The long way of a viscous vortex dipole

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 $O(Re^σ)$, where $σ< 1$ is arbitrary. This improves upon previous results which were essentially restricted to $σ= 0$. 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.

💡 Research Summary

The paper investigates the long‑range dynamics of a viscous vortex dipole in the plane, i.e. a pair of point vortices with opposite circulations that evolve under the two‑dimensional Navier–Stokes equations. The authors focus on the high‑Reynolds‑number regime (Re=\Gamma/\nu\gg1), where the vortex cores remain much smaller than the distance between their centers for a long time. They introduce two small parameters: the aspect ratio (\varepsilon(t)=\sqrt{\nu t}/d) (core size over separation) and the inverse Reynolds number (\delta=\nu/\Gamma). In this regime the advection time (T_{\rm adv}=d^{2}/\Gamma) is much shorter than the diffusion time (T_{\rm diff}=d^{2}/\nu), allowing the dipole to travel a distance far exceeding its own size before diffusion destroys the coherent structure.

The main result (Theorem 1.2) states that for any fixed (\sigma\in