Exact solutions to the oberbeck–boussinesq equations for convective stokes flows

To study large-scale convective flows (fluid motion in a thin layer), it is possible, for initial studies, to consider the Stokes approximation when integrating the Oberbeck–Boussinesq equation. The convective derivative in the momentum transfer equations and in the heat conduction equation is in this case assumed to be identically equal to zero. The paper discusses several approaches to constructing exact solutions for slow (creeping) flows of a non-uniformly heated fluid. Formulas for three-dimensional flows in the Lin–Sidorov–Aristov class are given for steady flows. The hydrodynamic fields are described by polynomials. Exact solutions are given for the velocity field nonlinearly depending on two spatial coordinates (longitudinal, or horizontal) with coefficients of nonlinear forms, which depend on the third coordinate. The study shows how it is possible to automate the computation of unknown coefficients for the formation of hydrodynamic fields (velocities and temperatures).