Flow of a Hydromagnetic Fluid through Porous Media between Permeable Beds under Exponentially Decaying Pressure Gradient
Prasad Balgangaghar 1, Kumar Amit 2
1Department of Mathematics, B N College Patna University
Patna-4, Bihar, India
e-mail: profbgprasad@gmail.com
2Department of Applied Mathematics
Birla Institute of Technology, Deoghar Campus
(Mesra) Jasidih-814142, Jharkhand, India
e-mail: amitkumar@bitmesra.ac.in
Received:
Received: 20 May 2010; revised: 31 March 2011; accepted: 1 April 2011; published online: 13 June 2011
DOI: 10.12921/cmst.2011.17.01.63-74
OAI: oai:lib.psnc.pl:740
Abstract:
An analytical solution of the flow of a hydromagnetic fluid through a porous medium between permeable beds is obtained and studied. The fluid is under an exponential decaying pressure gradient and the uniform magnetic field in a direction normal to the flow saturated porous medium is considered. Two governing equations, namely Navier-Stokes equations and Darcy’s law, are employed for the flow between and through the permeable beds, respectively. Injection and suction of the fluid through lower and upper permeable beds, respectively, with same velocity are allowed in the presence of porous medium and the uniform magnetic field. The velocity field and the volume flux are calculated analytically and presented graphically for different choices of the parameters exhibiting their phenomenal nature. Additionally if we replace the exponentially decaying pressure gradient by the pulsatile one and porousity of medium tends to zero, the results match excellently with those of Malathy and Srinivas [T. Malathy and S. Srinivas, Pulsatile flow of a hydromagnetic fluid between permeable beds, Int. Comm. In Heat and Mass Transfer 35, 681-688 (2008)].
Key words:
Darcy’s law, exponentially decaying pressure gradient, Hartman number, hydromagnetic fluid, injection and suction, permeable beds, porous medium
References:
[1] Y.C. Wang, Pulsatile flow in a porous channel. J. Appl. Mech. 38, 553-555 (1971).
[2] T. Malathy, S. Srinivas, Pulsatile flow of a hydromagnetic fluid between permeable beds. Int. Comm. In Heat and Transfer 35 681-688 (2008).
[3] K. Jagadeeswara Pillai, S. Vijaya Kumar, Varma, M. Syam Babu, Aligned magnetic effects through varying permeable bed. Proceedings Mathematical Sciences 96, 61-69 (2008).
[4] K. Vajravelu, P.V. Arunachalam, S. Sreenadh, Unsteady flow of two immiscible conducting fluids between two permeable beds, J. Math. Anal. Appl. 196 1105–1116 (1995).
[5] K. Vajravelu, K. Ramesh, S. Sreenadh, P.V. Arunachalam, Pulsatile flow between permeable beds. Int. J. Non-Linear Mech. 38, 999-1005 (2003).
[6] P. Chandra, J.S.V.R.K. Prasad, Pulsatile flow in circular tubes of varying cross-section with suction/injection, J. Aust. Math. Soc. Ser. B 35, 366-381 (1994).
[7] G.S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197-206 (1967).
[8] O.D. Makinde, P.Y. Mhone, Heat transfer to MHD oscillatory flow in a channel filled with porous medium. Rom. J. Phys. 50, 931-938 (2005).
[9] A. Raptis, C. Massalas, G. Tzivanidis, Hydromagnetic free convection flow through a porous medium between two parallel plates. Phys. Lett. 90A, 288-289 (1982).
[10] H.N. Hemida, M.N. Sabry, A. Abdel-Rahim, H. Mansour, Theoretical analysis of heat transfer in laminar pulsating flow. Int. J. Heat Mass Transfer 45, 1767-1780 (2002).
[11] A.S. Berman, Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232-1235 (1953).
[12] S. Srinivas, T. Malathy, P.L. Sachdev, On pulsatile hydromagnetic flow of an Oldroyd fluid with heat transfer. Eng. Trans. 55 (1), 79-94 (2007).
[13] H.A. Attia, Unsteady hydromagnetic Couette flow of a dusty fluid with temperature dependent viscosity and thermal conductivity under exponential decaying pressure gradient.
Comm. in Nonlinear Sc. And Num. Simulation 13, 1077- 1088 (2008).
[14] A.R. Rao, K.S. Deshikachar, MHD oscillatory flow of blood through channels of variable cross section. Int. J. Eng. Sci. 24, 1615-1628 (1986).
[15] G. Radhakrishnamacharya, M.K. Maiti, Heat transfer to pulsatile flow in a porous channel. Int. J. Heat Mass Transfer 20, 171-173 (1977).
[16] K.R. Rajagopal, L. Tao, Mechanics of mixtures, Series on Advances in Mathematics for Applied Sciences. Vol. 35, World Scientiffic Publishiing Co., Inc, River Edge, NJ, 1995.
An analytical solution of the flow of a hydromagnetic fluid through a porous medium between permeable beds is obtained and studied. The fluid is under an exponential decaying pressure gradient and the uniform magnetic field in a direction normal to the flow saturated porous medium is considered. Two governing equations, namely Navier-Stokes equations and Darcy’s law, are employed for the flow between and through the permeable beds, respectively. Injection and suction of the fluid through lower and upper permeable beds, respectively, with same velocity are allowed in the presence of porous medium and the uniform magnetic field. The velocity field and the volume flux are calculated analytically and presented graphically for different choices of the parameters exhibiting their phenomenal nature. Additionally if we replace the exponentially decaying pressure gradient by the pulsatile one and porousity of medium tends to zero, the results match excellently with those of Malathy and Srinivas [T. Malathy and S. Srinivas, Pulsatile flow of a hydromagnetic fluid between permeable beds, Int. Comm. In Heat and Mass Transfer 35, 681-688 (2008)].
Key words:
Darcy’s law, exponentially decaying pressure gradient, Hartman number, hydromagnetic fluid, injection and suction, permeable beds, porous medium
References:
[1] Y.C. Wang, Pulsatile flow in a porous channel. J. Appl. Mech. 38, 553-555 (1971).
[2] T. Malathy, S. Srinivas, Pulsatile flow of a hydromagnetic fluid between permeable beds. Int. Comm. In Heat and Transfer 35 681-688 (2008).
[3] K. Jagadeeswara Pillai, S. Vijaya Kumar, Varma, M. Syam Babu, Aligned magnetic effects through varying permeable bed. Proceedings Mathematical Sciences 96, 61-69 (2008).
[4] K. Vajravelu, P.V. Arunachalam, S. Sreenadh, Unsteady flow of two immiscible conducting fluids between two permeable beds, J. Math. Anal. Appl. 196 1105–1116 (1995).
[5] K. Vajravelu, K. Ramesh, S. Sreenadh, P.V. Arunachalam, Pulsatile flow between permeable beds. Int. J. Non-Linear Mech. 38, 999-1005 (2003).
[6] P. Chandra, J.S.V.R.K. Prasad, Pulsatile flow in circular tubes of varying cross-section with suction/injection, J. Aust. Math. Soc. Ser. B 35, 366-381 (1994).
[7] G.S. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197-206 (1967).
[8] O.D. Makinde, P.Y. Mhone, Heat transfer to MHD oscillatory flow in a channel filled with porous medium. Rom. J. Phys. 50, 931-938 (2005).
[9] A. Raptis, C. Massalas, G. Tzivanidis, Hydromagnetic free convection flow through a porous medium between two parallel plates. Phys. Lett. 90A, 288-289 (1982).
[10] H.N. Hemida, M.N. Sabry, A. Abdel-Rahim, H. Mansour, Theoretical analysis of heat transfer in laminar pulsating flow. Int. J. Heat Mass Transfer 45, 1767-1780 (2002).
[11] A.S. Berman, Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232-1235 (1953).
[12] S. Srinivas, T. Malathy, P.L. Sachdev, On pulsatile hydromagnetic flow of an Oldroyd fluid with heat transfer. Eng. Trans. 55 (1), 79-94 (2007).
[13] H.A. Attia, Unsteady hydromagnetic Couette flow of a dusty fluid with temperature dependent viscosity and thermal conductivity under exponential decaying pressure gradient.
Comm. in Nonlinear Sc. And Num. Simulation 13, 1077- 1088 (2008).
[14] A.R. Rao, K.S. Deshikachar, MHD oscillatory flow of blood through channels of variable cross section. Int. J. Eng. Sci. 24, 1615-1628 (1986).
[15] G. Radhakrishnamacharya, M.K. Maiti, Heat transfer to pulsatile flow in a porous channel. Int. J. Heat Mass Transfer 20, 171-173 (1977).
[16] K.R. Rajagopal, L. Tao, Mechanics of mixtures, Series on Advances in Mathematics for Applied Sciences. Vol. 35, World Scientiffic Publishiing Co., Inc, River Edge, NJ, 1995.
