Matched asymptotic expansions for free convection about an impermeable horizontal surface in a porous medium

Abstract

The problem of steady free convection in a porous medium adjacent to a horizontal impermeable heated surface, with wall temperature distribution w= ∞+ A λ(0≤λ<2), for ≥0 and = ∞ for <0, is investigated by the method of matched asymptotic expansions. The small parameter in the perturbation series is found to be the inverse one-third power of the Rayleigh number. For the first-order inner problem the governing equations reduced to the boundary layer approximations which have been solved previously. The effects of fluid entrainment, streamwise heat conduction and upward-drift induced friction are taken into consideration in the second and third-order theory for which similarity solutions are obtained. Numerical results for temperature and streamwise velocity profiles as well as the local Nusselt number at different local Rayleigh numbers and different prescribed wall temperature distributions are presented. It was found that the local Nusselt numbers as obtained from the boundary layer theory for λ=0.5 are accurate to the third-order, while those for λ=0 are accurate to the second-order. For other values of λ, the boundary layer theory underestimates the local Nusselt number slightly; the accuracy of the boundary layer theory decreases as the Rayleigh numbers decrease and as λ increases from λ=0.5.