A New Class of Exact Solutions of the Oberbeck–Boussinesq Equations Describing an Incompressible Fluid

A new class of exact solutions of the Oberbeck–Boussinesq equations for incompressible media is constructed taking into account body forces, heat sources (sinks), and Joule dissipation. The expressions for the velocities are quadratic forms with respect to two coordinates, generalizing the class of Lin–Sidorov–Aristov solutions. Temperature, pressure, and the field of body forces are described by forms of the fourth degree. The possibility of using this class for convective flows in the Stokes and Oseen approximation is considered, and the possibilities of a new class for describing rotating liquid masses are demonstrated. A simple example illustrates the complex structure of the velocity field for a creeping convective Couette-type fluid flow in a layer with a permeable boundary moving inhomogeneously.