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In this thesis two different models and numerical methods have been developed to investigate the dynamics of bubbles in viscoelastic fluids. In the interests of gaining crucial initial insights, a simplifed system of governing equations is first considered. The ambient fluid around the bubble is considered incompressible and the flow irrotational. Viscoelastic effects are included through the normal stress balance at the bubble surface. The governing equations are then solved using a boundary element method. With regard to spherical bubble collapse, the model captures the behaviour seen in other studies, including the damped oscillation of the bubble radius with time and the existence of an elastic-limit solution. The model is extended in order to investigate multi-bubble dynamics near a rigid wall and a free surface. It is found that viscoelastic effects can prevent jet formation, produce cusped bubble shapes, and generally prevent the catastrophic collapse that is seen in the inviscid cases.\ud \ud The model is then used to investigate the role of viscoelasticity in the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. The model predicts the cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies. \ud \ud A second model is developed with the intention of gaining further insight into the role of viscoelasticity and corroborating the finndings of the first model. This second\ud model employs the full compressible governing equations in a two dimensional domain. The equations are solved using the spectral element method, while the two phases are\ud represented by "marker particles". The results are in qualitative agreement with the first model and confirm that the findings presented are a faithful account of bubble\ud dynamics in viscoelastic fluids.
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