Vorticity internal transition layers for the Navier-Stokes equations

TL;DR

This paper analyzes the internal transition layers in the vorticity of incompressible Navier-Stokes flows with vortex patches, revealing sharp yet smooth variations in the vorticity near internal boundaries under certain conditions.

Contribution

It introduces an asymptotic expansion approach to describe the smooth internal layers in vorticity for Navier-Stokes equations with vortex patches, connecting to Euler flow dynamics.

Findings

Existence of sharp but smooth vorticity transition layers

Layer dynamics follow the flow of Euler equations

Layer thickness depends on viscosity and time

Abstract

We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t * nu is small, where nu is the viscosity coefficient.