Onset of thermal convection in a porous layer with mixed boundary conditions

Stability of a horizontal porous layer saturated by a fluid under conditions of heating from below is revisited. The aim is to extend the model for the boundary walls constraining the velocity and temperature by means of mixed, or third-kind, conditions. This model proved to be appropriate to describe departure from perfectly conducting or uniform heat flux conditions for the temperature, and impermeable or perfectly permeable conditions for the velocity. Linearised equations for general normal modes perturbing the basic rest state are obtained. The principle of exchange of stabilities is proven. The perturbation equations are solved analytically to deduce the dispersion relation at neutral stability, as well as to draw neutral stability curves and to yield the critical values of the Darcy-Rayleigh number and of the wave number for the onset of the instability. Specially interesting regimes are analysed in detail including the degenerate case, where the velocity and temperature boundary conditions feature the same mixing parameters, and the symmetric case, where identical conditions are prescribed on both boundaries.