Thermal convection in a linearly viscous fluid overlying a bidisperse porous medium

A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macro pores but also cracks or fissures in the skeleton which give rise to micro pores. In this article we develop and analyse 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. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are found. These depend on the ratio ${\hat d}$ of the depth $d$ of the fluid layer to the depth $d_m$ of the porous layer. We show that there is a critical value of ${\hat d}$ such that below this value convective motion initiates in the porous layer whereas for ${\hat d}$ above this value the convective instability commences in the fluid layer.