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Title:  Stationary Configurations of Point Vortices (Morse Theory, Hamiltonian Systems, Fluid Mechanics) 
Author(s):  O'neil, Kevin Anthony 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 selfsimilar 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 powerlaw systems. Arrangements of three vortices which collide in finite time are well known. Onedimensional families of such configurations are shown to exist for n vortices, n (GREATERTHEQ) 3. Stationary configurations of four vortices are examined in detail. 
Issue Date:  1985 
Type:  Text 
Description:  96 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1985. 
URI:  http://hdl.handle.net/2142/71234 
Other Identifier(s):  (UMI)AAI8521851 
Date Available in IDEALS:  20141216 
Date Deposited:  1985 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois