Long-wave instability of a multicomponent fluid layer with the Soret effect

Abstract

The long-wave instability of a vertical multicomponent fluid layer is investigated. The basic state is a plane parallel flow with linear profiles of temperature and composition in the cross section. The compositional stratification is induced by the Soret effect. The problem is reduced to that without cross-diffusion effect by a linear change of composition and separation ratios that characterize the Soret separation. This is the first report where such transformation is derived for the equations of multicomponent convection with thermal diffusion. It simplifies the system reducing the number of control parameters and can be applied to a variety of multidiffusive convection problems (e.g., Rayleigh–Bénard configuration). It is shown that the long-wave instability is caused by the interplay between the basic flow and concentration waves with long scale in the vertical direction. In the general case of n-component mixture, several stable regions in the parameter space are identified. A complete parametric study for ternary fluid revealed complex instability scenarios in the long-wave limit: monotonic instability in a certain range of Grashof numbers, the transition from monotonic to oscillatory instability with increasing Grashof, etc. The physical mechanisms underlying this complex behavior are discussed and explained. In addition to the long-wave stability analysis, we performed numerical calculations for finite wave numbers. These calculations showed that the monotonic long-wave perturbations represent the most unstable modes.