Molecular diffusion and Boussinesq effects in heterogeneous fluid convection

Abstract

The modelling of molecular diffusion in heterogeneous fluids, satisfying the Onsager hypothesis, always read as diffusion equations involving cross effects. In the present paper, we show how the double diffusive effects allow us to write uniformly valid asymptotic Boussinesq equations when the medium is deep or, even, unbounded. This formulation introduces a small parameter ε, namely, the ratio of the vertical scale of the motion and a vertical scale related to the static state of the medium. The diffusion equations reduce, in inert mixtures, to classical Fick's laws if we write the equations at the first order in ε. At the second order, cross effects appear. In non-inert mixtures, cross effects appear at the first order. By way of applications, we present the cases of diffusion equations in moist unsaturated air (inert ideal mixture), in salt sea-water (inert non-ideal mixture) and in moist saturated air (non-inert mixture).