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On stagnation points and streamline topology in vortex flows

Published online by Cambridge University Press:  16 July 2002

HASSAN AREF
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
MORTEN BRØNS
Affiliation:
Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark

Abstract

The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

Type
Research Article
Copyright
© 1998 Cambridge University Press

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