Nonlinear dynamics of gas bubbles in liquids

Gas bubbles in liquids are fascinating nonlinear oscillators that have long been exploited for industrial, scientific and medical applications. This thesis develops and analyzes mathematical models for three problems in nonlinear dynamics of bubbles: the interaction of two translating and pulsating spherical bubbles, acoustic cavitation of a single spherical bubble, and nonspherical oscillations of a bubble. A rigorous mathematical basis is also provided for a transport theorem which is used to obtain equations of motion for bubbles in potential flow. The common thread throughout is that concepts and techniques from dynamical systems theory provide significant insight into each problem.; For the two-bubble problem, a new model is developed that describes the translation and radial oscillations of two bubbles. For weak acoustic forcing, an averaging analysis of the model identifies and explains patterns of translational motion observed in recent experiments. At lowest order, the model recovers the classical secondary Bjerknes force, according to which two weakly pulsating bubbles attract or repel depending upon whether their pulsations are in or out of phase. For strong acoustic forcing, it is observed that phase-locking of the pulsations can cause the direction of the mutual force to be opposite that predicted by classical theory.; The acoustic cavitation problem, in this thesis, is to determine a dynamic pressure threshold predicting the onset of nonlinear bubble growth followed by violent collapse in a time-periodic pressure field. This is achieved, for a wide range of bubble sizes, through a normal form analysis of the classical Rayleigh-Plesset equation, which governs the radial oscillations of a spherical bubble. The new dynamic threshold is seen to be the culmination of a period doubling cascade of bifurcations in the bubble response.; For the problem of nonspherical pulsations of a single bubble, equations of motion are developed and used to study the parametric stability of the bubble interface. When shape modes couple resonantly to the spherical (volume) mode, an exchange of energy can occur to the shape modes, causing them to grow and possibly break up the bubble. The issue of stability is addressed by a Floquet analysis of the dynamic equations.