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Lecturer: Michele Dolce (EPFL)
Title: Dynamics of vortex pairs at large Reynolds number in 2D fluids
Abstract: The evolution of two point vortices in a 2D inviscid fluid in the whole plane, whether counter-rotating or co-rotating, is explicitly determined by the Helmholtz-Kirchhoff system. One translates on a straight line with a constant speed, while the latter rigidly rotates around each other. At a large but finite Reynolds number, vortex core sizes grow due to diffusion, revealing immensely rich phenomena stemming from this simple initial configuration.
The goal of the lectures is to validate an asymptotic expansion describing the dynamics of the viscous vortex pair. This will be done in terms of small adimensional parameters, particularly useful to isolate inviscid and viscous effects. The expansion is performed at a sufficiently high order where corrections to the Helmholtz-Kirchhoff motion are revealed. This in agreement with observations in the applied literature.
We then show that the exact solution remains close to our approximation over a time interval that increases boundlessly as the Reynolds number goes to infinity. The proof relies on stability estimates derived from Arnold’s variational characterization of the steady states of the 2D Euler equation, as recently revised by Gallay and Sverak and applied to viscous fluids.
The lectures are based on an ongoing project with T. Gallay.
Schedule of the course: