The Long Way of a Viscous Vortex Dipole

We consider the evolution of a viscous vortex dipole in R2 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(Reσ), where σ