Unstable buoyant flow in a vertical porous layer with convective boundary conditions

The instability of natural convection in a vertical porous layer is analysed. The plane parallel boundaries of the vertical layer are modelled as open and subject to Robin-type temperature conditions. The latter conditions describe heat transfer to the external environment, with a finite conductance. The basic state is given by a stationary fully-developed flow with linear velocity and temperature profiles. Instability arises when the Darcy-Rayleigh number exceeds its critical value. This value depends on the Biot number associated with the temperature boundary conditions. The most unstable normal modes turn out to be transverse. By solving numerically the stability eigenvalue problem, it is shown that the critical Darcy-Rayleigh number is a decreasing function of the Biot number when the Biot number is sufficiently small. For larger Biot numbers, a minimum is attained, and then the critical Darcy-Rayleigh number becomes an increasing function of the Biot number.