Crowdy, Darren G. (1998) Exact solutions for twodimensional Stokes flow. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd01182008133427
Abstract
This thesis comprises three parts. The principal topic is presented in Part I and concerns the problem of the freeboundary evolution of two dimensional, slow, viscous (Stokes) fluid driven by capillarity. A new theory of exact solutions is presented using a novel global approach involving complex line integrals around the fluid boundaries. It is demonstrated how the consideration of appropriate sets of geometrical line integral quantities leads to a concise theoretical reformulation of the problem. All previously known results for simplyconnected regions are retrieved and the analytical form of the exact solutions formally justified. For appropriate initial conditions, an infinite number of conserved quantities is identified. An important new general result (herein called the theorem of invariants) is also demonstrated.
Further, using the new theoretical reformulation, an extension to the case of doublyconnected fluid regions with surface tension is made. A large class of exact solutions for doublyconnected fluid regions is found. The method combines the new theoretical approach with elements of loxodromic function theory. To the best of the author's knowledge, this thesis provides the first known examples of exact solutions for Stokes flow in a doublyconnected topology. The theorem of invariants is extended to the doublyconnected case.
Finally analytical arguments are presented to demonstrate the existence, in principle, of a class of exact solutions for geometrically symmetrical fourbubble configurations.
In Part II, the most general representation for local solutions to the two dimensional elliptic and hyperbolic Liouville equations is formally derived.
In Part III, some analytical observations are presented on solutions to the linearized equations for small disturbances to the axisymmetric Burgers vortex. The relevance to the (as yet unsolved and little studied) problem of the linear stability of Burgers vortex to axiallydependent perturbations is argued and discussed.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Major Option:  Applied And Computational Mathematics 
Thesis Committee: 

Defense Date:  26 August 1997 
Record Number:  etd01182008133427 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd01182008133427 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  226 
Collection:  CaltechTHESES 
Deposited By:  Imported from ETDdb 
Deposited On:  14 Feb 2008 
Last Modified:  25 Dec 2012 14:58 
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