A Study of Thermal Convection in Two Porous Layers Governed by Brinkman's Model in Upper Layer and Darcy's Model in Lower Layer

Authors: M. S. Al-Qurashi

Abstract:

This work examines thermal convection in two porous layers. Flow in the upper layer is governed by Brinkman-s equations model and in the lower layer is governed by Darcy-s model. Legendre polynomials are used to obtain numerical solution when the lower layer is heated from below.

Keywords: Brinkman's law, Darcy's law, porous layers, Legendre polynomials, the Oberbeck-Boussineq approximation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1074783

References:

[1] Benard, H., Les tourbillons cellulaires dans une nappe liquide., Revue genral des sciences pures et appliquess, 11, 1216-1271 and 1309-1328 (1900).
[2] Benard, H., Les tourbillons cellulaires dans une nappe liquide transportant de la chalaur par convection en regime permanent., Ann. Chim. Phys., 23, 62-114 (1901).
[3] Chandrasekhar, S., The instability of a layer of fluid heated below and subject to Coriolis forces., Proc. Roy. Soc.(London) Vol. 217, 306-327 (1953).
[4] Chandrasekhar, S. and Elbert, D. D., The instability of a layer of fluid heated below and subject to Coriolis forces.II., Proc. Roy. Soc.(London). No. 231, 198-210 (1955).
[5] Nield, D. A., Onset of themohaline convection in a porous medium, water resources Res., 4, 553-560 (1968).
[6] Taunton, J. & Lightfoot, E., Thermohaline instability and salt fingers in a porous medium, Phys. Of fluid, 15, No. 5, 784-753 (1972).
[7] Sun, W. J., Convection Instability in Superposed Porous and Free Layers, Ph.D. Dissertation, University of Minnesota, Minneapolis. MN. 1973.
[8] Nield, D. A., Onset of Convection in a fluid layer overlying a porous me dium, J. Fluid Mech., 81, 513-522 (1977).
[9] Brinkman, H. C., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,. Appl. Sci. Res. Al. 27-34 (1947).
[10] Chen, F. & Chen, C.F., Onset of finger convection in a horizontal porous layer underlying a fluid layer., J. of Heat Tran., 110, 403-409 (1988).
[11] Lindsay K. A. & Ogden R. R., A Practical Implementation of Spectral Methods Resistant To The Generation of Spurious Eigenvalues Paper Nn. 92/23 University of Glasgow, Department of Mathematics preprint series (1992).
[12] Lamb, C. J., Hydromagnetic Instability In the Earth-s Core, Ph.D. thesis, Department of Mathematics, University of Glasgow 1994.
[13] Bukhari, A. f., A Spectral Eigenvalue Method for Multi-layered Continuua., Ph.D. thesis, University of Glasgow, Scot. 1996.
[14] Straughan, B., Surface-tension-driven convection in a fluid overlying a porous layer, J. Comput. Phys. 170, 320-337 (2001).
[15] Straughan, B., Effect of property variation and modelling on convection in a fluid overlying a porous layer, Int. J. Numer. Anal. Meth. Geomech. 26, 75-97 (2002).
[16] Allehiany, F. , Benard convection in a horizontal porous layer permeated by a conducting fluid in the presence of magnetic field and coriolis forces, An M. Sc. Thesis Umm Al- Qura University 2003.
[17] AL-Qurashi, M.S. & Bukhari, Abdul-Fattah K., Salt-finger convection in a horizontal porous layer superposes a fluid layer affected by rotation and vertical linear magnetic field on both layers., Advances and Applications in Fluid Mechanics, vol.11, 119-146,Apr.(2012).