Penetrative convection and multi-component diffusion in a porous medium

Abstract

Linear instability and nonlinear energy stability analyses are developed for the problem of a fluid-saturated porous layer stratified by penetrative thermal convection and two salt concentrations. Unusual neutral curves are obtained, in particular non-perfect `heart-shaped' oscillatory curves that are disconnected from the stationary neutral curve. These curves show that three critical values of the thermal Rayleigh number may be required to fully describe the linear stability criteria. As the penetrative effect is increased, the oscillatory curves depart more and more from a perfect heart shape. For certain values of the parameters it is shown that the minima on the oscillatory and stationary curves occur at the same Rayleigh number but different wavenumbers, offering the prospect of different types of instability occurring simultaneously at different wavenumbers. A weighted energy method is used to investigate the nonlinear stability of the problem and yields unconditional results guaranteeing nonlinear stability for initial perturbations of arbitrary sized amplitude.