Analytic solutions of stationary complex convection describing a shear stress field of different signs

We study layered convection of a viscous incompressible fluid. The flow of an incompressible medium is described by the overdetermined system of the Oberbeck-Boussinesq equations. An exact solution of the overdetermined system of equations is found. The solution belongs to the Lin-Sidorov-Aristov class. In this class the velocities are homogeneous with respect to the horizontal variables. The pressure and temperature fields are linear functions of the coordinates $x$ and $y$. The use of the Lin-Sidorov-Aristov class preserves the nonlinearity of the motion equations only in the heat equation. The boundary value problem is studied for the Benard-Marangoni convection with heat transfer at the free boundary. The heat transfer is determined by the Newton-Richman law. The convective motion of a fluid is characterized by the existence of a layer thickness at which the friction force (the shear stress) vanishes at an interior point of the fluid layer. We give constraints on the control parameters that determine the no-slip conditions for the layers in the cases of thermal and solutal convective flows.