Local Thermal Non-equilibrium Analysis of the Instability in a Vertical Porous Slab with Permeable Sidewalls

Buoyant flow in a fluid-saturated porous vertical slab with isothermal and permeable boundaries is performed. Two reservoirs, maintained at different uniform temperatures, confine the slab. The permeable plane boundaries of the slab are modelled by imposing a condition of hydrostatic pressure. Darcy’s law and the Oberbeck–Boussinesq approximation are employed. The hypothesis of local thermal equilibrium between the fluid and the solid phase is relaxed. A two-temperature model is adopted, so that two local energy balance equations govern the heat transfer in the porous slab. The basic stationary buoyant flow consists of a single convective cell of infinite height. The time evolution of normal mode perturbations superposed onto the basic state is investigated in order to determine the onset conditions for thermal instability. A pressure–temperature formulation is employed. Major asymptotic cases are investigated. It is shown that departure from local thermal equilibrium implies in general a destabilisation of the basic stationary flow.