Asymmetric and Moving-Frame Approaches to Navier-Stokes Equations

TL;DR

This paper introduces asymmetric and moving-frame methods to solve the 3D Navier-Stokes equations, producing various explicit solutions including blow-up, shock, and nozzle flow solutions, advancing understanding of complex fluid behaviors.

Contribution

It presents novel asymmetric and moving-frame techniques for solving Navier-Stokes equations, yielding explicit solutions with applications to turbulence, shock waves, and nozzle flows.

Findings

Derived seven families of non-steady rotating asymmetric solutions.

Identified solutions that blow up on moving planes, useful for turbulence studies.

Developed solutions modeling shock waves and nozzle flows.

Abstract

In this paper, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent variables and a method of moving frame for solving the three dimensional Navier-Stokes equations. Seven families of non-steady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blow up at any point on a moving plane with a line deleted, which may be used to study turbulence. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in study of shock waves. Another family of solutions are partially cylindrical invariant, contain two parameter functions of $t$ and structurally depend on two arbitrary polynomials, which may be used to describe incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spacial variables.