Convective instability induced by internal and external heating in a fluid saturated porous medium

A linear stability analysis is employed to study the appearance of thermal instabilities in a fluid saturated porous medium. In particular, focus is placed on the onset of the convective instability. The bounding walls are impermeable and subject to different temperature boundary conditions. An external heat flux is prescribed at the lower wall whereas the upper wall is subject to a third kind boundary condition, parametrized by a Biot number. A uniform internal heat source is imposed. A modal analysis is performed and the resulting eigenvalue problem is solved numerically using a shooting method. The principle of exchange of stabilities is used as a guide to construct approximate analytical correlations for the marginal and critical stability curves, whose errors are always below O(10%). They show that convection at the upper wall always has a stabilizing effect whereas internal heating always has a destabilizing effect. Finally, it is shown that the solutions for perfectly and imperfectly insulated upper walls differ significantly from one another, indicating the existence of a singular behavior in the zero Biot limit.