| Date: Friday, January 07, 2011
Title: On Some Free-Boundary Problems in Vortex Dynamics
Abstract: In this work we focus on a class of solutions of 2D steady-state Euler equations characterized by finite-area vortices embedded in a potential flow. Such solutions are relevant as they arise in the context of the Prandtl-Batchelor theory as possible limiting solutions of actual Navier-Stokes flows. Our goal is to devise analytical and computational methods that will allow us to study different properties of such distributed vortex equilibria, namely, continuity with respect to parameters, stability and design of vortex flows with optimal properties. Following a survey of relevant results concerning systems of point vortices, we will present a framework applicable to the study of Euler flows with distributed vortex equilibria. Recognizing that such flows are in fact described by equations of the free-boundary type, the proposed approach is based on methods of the shape-differential calculus which allows one to deal with PDE problems defined on variable domains. As regards vortex stability, we will argue that the classical results of Kelvin (1880) and Love (1893) arise as special cases in our approach. We will also show a number of computational results concerning the "vortex design problem" where equilibrium vortex configurations with prescribed properties are designed by optimizing the velocity boundary conditions. The presentation will contain elements of rigorous mathematical analysis alongside with results of large-scale numerical computations.
Speaker: Bartosz Protas
Institution: McMaster University
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