Rayleigh-Benard Instability in a Horizontal Porous Layer Affected by Rotation

Abdullah Ahmad Abdullah^*, Abeer Habeebullah Bakhsh

Department of Mathematical Sciences, Umm Al-Qura University, Makkah, Saudi Arabia.

DOI: 10.4236/am.2015.614202                

Abstract

This study examines the Benard convection of an infinite horizontal porous layer permeated by an incompressible thermally conducting viscous fluid in the presence of Coriolis forces. The porous layer is controlled by the Brinkman model. Analytical and numerical solutions are obtained for the cases of stationary convection and overstability. The critical thermal Rayleigh numbers are obtained for different values of the permeability of porous medium, Chandrasekhar number and Taylor number for different boundary conditions. The related eigenvalue problem is solved using the Chebyshev polynomial Tau method.

Keywords

Benard Problem, Porous Medium, Rotation, Stationary Convection, Overstability

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Benard, H. (1900) Les tourbillions cellaires dans une nappe liquide. Rev. Gén. Sci. Pures Appl., 11, 1216-1271, 1309-1328.
[2]	Benard, H. (1901) Les Tourbillons Cellulaires dans une Nappe Liquide Transportant de la Chaleur par Convection en Régime Permanent. Annales de Chimie et de Physique, 23, 62-144.
[3]	Rayleigh, R. (1916) On Convection Currents in a Horizontal Layer of Fluid When the Higher Temperature Is on the Underside. Philosophical Magazine, 32, 529-546. http://dx.doi.org/10.1080/14786441608635602
[4]	Chandrasekhar, S. (1953) The Instability of a Layer of Fluid Heated from Below and Subject to Corilois Forces. Proceedings of the Royal Society of London A, 217, 306-326. http://dx.doi.org/10.1098/rspa.1953.0065
[5]	Chandrasekhar, S. and Elbert, D. (1955) The Instability of Layer of Fluid Heated Below and Subject to Coriolis Forces. II. Proceedings of the Royal Society of London A, 231, 198-210. http://dx.doi.org/10.1098/rspa.1955.0166
[6]	Horton, C. and Rogers, F. (1945) Convection Currents in a Porous Medium. Journal of Applied Physics, 16, 367-370. http://dx.doi.org/10.1063/1.1707601
[7]	Lapwood, E. (1948) Convection of a Fluid in a Porous Medium. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 508-521. http://dx.doi.org/10.1017/S030500410002452X
[8]	Wooding, R. (1960) Rayleigh Instability of Thermal Boundary Layer in Flow through a Porous Medium. Journal of Fluid Mechanics, 9, 183-192. http://dx.doi.org/10.1017/S0022112060001031
[9]	Elder, J. (1967) Steady Free Convection in a Porous Medium Heated from Below. Journal of Fluid Mechanics, 27, 29-48. http://dx.doi.org/10.1017/S0022112067000023
[10]	Brinkman, H. (1947) A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Denseswarm of Particles. Applied Scientific Research, Al, 27-34.
[11]	Brinkman, H. (1949) On the Permeability of Media Consisting of Closely Packed Porous Particles. Applied Scientific Research, l, 81-86.
[12]	Yamamoto, K. and Iwamura, N. (1976) Flow with Convection Acceleration through a Porous Medium. Journal of Engineering Mathematics, 10, 41-54. http://dx.doi.org/10.1007/BF01535425
[13]	Rudraiah, N., Veerappa, B. and Rao, S. (1980) Effects of Nonuniform Thermal Gradient and Adiabatic Boundaries on Convection in Porous Media. Journal of Heat Transfer, 102, 254-260. http://dx.doi.org/10.1115/1.3244269
[14]	Georgiadis, J. and Catton, I. (1986) Prandtl Number Effect on Bénard Convection in Porous Media. Journal of Heat Transfer, 108, 284-290. http://dx.doi.org/10.1115/1.3246917
[15]	Kladias, N. and Prasad, V. (1990) Flow Transition in Buoyancy Induced Non-Darcy Convection in Porous Medium-Heated from Below. Journal of Heat Transfer, 112, 675-684. http://dx.doi.org/10.1115/1.2910440
[16]	Pradeep, S. and Sri Krishna, C. (2001) Rayleigh-Benard Convection in a Viscoelastic Fluid Filled High-Porosity Medium with Non Uniform Basic Temperature Gradient. IJMMS, 25, 609-619.
[17]	Hill, A. (2004) Convection Induced by the Selective Absorption of Radiation for the Brinkman Model. Continuum Mechanics and Thermodynamics, 16, 43-52. http://dx.doi.org/10.1007/s00161-003-0140-6
[18]	Ramambason, D. and Vasseur, P. (2007) Influence of a Magnetic Field on Natural Convection in a Shallow Porous Enclosure Saturated with a Binary Fluid. Acta Mechanica, 191, 21-35. http://dx.doi.org/10.1007/s00707-007-0444-x
[19]	Gaikwad, S., Malashetty, M. and Prasad, K. (2009) An Analytical Study of Linear and Nonlinear double Diffusive Convection in a Fluid Saturated Anisotropic Porous Layer with Soret Effect. Applied Mathematical Modelling, 33, 3617-3635. http://dx.doi.org/10.1016/j.apm.2008.12.013
[20]	Hoshoudy, G. (2011) Rayleigh-Taylor Instability with General Rotation and Surface Tension in Porous Media. Arabian Journal for Science and Engineering, 36, 621-633. http://dx.doi.org/10.1007/s13369-011-0051-y
[21]	Nield, D. and Bejan, A. (2013) Convection in Porous Media. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4614-5541-7
[22]	Abdullah, A. and Lindsay, K. (1990) Benard Convection in a Non-Linear Magnetic Fluid. Acta Mechanica, 85, 27-42. http://dx.doi.org/10.1007/BF01213540
[23]	Abdullah, A. and Lindsay, K. (1991) Some Remarks on the Computation of the Eigenvalues of Linear Systems. Mathematical Models and Methods in Applied Sciences, 1, 153-165. http://dx.doi.org/10.1142/S0218202591000095
[24]	Hassanien, I., Abdullah, A. and Gorla, R. (1998) Numerical Solutions for Heat Transfer in Amicropolar Fluidover a Stretching Sheet. Journal of Applied Mechanical Engineering, 3, 377-391.
[25]	Straughan, B. (2002) Effect of Property Variation and Modelling on Convection in a Fluid Overlying a Porous Layer. International Journal for Numerical and Analytical Methods in Geomechanics, 26, 75-97. http://dx.doi.org/10.1002/nag.193
[26]	Banjer, H. and Abdullah, A. (2012) Thermal Instability in Superposed Porous and Fluid Layers in the Presence of a Magnetic Field Using Brinkman Model. Journal of Porous Media, 15, 1-10.