Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer

A new exact solution of the two-dimensional Oberbeck–Boussinesq equations has been found. The analytical expressions of the hydrodynamic fields, which have been obtained, describe convective Couette flow. Fluid flow occurs in the case of nonuniform distribution of velocities and the quadratic heat source at the upper boundary of an infinite layer of viscous incompressible fluid. Two characteristic scales have been introduced for finding the exact solutions of the Oberbeck–Boussinesq equations. Using the anisotropic layer allows one to explore large-scale flows of liquids for large values of the Grashof number. A connection is shown between solutions describing the quadratic heating of boundaries with boundary problems concerned with motions of fluids in which the temperature is distributed linearly. Analysis of polynomial solutions describing the natural convection of the fluid is presented. The existence of points at which the hydrodynamic fields vanish inside the fluid layer. Thus, the above class of exact solutions allows us to describe the counterflows in the fluid and the separations of pressure and temperature fields.