Mathematics & Physics 2016, 9(1), 30–36 УДК 532.5 On One Two-dimensional Stationary Flow of a Binary Mixture and Viscous Fluid in a Plane Layer Marina V.Efimova∗ Institute of Computational modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Russia Received 12.10.2015, received in revised form 10.11.2015, accepted 20.12.2015 Nonlinear model of convection in Oberbeck-Boussinesq approximation describing the flat joint motion of a binary mixture and viscous fluid with a common interface is investigated. <...> It is important that the longitudinal temperature gradient and the concentration is quadratic dependence on the coordinate x. <...> Stationary solution of the system is built. <...> Convection is one of the most common of hydrodynamic phenomena in nature. <...> Study of convection is an important part of the theoretical fluid mechanics. <...> Convective processes influence the thermal conditions in the oil storage tanks, chemical process technology and others. <...> Problems for thermal convection are very complex because of the diversity of cavities and thermal boundary conditions for the nonlinear system of partial differential equations. <...> Solutions of the Oberbeck-Boussinesq equations with a linear dependence of temperature on one of the space coordinates firstly were studied by G. A. Ostroumov [1]. <...> The exact solution described plane stationary flow in a strip under action of longitudinal temperature gradient and transversal gravity field, was obtained by R. V. Birikh [2]. <...> The existence of solutions with nonlinear dependence of density on temperature and concentration is proved in [4] where two boundary value problems with exponential temperature distribution on the walls are solved. <...> In [5–7] the exact solutions of the three-dimensional convection problem for two immiscible viscous, incompressible fluids in a channel with a rectangular cross section in the presence of the interface and under the influence of a longitudinal temperature gradient are studied. <...> In this paper we are built the exact solution of the two-layer convection system with longitudinal temperature gradient at the solid walls and shear force of gravity. 1. <...> Problem formulation Let us consider the joint motion of a binary mixture and viscous fluid with a general interface. <...> Suppose that Ω1 = { |x| < ∞, 0 < y < l1} is the region occupied by a binary mixture and ∗efmavi@icm.krasn.ru ⃝ Siberian Federal University. <...> All rights reserved c – 30 – Marina V.Efimova On One Two-dimensional Stationary Flow <...>