Static and steady-state bubbles in the channel

This dissertation devotes its attention to the mathematical theory of bubbles moving in a channel. In particular, it deals with static bubbles and add new information about the shape of a single steady-state bubble which appeared in "the selection problem of moving bubbles." The selection problem first arose in the experimental work of Saffman and Taylor. They were interested in the motion and the shape of bubbles and fingers in a Hele-Shaw cell. They introduced an analytical approach to this nonlinear problem when surface tension and fluid viscosity are ignored.

Chapter 1. A mathematical formulation for static bubbles in the channel is presented. The method of {\it reduction to symmetry} is employed to derive an explicit solution for a finite number of bubbles in terms of Green's Function and Harmonic Measures. The theory of Quadratic Differentials is applied to study the flow in a simpler way. Qualitative properties of the flow and open problems on static bubbles are also discussed.

Chapter 2. This chapter focuses on geometric characteristics of steady-state bubbles. The case of a single bubble is considered and various geometric properties are discussed analytically. The formulas for maximal length, width, and area of the stead-state bubble are obtained in terms of the parameters of the problem. Monotonicity of the area for a single moving bubble in the channel is also discussed. Future research and open problems relating to the selection-problem of bubbles in the channel are mentioned briefly.