Existence and stability of viscous vortices

TL;DR

This paper reviews mathematical results on the existence and stability of viscous vortices in various geometries, highlighting their role in fluid dynamics and turbulence.

Contribution

It provides a comprehensive review of rigorous mathematical findings on viscous vortex existence and stability in simple geometries.

Findings

Existence of viscous vortex solutions in simple geometries.

Stability conditions for viscous vortices.

Relevance of vortices in turbulence and flow dynamics.

Abstract

Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real flows vortex lines are usually not straight, and can even form closed curves, as in the case of axisymmetric vortex rings which are very common in nature and in laboratory experiments. The aim of this chapter is to review a few rigorous results concerning existence and stability of viscous vortices in simple geometries.