The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium

Abstract

The Brinkman model is used for the theoretical study of boundary effects in a natural convection porous layer adjacent to a semi-infinite vertical plate with a power law variation of wall temperature, i.e. T ̄ wα x ̄ λ . It is shown that the dimensionless governing equations based on this model contain two parameters ϵ, = (Ra) − 1 2 and σ = (Da/φ) 1 2 where Ra and Da are the Rayleigh number and the Darcy number based on a reference length and φ is the porosity. For the limit of ϵ → 0, σ → 0 and σ ⪡ ϵ, a perturbation solution for the problem is obtained based on the method of matched asymptotic expansions. The physical problem can then be visualized to be consisting of three layers : an inner viscous sublayer, adjacent to the heated surface, with a thickness of the order of 0(σ) ; a middle thermal layer of thickness of the order of 0(ϵ) ; and the outer potential flow region with thickness of the order of 0(1). It is found that the first-order problem of the thermal layer is identical to that based on Darcy's law with slip flow, whose solution was obtained previously. Composite solutions for stream function and temperature, uniformly valid everywhere in the flow field, are constructed from the solutions for the thermal layer and the viscous sublayer. A new parameter Pn x , defined as Pn x = (Ra xDa x) 1 2 with Ra x and Da x denoting the local Rayleigh number and the local Darcy number based on x̄, is found to be a measure of the boundary effect. It is shown that the viscous effect on the boundary has a drastic effect on the streamwise velocity component near the wall with a lesser effect on heat transfer characteristics. The boundary effect slows down the buoyancy-induced flow with a resulting decrease in heat transfer. The local Nusselt number is found to be of the form Nu x(Ra x) 1 2 = C 1− C 2Pn x where the values of C 1 and C 2 depending on λ. For an isothermal vertical plate (λ = 0), the first-order correction to the local Nusselt number is identically zero, i.e. C 2 = 0. In general, the boundary effect on the local Nusselt number becomes more pronounced as the value of Pn x , Ra x or Da x is increased.