Exact solution to the velocity field description for Couette–Poiseulle flows of binary liquids

Exact solution of the Oberbeck–Boussinesq equations for describing steady flows of binary Poiseuille-type fluids is proposed and studied. The fluid motion is considered in the infinite horizontal layer. Shear flows are described by overdetermined system of equations. Nontrivial exact solution for the Oberbeck–Boussinesq system exists in the class of velocities with two vector components and depends only on the transverse coordinate. This structure of the velocity vector coordinates ensures naturally the fulfillment of the continuity equation as an “extra” equation. The pressure field, the temperature field, and the concentration field of the dissolved substance are described by linear functions of horizontal (longitudinal) coordinates with coefficients that functionally depend on the third coordinate. Fluid layer, as it is shown, can have two points where the velocity becomes zero. In this case, the spiral flow is realized (the hodograph of the velocity vector has a turning point).