The long way of a viscous vortex dipole

TL;DR

This paper analyzes the evolution of viscous vortex dipoles in two dimensions at high Reynolds numbers, providing an accurate asymptotic approximation and justifying corrections to their translation speed.

Contribution

It introduces a two-parameter asymptotic expansion for viscous vortex dipoles and extends the validity of the approximation to longer times than previous results.

Findings

Approximate solution remains close to the exact Navier-Stokes solution for time $O(Re^\sigma)$.

Provides a rigorous justification for the correction to the dipole's translation speed due to finite size.

Improves understanding of vortex dipole dynamics at high Reynolds numbers.

Abstract

We consider the evolution of a viscous vortex dipole in $R^2$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $Re >> 1$, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length $O(Re^\sigma)$, where $\sigma < 1$ is arbitrary. This improves upon previous results which were essentially restricted to $\sigma = 0$. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.