EXACT SOLUTIONS TO THE OBERBECK–BOUSSINESQ EQUATIONS FOR DESCRIBING MULTILAYER FLUID FLOWS IN THE STOKES APPROXIMATION

The flow of viscous incompressible fluids in engineering devices, in technological and natural processes is characterized by the stratification of hydrodynamic fields. Conventionally, the stratification of the velocity field and the pressure field is studied for isothermal flows. If fluid motion occurs in a thermal field, temperature is an important characteristic of an incompressible fluid. Convective fluid flow has a very complex topological structure of hydrodynamic fields due to the temperature dependence of density. As is known, in the description of convection in the Boussinesq approximation, the dependence of density on the spatial coordinate and time is ignored in the continuity equation, which is then transformed into the incompressibility equation. Field and experimental observations of fluid flow allow us to identify flow regions with discrete density distribution along one of the coordinates. Such fluids are referred to as stratified fluids in the scientific literature. Their mathematical description is significantly complicated since it is necessary to solve the Oberbeck–Boussinesq equations for each layer and join the analytical or numerical solutions between the layers and the boundaries. For applied studies of convective flows, the Stokes approximation for the total derivative of a vector or scalar function is often introduced. The paper considers the construction of exact Lin–Sidorov–Aristov solutions for describing slow (creeping) flows of a non-uniformly heated stratified fluid. In this case, hydrodynamic fields are described by special polynomials with functional arbitrariness. It is shown how the calculations of unknown coefficients can be automated to form hydrodynamic fields (velocities and temperatures). For steady-state Stokes-type flows, an exact solution of the Oberbeck–Boussinesq system is written out explicitly (by means of formulas).