University of Illinois Urbana-Champaign

Stationary Configurations of Point Vortices (Morse Theory, Hamiltonian Systems, Fluid Mechanics)

O'neil, Kevin Anthony

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Description

Title
Stationary Configurations of Point Vortices (Morse Theory, Hamiltonian Systems, Fluid Mechanics)
Author(s)
O'neil, Kevin Anthony
Issue Date
1985
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion, and are of considerable physical interest.
  • It is known that a configuration of vortices in equilibrium must have total angular momentum 0. A converse is proved, namely that for almost every choice of circulations with zero angular momentum, there are exactly (n - 2)! equilibrium configurations. A similar statement is proved for rigidly translating configurations with total circulation zero. The proofs involve ideas from algebraic geometry.
  • Relative equilibria (rigidly rotating configurations) were studied by Palmore in the case of positive circulations. Upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power-law systems.
  • Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for n vortices, n (GREATERTHEQ) 3.
  • Stationary configurations of four vortices are examined in detail.
Graduation Semester
1985
Type of Resource
text
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http://hdl.handle.net/2142/71234

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