Analysis of natural convection about a vertical plate embedded in a porous medium

Abstract

Buoyancy-driven fluid flow and heat transfer about a vertical flat plate embedded in a porous medium are analyzed in depth for two general cases. These are the constant wall temperature case and the constant wall heat flux case. By an order of magnitude analysis it is shown that there are two governing dimensionless parameters which are related to the thermal and viscous effects. These are Ra L − 1 2 and Da L 1 2 for the constant wall temperature case and Ra L ∗− 1 3 and Da L 1 2 for the constant wall heat flux case. For each case, two important categories are considered. For the constant wall temperature case these categories are shown to be physically related to either Da L 1 2 ⪡ Ra L − 1 2 ( Da L 1 2 ⪡ Ra L ∗− 1 3 for the constant wall heat flux) or Da L 1 2 ⪢ Ra L − 1 2 ( Da L 1 2 ⪢ Ra L ∗− 1 3 for the constant wall heat flux). The governing equations are solved analytically using the method of matched asymptotic expansions along with the modified Oseen method. The numerical solution of the governing equations based on the similarity transformations is also obtained. It is found that the rate of heat transfer depends on the modified Rayleigh number for Da L 1 2 ⪡ Ra L − 1 2 (or Da L 1 2 ⪡ Ra L ∗− 1 3 ) while the Nusselt number depends on the product of the Rayleigh number and the porosity for Da L 1 2 ⪡ Ra L − 1 2 (or Da L 1 2 ⪡ Ra L ∗− 1 3 ). The heat transfer characteristics for the latter are shown to be independent of the permeability of the porous medium and approaching those of the regular fluid with a high Prandtl number. A complete physical description of the problem is presented throughout the analysis.