Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s) M. Sathiyamoorthy a , Tanmay Basak b , S. Roy c , I. Pop d, * a Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi 835 215, India b Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India c Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India d Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania Received 18 September 2006 Available online 6 December 2006 Abstract In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bot- tom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy–Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-qua- dratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 10 3 6 Ra 6 10 6 , Darcy number Da, 10 5 6 Da 6 10 3 , and Prandtl number Pr, 0.2 6 Pr 6 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Penalty finite element method; Natural convection; Square cavity; Non-uniform heating 1. Introduction Natural convection in fluid saturated porous media has received considerable attention over the last several years and non-Darcy effects on natural convection in porous media have a great deal of attention in recent years. This is due to a large number of technical applications, such as, fluid flow in geothermal reservoirs, separation processes in chemical industries, dispersion of chemical contaminants through water saturated soil, solidification of casting, migration of moisture in grain storage system, crude oil production, etc. Comprehensive literature survey con- cerned with this subject is given by Kaviany [1], Nield and Bejan [2], Ingham and Pop [3,4], Vafai [5,6], Pop and Ingham [7], Bejan and Kraus [8], Ingham et al. [9] and Bejan et al. [10]. The buoyancy driven convection in a porous cavity heated differentially in the horizontal side has been analyzed by Walker and Homsy [11] by a number of different tech- niques. The results obtained are fairly good agreement with each other as well as experimental results. The Brnikman- extended Darcy model has been considered by Tong and Subramanian [12], and Lauriat and Prasad [13] to examine the buoyancy effects on free convection in a vertical cavity. This model has been introduced by Brinkman [14] in order to account for the transition from Darcy flow to highly vis- cous flow (without porous matrix), in the limit of extremely high permeability. However, Brinkman model does not account adequately for the transition from porous medium flow to pure fluid flow as the permeability of the porous medium increases. A model that bridges the entire gap between the Darcy and Navier Stokes equations is the Darcy–Forchheimer model which was developed by Vafai and Tien [15]. It is known that the Darcy’s law is an empir- ical formula relating the pressure gradient, the gravitational force and the bulk viscous resistance in porous media. Thus 0017-9310/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2006.10.010 * Corresponding author. E-mail address: pop.ioan@yahoo.co.uk (I. Pop). www.elsevier.com/locate/ijhmt International Journal of Heat and Mass Transfer 50 (2007) 1892–1901