Vanishing viscosity limit for concentrated vortex rings

TL;DR

This paper investigates the behavior of viscous incompressible fluids with concentrated vortex rings as viscosity and ring thickness approach zero, demonstrating that vortex rings persist and translate uniformly in the limit.

Contribution

It establishes the vanishing viscosity limit for concentrated vortex rings, showing their persistence and uniform translation without swirl in an axial symmetric setting.

Findings

Vorticity remains concentrated in disjoint rings in the limit.

Each vortex ring translates along the axis with constant speed.

The result applies as both viscosity and ring thickness tend to zero.

Abstract

We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl, when the initial vorticity is very concentrated in $N$ disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in $N$ disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.