Non-linear stability and convection for laminar flows in a porous medium with Brinkman law

The non-linear stability of plane parallel shear flows in an incompressible homogeneous fluid heated from below and saturating a porous medium is studied by the Lyapunov direct method. In the Oberbeck-Boussinesq-Brinkman (OBB) scheme, if the inertial terms are negligible, as it is widely assumed in literature, we find global non-linear exponential stability (GNES) independent of the Reynolds number R. However, if these terms are retained, we find a restriction on R (depending on the inertial convective coefficient) both for a homogeneous fluid and a mixture heated and salted from below. In the case of a mixture, when the normalized porosity epsilon is equal to one, the laminar flows are GNES for small R and for heat Rayleigh numbers less than the critical Rayleigh numbers obtained for the motionless state. Copyright (C) 2003 John Wiley Sons, Ltd.