Аннотация:The unsteady Navier–Stokes equations with two space variables have been studied. It has been shown that the corresponding fourth-order nonlinear equation for the stream function, derived from the Navier–Stokes equations, has three independent variables and allows functional separable solutions described by a system of three partial differential equations in two independent variables. A number of exact solutions, which generate new classes of exact solutions of the Navier–Stokes equations, have been obtained. All these solutions involve two or more arbitrary functions of a single argument and a few free parameters. Many solutions are expressed in terms of elementary functions if the arbitrary functions are also elementary. The results obtained can be used to solve certain model problems and make it possible to effectively estimate the domain of applicability and accuracy of numerical, asymptotic, and approximate analytical methods in hydrodynamics.