Interacting vortex pairs in inviscid and viscous planar flows

Abstract

The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the vanishing viscosity limit, of the solution of the two-dimensional Navier-Stokes equation with point vortices as initial data. In such a situation, it is known (Gallay, 2011) that the solution behaves to leading order like a linear superposition of Oseen vortices whose centers evolve according to the point vortex system, but higher order corrections can also be computed which describe the deformation of the vortex cores due to mutual interactions. The second result is the construction by D. Smets and J. van Schaftingen of solutions of the two-dimensional Euler equation. These solutions are stationary in a uniformly rotating or translating frame, and converge either to a single vortex or to a vortex pair as the size parameter $\epsilon$ goes to zero. We consider here the particular case of a pair of identical vortices, and we show that the solution of the weakly viscous Navier-Stokes equation is accurately described at time $t$ by an approximate steady state of the rotating Euler equation which is a desingularized solution in the sense of Smets and van Schaftingen, with Gaussian profile and size $\epsilon = \sqrt{\nu t}$.