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Abstract-Existence conditions are investigated for the unidirectional flows of binary mixtures describable by the equations of motion in the Oberbeck-Boussinesq approximation with account for the thermodiffusion effect. Possible solutions are classified, known and novel classes of exact solutions being distinguished. For the solutions obtained different formulations of boundary-value problems are proposed. Flows between two rigid walls with given heat distribution laws are described.Keywords: Oberbeck-Boussinesq model, thermodiffusion, unidirectional flow, binary mixture. DOI: 10.1134/S0015462816020022The study is devoted to investigating the thermal diffusion convection equations in the Oberbeck-Boussinesq approximation for unidirectional binary mixture flow. The latter can be realized in a channel with walls of high heat conductivity and ends different in temperature, sufficiently long in the horizontal direction. For a homogeneous fluid the mathematical model of such flow was proposed in book [1], the solution being obtained and interpreted for constant temperatures gradients at the walls in [2]. Later, this solution was obtained once more [3,4] and generalized to include flows in geometrically different domains [5,6]. Its unsteady analogs were also considered [7,8].For the nonlinear dependence of density on temperature and concentration the problem of solution existence was investigated within the framework of a similar model in [9].In the present study, the conditions of existence of the flows described are investigated in detail for a binary mixture with account for thermodiffusion. In contrast to the solutions of equations for purely thermal convection, where the linear temperature distribution at the wall requires the linearity of temperature distribution inside the layer, within the framework of the model considered the dependence of temperature and concentration on the longitudinal coordinate is generally quadratic, which in its turn suggests the quadratic law for the temperature distribution along the horizontal wall. For the system in question all possible exact solutions are exhausted by three cases, depending on the integration of the equations that link the density function and the parameters of state. For the solutions constructed, we propose some formulations of boundary-value problems for binary mixture flow between two rigid walls at which the temperature distribution is specified or there is no heat flux, as well as on the assumption that the layer is bounded from above by a free boundary. Examples are presented for two mixtures, which demonstrate the fluid behavior under the temperature gradient that acts by a given law along the horizontal walls.It should be noted that in [10] for the system investigated the group classification problem was solved with respect to the thermal and concentration expansion coefficients, thermal diffusivity, and the diffusion and thermodiffusion parameters. The solution of that problem resulted in the list of admissible differential operators whose action leads ...