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 . 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 , 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 , 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 much more complex. The original features of the paper are the following: we study , providing for the linear and nonlinear analyses; we study 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 ; we give some estimates of asymptotical stability; we study linear instability and nonlinear stability also with the presence of the , i.e. for a finite Va.