Vortices and invariants surfaces generated by symmetries for the 3D Navier-Stokes equations

TL;DR

This paper explores how symmetries of the 3D Navier-Stokes equations generate vortex solutions, classifies these vortices algebraically, and identifies invariant surfaces and new vortex structures.

Contribution

It introduces an algebraic classification of vortices based on Lie symmetries and uncovers new vortex configurations related to symmetry-invariant boundary conditions.

Findings

Infinite-dimensional symmetry algebra for Navier-Stokes

New vortex solutions generalizing known configurations

Invariant surfaces corresponding to boundary conditions

Abstract

We show that certain infinitesimal operators of the Lie-point symmetries of the incompressible 3D Navier-Stokes equations give rise to vortex solutions with different characteristics. This approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector. The symmetry algebra associated with the Navier-Stokes equations turns out to be infinite- dimensional. New vortical structures, generalizing in some cases well-known configurations such as, for example, the Burgers and Lundgren solutions, are obtained and discussed in relation to the value of the dynamic angle. A systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed, and the corresponding invariant surfaces are recognized.