Exact Solutions for Three-Dimensional Nonlinear Flows of a Viscous Incompressible Fluid

Exact solutions for three-dimensional nonlinear flows of a viscous incompressible fluid are examined. These solutions belong to Lin's class of solutions, which are velocities linearly dependent on a part of coordinates. This allows these solutions to be used for the description of large-scale currents of the World Ocean. The obtained exact solution describes the flow of a vertical vortex fluid. A vertical twist in the fluid arises from the consideration of inertia forces and the nonuniform velocity distribution on the free boundary of the fluid layer. This solution allows one to describe counterflows of an incompressible fluid for flows in a thin layer. Thus, the obtained exact solution of the Navier-Stokes equations describes a new mechanism of momentum transfer in a fluid. The obtained solutions are analyzed in this paper. The existence of stagnation points for the flow of a vertical vortex fluid in an infinite layer with permeable boundaries is shown.