Application of Lie symmetry analysis to the equations of convection in binary and multicomponent mixtures with Soret effect

Abstract:
The models of convection in binary and multicomponent fluids with Soret effect are considered. The Soret effect (or thermal diffusion) refers to the molecular transport of mass due to the temperature gradient in a mixture. The corresponding models include Navier-Stokes as well as heat and mass transfer equations. These equations are investigated by symmetry analysis method. A concise review of results obtained in binary fluid case is presented. It includes symmetry properties of the governing equations, equivalence transformations, classification of invariant submodels, examples of exact solutions, and their physical interpretation. The symmetries of equations for binary fluid are extended to the system that describes multicomponent heat and mass transfer. The generalization of some invariant solutions is performed and their physical interpretation is discussed. In particular, the exact solution describing the behaviour of multicomponent fluid in thermogravitational column is obtained (thermogravitational column is an experimental setup for measuring transport coefficients). Two important transformations of equations are found in multicomponent case. The first transformation provides a connection between equations with and without Soret effect. The second transformation excludes cross-diffusion terms in mass transfer equations by introducing new concentrations and thermal diffusion coefficients. It preserves boundary conditions in many problems, which are important in applications (e.g. Rayleigh-Benard configuration, thermogravitational column) and leaves invariant some key dimensionless parameters. The exclusion of cross-diffusion terms reduces the number of control parameters and essentially simplifies a particular problem. Application of this transformation to stability problems is discussed.