Indirect boundary element methods for modelling bubbles under three dimensional deformation

The nonlinear behaviour of gas and vapour bubbles is a complex phenomenon which plays a signi cant role in many natural and man-made processes. For example, bubbles excited by an acoustic eld play important roles in lithotripsy, drug delivery, ultrasonic imaging, surface cleaning and give rise to the phenomenon of sonoluminescence (light emission from a bubble excited by sound). In such contexts, the oscillation of even a single bubble is not yet fully understood, let alone the behaviour of multiple bubbles interacting with each other. An essential part of understanding such problems is un- derstanding the complex and sometimes unpredictable coupling between the oscillation of the bubble volume and the bubble shape, a problem requiring experimental research, theoretical work and numerical studies. In this Thesis we focus on numerical simulation of a single gas bubble oscillating in a free liquid. Previously, such numerical simulations have al- most exclusively assumed axisymmetry and small amplitude oscillations. To avoid these assumptions we build upon and extend previous boundary ele- ment methods used for three dimensional simulations of other bubble prob- lems. We use high order elements and parallel processing to yield an indirect boundary element method capable of capturing ne surface e ects on three dimensional bubbles subjected to surface tension, over extended periods of time. We validate the method against the classical Rayleigh-Plesset equation for spherical oscillation problems before validating the indirect boundary el- ement method and the method used by Shaw (2006), against each other, on several small amplitude axisymmetric oscillation problems. We then proceed to study near-resonant non-axisymmetric shape oscillations of order 2 and 4 and the e ect these oscillations have on higher order modes, with a level of detail we believe has not been achieved in a non-axisymmetric study before. We also con rm some predictions made by Pozrikidis' on resonant interac- tions between the second order modes and the volume mode in addition. Finally we study the spherical instability of a bubble trapped in a uniform acoustic eld, demonstrating, as expected, that instabilities show up in all resonant shape modes, including non-axisymmetric ones.