Long-time dynamics of viscous Gröbli triangles

Abstract

In 1877 Gröbli discovered that three point vortices in the plane can move self-similarly, forming a triangle that rotates while either expanding indefinitely or collapsing in finite time. We study the viscous counterpart of these configurations, namely the two-dimensional Navier--Stokes evolution starting from three point vortices arranged in a Gröbli triangle. In the expanding case, viscosity spreads the vortex cores at the same rate as the triangle grows, so the vortices remain well separated relative to their size. Building on this observation, we show that at high Reynolds number the solution stays close to three Lamb--Oseen vortices centered near the vertices of the inviscid triangle for times that grow exponentially with a power of the Reynolds number. In the collapsing case, the same description holds up to times close to the inviscid collision. As an application of the long-time result in the expanding case, we obtain a vanishing viscosity realization of the vortex burst, a nonuniqueness mechanism for point vortices in which a single vortex splits into an expanding Gröbli triangle.