A Darcy-Brinkman Model for Penetrative Convection in LTNE

The aim of this paper is to investigate the onset of penetrative convection in a Darcy-Brinkmann porous medium under the hypothesis of local therma non-equilibrium. For the problem at stake, the strong form of the principle of exchange of stabilities has been proved, i.e. convective motions can occur only through a secondary stationary motion. We perform linear and nonlinear stability analyses of the basic state motion, with particular regard to the behaviour of the stability thresholds with respect to the relevant physical parameters characterizing the model. The Chebyshev-$\tau$ method and the shooting method are employed and implemented to solve the differential eigenvalue problems arising from linear and nonlinear analyses to determine critical Rayleigh numbers. Numerical simulations prove the stabilising effect of upper bounding plane temperature, Darcy's number and the interaction coefficient characterising the local thermal non-equilibrium regime.