The onset of Darcy–Brinkman convection in a porous layer: an asymptotic analysis

In highly porous media boundary (Brinkman) effects are important near impermeable surfaces. We investigate in detail how these effects modify the well-known criterion for the onset of convection of a Boussinesq fluid in a porous medium where Darcyʹs law applies. It is known that boundary effects serve to raise the critical Darcy–Rayleigh number as D, the Darcy number, increases. For many porous media the value of D is small and this causes severe numerical difficulties in solving the perturbation equations. We extend an earlier numerical study by Walker and Homsy [A.S.M.E. J. Heat Transfer 99 (1977) 338] by performing an asymptotic analysis of the singular perturbation problem which arises in the small-D limit. Excellent agreement is obtained between the asymptotic and numerical results.