Effects of Neumann and Robin boundaries on the thermal instability

The thermo convective instability of the Darcy-Benard problem (DB) using Robin (third-kind) thermal conditions is investigated here. We consider a viscous Newtonian fluid saturating a porous layer in which the layer is sandwiched between two impermeable boundaries. The upper and the lower walls are modelled in the form of the Neumann (second-kind) and the Robin (third-kind) thermal conditions, respectively. The difference in the temperature distribution between both phases allows the lack of a local thermal equilibrium model to be present. As a consequence, the third kind of thermal condition brings about one extra dimensionless parameter of the Biot number to the usual one of the inter-heat transfer coefficient and the thermal conductivity ratio. The normal modes method adopted in a linear stability analysis gives rise to perturbed governing equations. The eigenvalue problem is handled numerically as a result of the perturbed governing equations leading to the marginal stability condition.