Natalya Burmasheva

In this paper, we report on several classes of exact solutions for describing the convective flows of multilayer fluids. We show that the class of exact Lin – Sidorov – Aristov solutions is an exact solution to the Oberbeck – Boussinesq system for a fluid discretely stratified in density and viscosity. This class of exact solutions is characterized by the linear dependence of the velocity field on part of coordinates. In this case, the pressure field and the temperature field are quadratic forms. The application of the velocity field with nonlinear dependence on two coordinates has stimulated further development of the Lin – Sidorov – Aristov class. The values of the degrees of the forms of hydrodynamical fields satisfying the Oberbeck – Boussinesq equation are determined. Special attention is given to convective shear flows since the reduced Oberbeck – Boussinesq system will be overdetermined. Conditions for solvability within the framework of these classes are formulated.

In this paper we consider the steady inhomogeneous shear flow of a viscous incompressible fluid taking into account the possibility of solid-body rotation of a representative volume. Mathematically, the contribution of couple stresses manifests itself in an increase in the order of the system of governing differential equations. We discuss problems of the existence of an exact solution within the framework of the class of functions linear in some of the coordinates. It is shown that the problem of overdetermination of the system of equations, which is traditional for models describing shear flows, does not arise for the chosen class of solutions. An exact solution is constructed for the velocity field of the flow. Also, an exact solution of the boundary-value problem describing adhesion and superadhesion on the boundaries of the flow region is analyzed in dimensionless form. It is shown that these exact solutions are capable of describing stagnation regions observed in real fluids and the effect of increase in velocities.