Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery

In this paper, we study the steady creeping convective flow of a viscous incompressible fluid in the thin infinite layer. The study of the fluid flow is based on the exact solutions class for the Oberbeck–Boussinesq equations in the Stokes approximation using. The velocity field is described by the Hiemenz exact solution. The temperature field and the pressure field linearly depend on the horizontal (longitudinal) coordinate, it corresponds to the Ostroumov–Birich exact solutions class. The convective motion of a viscous incompressible fluid was induced by tangential stresses on the upper permeable (porous) boundary and thermal source definition at the lower boundary. In addition, the heat exchange according to the Newton–Richmann law takes into account at the upper boundary. The obtained exact solutions describe counterflows in fluids. The stagnant points number in the fluid layer does not exceed three. The formation of counterflows in the fluid is accompanied by sucking and injection of the fluid through the permeable boundary. The larger number of stagnant points presence forms a cellular structure of the streamlines. In addition, the velocity field, which obtained in the solution of the boundary value problem is characterized by localization of the flow near the boundary of the fluid layer (boundary layer). The exact solutions obtained in this paper can be used for the nonlinear Oberbeck–Boussinesq system solving. The Grashof number can take large values, which depends on the geometric anisotropy index for the linearized Oberbeck–Boussinesq system.