Exact solutions to thermal diffusion equations for Stokes slow flows

The article considers a class of exact solutions for describing Stokes slow flows of binary fluids. The family of exact solutions is constructed on the basis of the Lin–Sidrov–Aristov ansatz for the velocity field. The velocity field has a wide functional arbitrariness. It depends linearly on two coordinates (horizontal or longitudinal). The coefficients of the linear forms are functions of two variables from the third (vertical or transverse) coordinate and time. The pressure field, the temperature field, and the field of dissolved substance concentration are quadratic forms. In other words, the study takes into account not only horizontal gradients, but also the curvature of the hydrodynamic fields. The constructed exact solution describes thermal diffusion with both Soret and Dufour cross dissipative effects. A system of equations for describing unsteady flows is derived, which consists of heat conduction equations and gradient equations. Formulas of hydrodynamic fields are given to describe the Stokes slow steady-state flow of a binary fluid.