Darcian free convective flow about an impermeable horizontal surface bounded by a vertical wall

Abstract

The problem of steady free convection in a porous medium adjacent to a horizontal impermeable heated surface, with wall temperature T ̄ W = T ̄ 0 + A x ̄ λ (0 ⩽ λ < 2) for x ̄ ⩾ 0 and T ̄ = T 0 for − d ⩽ x ̄ < 0 and a vertical wall with temperature T ̄ W = T ̄ 0 for x ̄ = − d, y ̄ > 0 , is investigated by the method of matched asymptotic expansions. The first—and second—order boundary layer equations and the outer inviscid flow equations are studied to find the effects of large, but finite values of the Rayleigh number. The small parameter in the perturbation series is found to be the inverse one-third power of the Rayleigh number. It is found that λ = 1 2 is a critical value and therefore results are presented in detail for the two cases of λ = 0 and 1. It is found that the first-order boundary layer solutions overestimate the local Nusselt number except when λ > 1 2 and only then for short distances along the heated surface. Further, the horizontal velocity of the fluid on the heated surface is overestimated by the first-order boundary layer theory except for λ < 1 2 and for x̄ sufficiently small.