Nonlinear isobaric flow of a viscous incompressible fluid in a thin layer with permeable boundaries

Abstract

A new exact solution of the Navier-Stokes equations system is investigated. This solution describes an isobaric three-dimensional nonlinear flow of a viscous incompressible fluid in an infinite horizontal layer with permeable boundaries. Permeable layer boundaries allow one to realize fluid suction or injection in a vertical direction. Thus, a generalization of the non-uniform layered Couette-type flow to the three-dimensional case is obtained. The announced exact solution belongs to the Lin class. The velocity field is a linear form with respect to two spatial horizontal coordinates with coefficients depending on the third (transverse) coordinate in this class. The obtained exact solution describes a three-dimensional flow of a vertically vortex fluid, which can be used to describe large-scale processes in oceanology and in atmospheric physics. The obtained exact solution describes a large-scale flow of a vertical vortex fluid in the thin layer approximation. Vertical twist in a non-rotating fluid arises due to the inclusion of inertial forces in the motion equations and the velocities inhomogeneous distribution on the upper non-deformable permeable boundary of the layer. A non-uniform velocity field is studied using the Navier slip condition on the lower boundary. Additionally, the case of equality to zero of the slip length (sticking condition) is analyzed. The velocity field is studied for an arbitrary value of the Navier parameter. The obtained exact solution allows one to describe the counterflows of a viscous incompressible fluid. The obtained solution is analyzed and the existence of stagnation points in the vertical vortex fluid flow in an infinite layer with permeable boundaries is shown. Only one stagnation point is recorded in the fluid flow, when the no-slip and Navier slip conditions at the lower boundary are realized. Thus, the obtained exact solutions of the Navier-Stokes equations describe a new angular momentum transfer mechanism in a fluid. This exact solution illustrates the existence of vertical vorticity in a non-rotating fluid. Vertical twist is induced by a non-uniform velocity field at the boundaries of the fluid layer.