Point vortex interactions
Abstract
The dynamics of point vortices with a view towards the onset of chaos in these systems were studied. It was conjectured that for a given geometry and external flow, a chaotic threshold N* exists such that NN* Quasiperiodic Motion NN* Chaotic Motion where N is the number of vortices. The conjecture (and the determination of N*) for a few special cases is proven. The existence of Horseshoes in the Poincare map of the perturbed system via Melnikov function technique are shown. Such a chaotic threshold suggests the nonintegrability of the two dimensional Euler Equations, in which the vortex equations are imbedded. It is shown that a vortex moving in a bounded, simply connected region of the plane has precisely one equilibrium point that is elliptic and Liapunov stable. The problem of two vortices inside a circle is solved exactly and the role of Morse theory in estimating special solutions is illustrated.
 Publication:

Ph.D. Thesis
 Pub Date:
 December 1985
 Bibcode:
 1985PhDT........53S
 Keywords:

 Euler Equations Of Motion;
 Liapunov Functions;
 Morse Potential;
 Poincare Spheres;
 Vortex Generators;
 Vorticity;
 Cavitation Flow;
 Differential Equations;
 Kinetic Theory;
 Velocity Measurement;
 Fluid Mechanics and Heat Transfer