A procedure for the construction of non-stationary Riccati-type flows for incompressible 3D Navier-Stokes equations

TL;DR

This paper develops a method to construct non-stationary Riccati-type flows for the 3D incompressible Navier-Stokes equations, providing explicit solutions that decay exponentially over time, enhancing analytical understanding of fluid dynamics.

Contribution

It introduces a novel analytical procedure to derive explicit non-stationary solutions of the Navier-Stokes equations using Riccati equations, simplifying the solution process.

Findings

Derived explicit non-stationary solutions that decay exponentially as time approaches infinity.

Presented a simplified analytical framework for Riccati-type flows in incompressible 3D Navier-Stokes equations.

Demonstrated the solutions as a class of perturbations absorbed exponentially over time.

Abstract

In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid. But there is an essential deficiency of non-stationary solutions indeed. We explore the ansatz of derivation of non-stationary solution for the Navier-Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to the time-parameter t). But we find an elegant way to simplify it to the proper analytical presentation of exact solution (such a solution is exponentially decreasing to zero for t going to infinity). Also it has to be specified that the solutions that are constructed can be considered as a class of perturbation absorbed exponentially as t going to infinity by the null solution.