Stability of two-dimensional convection in a fluid-saturated porous medium

Abstract

The range of existence and the stability of spatially periodic solutions has been studied for steady and oscillatory two-dimensional convection in a fluid-saturated porous medium. We have analysed the limit where viscous effects are dominant and Darcy's law can be applied. A Galerkin method has been used to obtain the steady and the centrosymmetric oscillatory solutions that appear in nonlinear convection at Rayleigh numbers up to 20 times the critical value. Their stability boundaries to arbitrary infinitesimal perturbations have been obtained. Above a given Rayleigh number stable oscillatory solutions are possible at wavenumbers close to the critical values. The stability of this oscillatory state with respect to infinitesimal perturbations of any wavenumber has also been studied. The resulting temporal dynamics in the different unstable regimes is briefly discussed. We show the existence of 3:1 spatial resonances of the steady roll solutions and the existence of stable centrosymmetric and non-centrosymmetric oscillatory solutions.