Date of Award

Spring 2004

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

N. Aubry

Second Advisor

Denis L. Blackmore

Third Advisor

Lou Kondic

Fourth Advisor

Demetrius T. Papageorgiou

Fifth Advisor

Michael Siegel

Abstract

Stirring is a well-known means of fluid mixing due to the emergence of complex patterns in the flow, even at low Reynolds numbers. In this work, we consider a stirrer rotating along a circular trajectory at constant speed. The fluid flow, considered incompressible, inviscid and two dimensional (in a circular container), is modeled by a point vortex model consisting of a vortex rotating in a circular container at constant angular speed. The mixing problem is addressed by considering the Hamiltonian form of the advection equations formulated in a frame of reference moving with the vortex. The dynamics of passive fluid particles is considered using dynamical systems theory. The bifurcation diagram reveals the presence of degenerate fixed points and homoclinic/heteroclinic orbits, whose nature varies for different parameter values. By considering an initially concentrated set of marker particles and using the various structures of the phase space in the bifurcation diagram, we produce a complex dynamics which, in turn, can generate efficient mixing. The latter is studied using both numerical simulations and physical experiments. A perturbation study for one particular structure for the phase space shows the presence of a transverse homoclinic orbit as well as resonances, or a set of closed trajectories.

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Mathematics Commons

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