Existence and uniqueness theorems for a steady thermo-diffusive mixture in a mixed problem

Abstract

It is known ([l-3]) that, in a chemical homogeneous fluid, small changes of density, induced by temperature gradients, will induce convection currents. The most important case is certainly the BCnard problem of a fluid layer heated from below (see [l, 31 and the references therein). In the case of a binary fluid mixture there can be convective motions due to the density variations induced by temperature and concentration gradients (see [l-3]). The study of motions for fluid mixtures is of great importance for many applications such as oceanology, metereology, astrophysics, geophysics, etc. This problem has been studied within the scheme of the Oberbeck-Boussinesq equations (OB) by several authors [l-151$. Nevertheless, only few authors take into account the interactions between thermal diffusion and thermal-diffusive conduction, [2,4-6,11,14]. It is worth noting that, from the physical point of view, one should take into account the aforesaid interactions in every case in which the concentration is not “too small”. Up to this date, as far as we know, the wellposedness of this problem, in the case of mixed boundary conditions, has not yet been proved. The aim of the present paper is to prove existence and uniqueness theorems for steady solution of the OB equations with mixed boundary conditions, when the thermo-diffusive interactions are not neglected. The region of a motion 9 c R3 is assumed to be bounded with a smooth boundary which is in part rigid and in part “free but invariable” (i.e. a system of coordinates can be chosen in which it does not change). On the rigid boundary we give the usual adherence conditions for the velocity field and we assign the temperature and concentration fields. On the free boundary we give the usual slip conditions for the velocity field (which correspond to assign the normal component of the velocity and the tangential component of the stress vector) and we assign the heat and mass fluxes. This problem is a mixed boundary value problem and represents one of the first steps towards more general problems of the motion of fluid mixtures in a domain with free boundary. This kind of problem has been studied-for an homogeneous isothermal fluid-by several authors (see e.g. [3,16-201 and the references in [IS]).