Exact solutions for two-dimensional Stokes flow

Citation

Crowdy, Darren G. (1998) Exact solutions for two-dimensional Stokes flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3jys-vr33. https://resolver.caltech.edu/CaltechETD:etd-01182008-133427

Abstract

This thesis comprises three parts. The principal topic is presented in Part I and concerns the problem of the free-boundary 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 simply-connected 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 doubly-connected fluid regions with surface tension is made. A large class of exact solutions for doubly-connected 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 doubly-connected topology. The theorem of invariants is extended to the doubly-connected case.

Finally analytical arguments are presented to demonstrate the existence, in principle, of a class of exact solutions for geometrically symmetrical four-bubble 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 axially-dependent perturbations is argued and discussed.

Thesis Files