Inclined convection in a porous Brinkman layer: linear instability and nonlinear stability

In this article, we deal with thermal convection in an inclined porous layer modelled by the Brinkman Law. Inertial effects are taken into account, and the physically significant rigid boundary conditions are imposed. This model is an extension of the work by Rees & Bassom (Rees & Bassom 2000 Acta Mech. 144, 103-118 (doi:10.1007/BF01181831)), where Darcy's Law is adopted, and only linear instability is investigated. It also completes the work of Falsaperla & Mulone (Falsaperla & Mulone 2018 Ric. Mat. 144, 1-17 (doi:10.1007/s11587-018-0371-2)), where the case of stress-free boundary conditions is studied and the inertial terms are absent. In this model, the basic laminar solution for the velocity is a combination of hyperbolic and polynomial functions, which makes the linear and nonlinear analysis muchmore complex. The original features of the paper are the following: we study three-dimensional perturbations, providing critical surfaces for the linear and nonlinear analyses; we study nonlinear stability with the Lyapunov method and, for the first time in the case of inclined layers, we compute the critical nonlinear Rayleigh regions by solving the associated variational maximum problem; we give some estimates of global nonlinear asymptotical stability; we study linear instability and nonlinear stability also with the presence of the inertial term, i.e. for a finite Va.