Exact Solutions to the Navier–Stokes Equations for Nonlinear Viscous Flows by Undetermined Coefficient Approach

Abstract

Exact solutions to the Navier–Stokes’s equations for nonlinear viscous flows are investigated. These solutions belong to Lin's class of solutions, which are velocities that are linearly dependent on a portion of coordinates. This enables these solutions to be used to understanding large-scale processes of nature for example in the ocean and atmospheric phenomena. The precise solution obtained describes the flow of a vertical vortex fluid. The consideration of inertia forces and the nonuniform velocity distribution on the fluid layer's free boundary results in a vertical twist in the fluid. This solution describes counterflows of an incompressible fluid for flows in a thin layer. As a result, the exact solution of Navier–Stokes’s equations for nonlinear viscous flows is obtained. This solution describes counterflows of an incompressible fluid for thin-layer flows. As a result, the exact solution of Navier-Stokes’s equations obtained describes a novel mechanism of momentum transfer in a fluid. The objective of this research is to obtain the solutions of the exact solution of Navier-Stokes’s equations and analyse the results. From the finding from the analysed results, it is demonstrated that stagnation points exist for the flow of a vertical vortex fluid in an infinite layer with permeable boundaries.