The energy method analysis of the Darcy–Bénard problem with viscous dissipation

A nonlinear analysis of the effect of viscous dissipation on the Rayleigh–Bénard instability in a fluid saturated porous layer is performed. The saturated medium is modelled through Darcy’s law, with the layer bounded by two parallel impermeable walls kept at different uniform temperatures, so that heating from below is supplied. While it is well known that viscous dissipation does not influence the linear threshold to instability, a rigorous nonlinear analysis of the instability when viscous dissipation is taken into account is still lacking. This paper aims to fill this gap. The energy method is employed to prove the nonlinear conditional stability of the basic conduction state. In other words, it is shown that a finite initial perturbation exponentially decays in time provided that its initial amplitude is smaller than a given finite value.