Large Amplitude Non-Spherical Bubbles

We consider the long-term evolution of an axisymmetric bubble and explore the ways in which it may develop. Linearised inviscid analysis is used to predict the stability of the bubble with a small disturbance while a nonlinear inviscid extension shows that the growth of unstable modes is ultimately limited by the formation of axisymmetric curvature singularities. The addition of surface tension is shown to delay, but not entirely prevent, these singularities. Our results are found to agree well with a viscous Boussinesq theory at least to early times. The inclusion of viscosity means that the development of the bubble structure is not limited by the creation of singularities, and the bubble may ultimately adopt one of a wide range of possible large-scale deformations. Among these, perhaps the most exotic are jet-like structures which can pinch off and break into several distinct parts. Spectral methods are employed to solve the inviscid and Boussinesq models while the linearised inviscid model admits a closed-form series solution.

Automated summary

We study the long-term evolution of an axisymmetric bubble and explore its possible development. Through linearized inviscid analysis, we predict the stability of the bubble under small disturbances, while a nonlinear inviscid extension reveals that the growth of unstable modes is ultimately limited by the formation of axisymmetric curvature singularities. The addition of surface tension delays but does not entirely prevent these singularities. Our results show good agreement with a viscous Boussinesq theory, especially at early times. The inclusion of viscosity allows for a wide range of possible large-scale deformations, with the most exotic being jet-like structures that can pinch off and break into distinct parts. Spectral methods are used to solve the inviscid and Boussinesq models, while the linearized inviscid model admits a closed-form series solution.