Thermal Convection in a Darcy Porous Medium with Anisotropic Spatially Varying Permeability

We consider the problem of thermal convection in a saturated porous material of Darcy type. We allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For a particular numerical result we consider the case of permeability tensor (mathbf{K}=operatorname{diag} { {K_{amalg},K_{amalg},K_{perp}(z) } }) . Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one.