Penetrative convection in a bi-disperse porous medium

The present paper investigates penetrative convection in a bi-disperse porous medium. In particular, penetrative convection is modeled via a quadratic dependence on the temperature for the fluid density. For the problem under examination, convection can occur only through a secondary stationary motion since the validity of the principle of exchange of stabilities is proved. Via linear instability analysis of the conduction solution, the critical Rayleigh number for the onset of penetrative convection is determined. Moreover, the nonlinear stability is investigated via weighted energy method. In order to analyze the behavior of the stability and instability thresholds, numerical simulations are performed through Chebyshev- (Figure presented.) method, proving the stabilizing effect of the upper boundary layer temperature on the onset of convective motion.