In the paper, several new exact solutions of the Navier–Stokes equations for description of unsteady and non-stationary inhomogeneous shear gradient flows are constructed. Flows of viscous incompressible fluid of “two and a half” dimension are considered. Such flows are characterized by a two-dimensional velocity field with dependence on three spatial coordinates and time. The quadratically nonlinear system of partial differential equations consisting of the momentum transfer equations and the incompressibility equation for description of shear flows of “two and a half” dimension is overdetermined. For this system in polynomial classes of exact solutions, nontrivial exact solutions of the overdetermined system of equations of hydrodynamics of inhomogeneous shear flows are constructed. Description of unsteady flows is based on modification of the Fourier variable separation method. Main attention is paid to the Lin – Sidorov – Aristov class with linear dependence of the velocity field and pressure field on two spatial variables. In addition, it is shown how a class of exact solutions with a nonlinear dependence on two spatial coordinates can be used to describe non-uniform shear flows of a vertically swirling fluid without preliminary swirling. A method for replicating (multiplying) exact solutions for hydrodynamic equations is given.
Prosviryakov et al. (Mon,) studied this question.