I consider the hydrodynamic stability of imploding ideal gases as an idealized model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes (short-wavelength incompressive modes elongated in the direction of the mean flow), a second-order ordinary differential equation is derived that can be used to assess the stability of any time-dependent flow with planar, cylindrical or spherical symmetry. Upon further restricting the analysis to homologous flows, it is shown that a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, both entropy and vorticity fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and, in the absence of dissipative effects, is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor $(2C)^{|N_{0}|t_{i}}$, where $C$ is the convergence ratio of the implosion, $N_{0}$ is the initial buoyancy frequency and $t_{i}$ is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor $\exp ({\rm\pi}|N_{0}|t_{s})$, where $N_{0}$ is the buoyancy frequency at stagnation and $t_{s}$ is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this additional growth due to compression results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. The above analytical results are verified by evolving the complete set of linear equations as an initial value problem, and it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode $\ell \gtrsim 100$ for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of ${\sim}30$ zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of ${\sim}(12/\ell )^{\circ }$, or ${\lesssim}0.1^{\circ }$ to resolve the fastest-growing modes.

The interaction of a shockwave with a gas bubble in a liquid medium is of interest in a variety of areas, e.g. shockwave lithotripsy, cavitation damage and the study of sonoluminescence. This study employs a high-resolution front-tracking framework to numerically investigate this phenomenon. The modelling paradigm is validated extensively and then used to explore the parametric space of interest. We provide a comprehensive qualitative analysis of the collapse process, which we categorize into three phases, based on the principal feature dominating each phase. This results in the characterization of numerous previously unidentified features important in the evolution of the process and in the emergence of peak temperatures and pressures. For example, we discover that the peak pressure does not occur as a result of the impact of the main transverse jet (also called the re-entrant jet) but later in the collapse. We perform fully three-dimensional simulations, showing that three-dimensional instabilities are limited to the small-scale details of collapse, and continue by comparing collapse of cylindrical and spherical bubbles. We detail a parametric investigation varying the shock strength from 100 MPa to 100 GPa. A counter-intuitive discovery is that the maximum gas density decreases with increasing shock strength.

This paper investigates the behaviour of a non-spherical cavitation bubble in an acoustic standing wave. The study has important applications to sonochemistry and in understanding features of therapeutic ultrasound in the megahertz range, extending our understanding of bubble behaviour in the highly nonlinear regime where jet and toroidal bubble formation may be important. The theory developed herein represents a further development of the material presented in Part 1 of this paper (Wang & Blake, J. Fluid Mech. vol. 659, 2010, pp. 191–224) to a standing wave, including repeated topological changes from a singly to a multiply connected bubble. The fluid mechanics is assumed to be compressible potential flow. Matched asymptotic expansions for an inner and outer flow are performed to second order in terms of a small parameter, the bubble-wall Mach number, leading to weakly compressible flow formulation of the problem. The method allows the development of a computational model for non-spherical bubbles by using a modified boundary-integral method. The computations show that the bubble remains approximately of a spherical shape when the acoustic pressure is small or is initiated at the node or antinode of the acoustic pressure field. When initiated between the node and antinode at higher acoustic pressures, the bubble loses its spherical shape at the end of the collapse phase after only a few oscillations. A high-speed liquid bubble jet forms and is directed towards the node, impacting the opposite bubble surface and penetrating through the bubble to form a toroidal bubble. The bubble first rebounds in a toroidal form but re-combines to a singly connected bubble, expanding continuously and gradually returning to a near spherical shape. These processes are repeated in the next oscillation.

A small gas bubble in a liquid, when driven by intense ultrasound, collapses and emits light in a process called Single-Bubble Sonoluminescence (SBSL). While the dynamics of driven bubbles are well studied, less is known of the physical conditions in the gas or whether it is necessary to include ionisation in simpler studies of bubble dynamics. In this study, a model was derived from Rayleigh-Plesset dynamics, a van der Waals equation of state and the first law of thermodynamics (including interfacial heat transfer and ionisation). Stronger model ionisation reduced the maximum collapse temperature, and altered other collapse characteristics. Chaotic parameter regions are proximal to, but not coincident with, known stable SL regions. Resonant behaviour was only markedly affected by ionisation close to these chaotic regions.