Thermal Convection in a Linearly Viscous Fluid Overlying a Bidisperse Porous Medium

Abstract

A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macropores but also cracks or fissures in the skeleton which give rise to micropores. In this article, we develop and analyze a model for thermal convection where a layer of viscous incompressible fluid overlies a layer of bidisperse porous medium. Care has to be taken with the boundary conditions at the interface of the fluid and the porous material, and this aspect is investigated. We propose two Beavers–Joseph conditions at the interface and we argue that the parameters in these relations should be different since they depend on the macro or micro permeability, and these parameters are estimated from the original experiments of Beavers and Joseph. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are found. These can depend on the relative permeability between the macro and micropores, the Beavers–Joseph conditions appropriate to the macro or micropores, the ratio $\hat{d}$ of the depth $d$ of the fluid layer to the depth $d_m$ of the porous layer, or generally the nature of the bidisperse medium.