Local thermal non-equilibrium effects in the Darcy–Bénard instability of a porous layer heated from below by a uniform flux

The influence of the lack of thermal equilibrium between the solid phase and the fluid phase on the convective instability in a porous medium is studied. A horizontal layer with parallel and impermeable bounding walls is considered. The lower wall is assumed to be isoflux, and the upper wall isothermal. The basic motionless state is perturbed with small-amplitude disturbances, so that a linear analysis of the instability is carried out with a streamfunction-temperature formulation of the local balance equations. Then, the governing equations are solved for the normal modes, leading to an eigenvalue problem for the neutral stability. This eigenvalue problem is solved analytically, thus obtaining a dispersion relation for the neutral stability curves. The different parametric regimes are explored, including the asymptotic case of local thermal equilibrium and that of complete thermal uncoupling between the phases. The critical values of the Darcy–Rayleigh number and of the wave number are obtained. It is shown that, in the special case where the solid is much more thermally conductive than the fluid, the saturated porous medium is unstable for arbitrarily small values of the Darcy–Rayleigh number.