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 much more 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.
On the transition between slip and rigid boundary conditions between two solid media
