A New Class of Exact Solutions to the Navier–Stokes Equations with Allowance for Internal Heat Release

Abstract

New exact solutions to the three-dimensional Navier–Stokes equations, which take into account energy dissipation in the equation of heat transfer in a moving fluid, are presented. The flows of viscous incompressible fluids can both steady and unsteady. The new solutions are based on the Lin–Sidorov–Aristov class of exact solutions. The characteristic feature of the velocity field representation is that it is described by linear forms with respect to two coordinates (horizontal or longitudinal). The coefficients of the linear forms depend on the third coordinate (vertical, or transverse) and time. The fluid pressure and temperature are quadratic forms with a similar structure for the velocity. This family of exact solutions describes the flows of viscous incompressible fluids with a spatial acceleration. In other words, allowance is made for nonlinear effects of inertia forces, which are expressed through the convective derivative of the velocity and temperature vectors in the Navier–Stokes and the heat conduction equations, respectively. Due to the energy dissipation in fluids, two quadratically nonlinear effects compete. This significantly complicates the study of flows, because of which the paper presents formulas describing a creeping flow (the Stokes approximation) and Oseen motion. Thus, the study shows the possibility of constructing exact solutions to the motion equations with mechanical energy dissipation into thermal energy for the full Navier–Stokes equations and for their linearized analogs in the Stokes and Oseen approximations.