, Vol. 45, No. 2, 2010, pp. 175-185. HHUUdoi:org/10.1007/s11012-009-9235-xUU

[25] S. K. Ghosh, S. Rawat, O. Anwar Bég and Tasveer A. Bég, “Thermal Radiation Effects on Unsteady Hydromagnetic Gas Flow Along an Inclined Plane with Indirect Natural Convection,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 13, 2010, pp. 41-57.

[26] S. K. Ghosh, O. Anwar Bég, J. Zueco and Prasad V. R., “Transient Hydromagnetic Flow in a Rotating Channel Permeated by an Inclined Magnetic Field with Magnetic Induction and Maxwell Displacement Current Effects,” ZAMP: Journal of Applied Mathematics and Physics, Vol. 61, No. 1, 2010, pp. 147-169. HHUUdoi:org/10.1007/s00033-009-0006-2UU

[27] O. Anwar Bég, J. Zueco, S. K. Ghosh and A. Heidari, “Unsteady Magnetohydrodynamic Heat Transfer in A Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study,” Advances in Numerical Analysis, Vol. 2011, 2011, pp. 1-17.

[1] C. P. Malhotra, R. L. Mahajan, W. S. Sampath, K. L. Barth and R. A. Enzenroth, “Control of Temperature Uniformity during the Manufacture of Stable Thin-Film Pho- Tovoltaic Devices,” International Journal of Heat and Mass Transfer, Vol. 49, No. 17-18, 2006, pp. 2840- 2850. HHUUdoi:10.1016/j.ijheatmasstransfer.2006.02UU

[2] K. A. Antonopoulos and F. Democritou, “Experimental and Numerical Study of Unsteady Non-periodic Wall Heat Transfer under Step, Ramp and Cosine Temperature Perturbations,” International Journal of Energy Research, Vol. 18, No. 6, 1994, pp. 563-579. HHUUdoi:10.1002/er.4440180602UU

[3] M. A. Kuczmarski and S. A. Gokoglu, “Buoyancy Suppression in Gases at High Temperatures,” International Journal of Heat Fluid Flow, Vol. 28, No. 3, 2007, pp. 496-511.HHUUdoi:10.1016/j.ijheatfluidflow.2006.07.006UU

[4] J. A. Schetz, “On the Approximate Solution of Viscous-Flow Problems,” ASME Journal of Applied Mechanics, Vol. 30, 1963, pp. 263-268.

[5] J. A. Schetz and R. Eichhorn, “Unsteady Natural Convection in the Vicinity of a Doubly Infinite Vertical Plate,” Journal of Heat Transfer, Vol. 84, 1962, pp. 334-338.

[6] A. A. Hayday, D. A. Bowlus and R. A. McGraw, “Free Convection from a Vertical Flat Plate with Step Discontinuities in Surface Temperature,” ASME Journal of Heat Transfer, Vol. 89, 1967, pp. 244-250.

[7] M. Kelleher, “Free Convection from a Vertical Plate with Discontinuous Wall Temperature,” ASME Journal of Heat Transfer, Vol. 93, 1971, pp. 349-356. HHUUdoi:org/10.1115/1.3449830UU

[8] T. T. Kao, “Laminar Free Convective Heat Transfer Response along a Vertical Flat Plate with Step Jump in Surface Temperature,” Letters of Heat Mass Transfer, Vol. 2, No. 5, 1975, pp. 419-428. HHUUdoi:org/10.1016/0094-4548(75)90008-9UU

[9] S. Lee and M. M. Yovanovich, “Laminar Natural Convection from a Vertical Plate with a Step Change in Wall Temperature,” ASME Journal of Heat Transfer, Vol. 113, 1991, pp. 501-504. HHUUdoi:org/10.1115/1.2910591UU

[10] P. Chandran, N. C. Sacheti and A. K. Singh, “Natural con-Vection near a Vertical Plate with Ramped Wall Temperature,” Heat Mass Transfer, Vol. 41, No. 5, 2005, pp. 459-464. HHUUdoi:org/10.1007/s00231-004-0568-7UU

[11] S. C. Saha, C. Lei and J. C. Patterson, “On the Natural Convection Boundary Layer Adjacent to an Inclined Flat Plate Subject to Ramp Heating,” 16th Australian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Australia, 2007, pp. 121-124.

[12] R. C. Chaudhary and P. Jain, “Transient Free Convection Flow along an Infinite Vertical Oscillating Plate with a Step Change in Wall Temperature,” Acta Technica ?SAV, Vol. 52, 2007, pp. 187-204.

[13] B. Gebhart and L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects of Thermal and Mass Diffusion,” International Journal of Heat and Mass Transfer, Vol. 14, No. 12, 1971, pp. 2025-2050. HHUUdoi:org/10.1016/0017-9310(71)90026-3UU

[14] T. S. Chen, C. F. Yuh and A. Moustsoglou, “Combined Heat and Mass Transfer in Mixed Convection along Vertical and Inclined Plates,” International Journal of Heat and Mass Transfer, Vol. 23, No. 4, 1980, pp. 527-537. HHUUdoi:org/10.1016/0017-9310(80)90094-0UU

[15] O. Anwar Bég, Tasveer A. Bég, A. Y. Bakier and V. R. Prasad, “Chemically-Reacting Mixed Convective Heat and Mass Transfer along Inclined and Vertical Plates with Soret and Dufour Effects: Numerical Solutions,” International Journal of Applied Mathematics Mechanics, Vol. 5, No. 2, 2009, pp. 39-57.

[16] V. M. Soundalgekar, “Effects of Mass Transfer and Free Convection on the Flow Past an Impulsively Started Vertical Plate,” ASME Journal of Applied Mechanics, Vol. 46, No. 4, 1979, pp. 757-760. HHUUdoi:org/10.1115/1.3424649UU

[17] U. N. Dass, S. N. Ray and V. M. Soundalgekar, “Mass Transfer Effects on Flow Past an Impulsively Started Infinite Vertical Plate with Constant Mass Flux–an Exact Solution,” Heat and Mass transfer, Vol. 31, No. 3, 1996, pp. 163-167. HHUUdoi: 10.1007/bf02333314UU

[18] R. Muthucumaraswamy, P. Ganesan and V. M. Soundalgekar, “Heat and Mass Transfer Effects on Flow Past an Impulsively Started Vertical Plate,” Acta Mechanica, Vol. 146, No. 1-2, 2001, pp. 1-8. HHUUdoi:org/10.1007/BF01178790UU

[19] S. Ostrach, “An Analysis of Laminar Free Convection Flow and Heat and Mass Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force,” NACA 1111, Technical Report, 1952.

[20] H. S. Takhar and C. P. Perdikis, “Forced and Free Convective Flow of Water at 4?C through a Porous Medium,” International Communications in Heat and Mass Transfer, Vol. 21, 1986, pp. 605-609. HHUUdoi:org/10.1016/0735-1933(94)90005-1UUHHS. Takhar and P. C. Ram, “Magnetohydrodynamic Free Convection Flow of Water at 4?C through a Porous Medium,” International Communications in Heat and Mass Transfer, Vol. 21, No. 3, 1994, pp. 371-376.

[21] M. Ilicak and G. K. Vallis, “Simulations and Scaling of Horizontal Convection,” Tellus, under review, May 25th 2011.

[22] J. H. Siggers, R. R. Kerswell and N. J. Balmforth, “Bounds on Horizontal Convection,” Journal of Fluid Mechanics, Vol. 517, 2004, pp. 55-70. HHUUdoi:org/10.1017/S0022112004000497UU

[23] S. K. Ghosh, O. Anwar Bég and M. Narahari, “Hall Effects on MHD Flow in a Rotating System with Heat Transfer Characteristics,” Meccanica Journal, Vol. 44, No. 6, 2009, pp. 741-765. HHUUdoi:org/10.1007/s11012-009-9210-6UU

[24] S. K. Ghosh, O. Anwar Bég and J. Zueco, “Hydromagnetic Free Convection Rayleigh Flow with Induced Magnetic Field Effects,” Meccanica, Vol. 45, No. 2, 2010, pp. 175-185. HHUUdoi:org/10.1007/s11012-009-9235-xUU

[25] S. K. Ghosh, S. Rawat, O. Anwar Bég and Tasveer A. Bég, “Thermal Radiation Effects on Unsteady Hydromagnetic Gas Flow Along an Inclined Plane with Indirect Natural Convection,” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 13, 2010, pp. 41-57.

[26] S. K. Ghosh, O. Anwar Bég, J. Zueco and Prasad V. R., “Transient Hydromagnetic Flow in a Rotating Channel Permeated by an Inclined Magnetic Field with Magnetic Induction and Maxwell Displacement Current Effects,” ZAMP: Journal of Applied Mathematics and Physics, Vol. 61, No. 1, 2010, pp. 147-169. HHUUdoi:org/10.1007/s00033-009-0006-2UU

[27] O. Anwar Bég, J. Zueco, S. K. Ghosh and A. Heidari, “Unsteady Magnetohydrodynamic Heat Transfer in A Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study,” Advances in Numerical Analysis, Vol. 2011, 2011, pp. 1-17.