Nonlinear Stability for Thermal Convection in a Brinkman Porous Material with Viscous Dissipation

We investigate nonlinear stability in a model for thermal convection in a saturated porous material using Brinkman theory, taking into account viscous dissipation effects. There are (at least) two models for viscous dissipation available, and we include a derivation of one of these by assuming that the flow in the porous medium may be described by a theory for a mixture of an elastic solid and a linearly viscous fluid. A fully nonlinear stability result is provided when either of the viscous dissipation functions is taken into account, and it is shown that from the nonlinear energy stability viewpoint both models are, in a sense, equivalent.