Local thermal non-equilibrium effects in the Darcy-Bénard instability with isoflux boundary conditions

The Darcy-Bénard problem with constant heat flux boundary conditions is studied in a regime where the fluid and solid phases are in local thermal non-equilibrium. The onset conditions for convective instability in the plane porous layer are investigated using a linear stability analysis. Constant heat flux boundary conditions are formulated according to the Amiri-Vafai-Kuzay Model A, where the boundary walls are assumed as impermeable and with a high thermal conductance. The normal mode analysis of the perturbations imposed on the basic state leads to a one-dimensional eigenvalue problem, solved numerically to determine the neutral stability condition. Analytical solutions are found for the limit of small wave numbers, and in the regime where the conductivity of the solid phase is much larger than the conductivity of the fluid phase. A comparison with the corresponding results under conditions of local thermal equilibrium is carried out. The critical conditions for the onset of convection correspond to a zero wave number only when the inter-phase heat transfer coefficient is sufficiently large. Otherwise, the critical conditions correspond to a nonzero wave number.