Stability of an imploding spherical shock wave in a van der Waals gas II

The emission of light from a sonoluminescing bubble may dependon whether shock waves launched each acoustic cycle by the implodingsurface of the bubble focus on to a sufficiently small volumeat the bubble centre. This in turn may depend on whether theshock maintains its stability as it travels inwards. With thisapplication in mind, the linear stability of an imploding sphericalshock was studied in Part I, using a van der Waals equationof state for the gas. Conditions for instability were determined,but the subsequent fate of the perturbations of the bubble surfacewas unknown. Would the instabilities grow and persist at finiteamplitude or would they disappear during implosion? The answersto such questions are sought here by integrating the gas dynamicsequations using the finite-difference essentially non-oscillatorymethod of Shu and Osher. The shock is initiated by a nearlyspherical ‘piston’ and its subsequent evolution,including its finite-amplitude deviations from sphericity, isdetermined. Two types of behaviour are found depending on theparameter , where b is thevan der Waals excluded volume and is the initial uncompressed density of gas ahead of the shock.When is sufficiently large, an initially smooth shock front remains smooth as it focusesand, although it is impossible to continue the integrationsup to the moment of implosion, it appears that it will focuson a small volume at the centre of the bubble. This is in sharpcontrast to what happens at smaller values of for which the initial distortion of the shock front,if sufficiently large, becomes and remains polygonal shaped.This is consistent with experimental results for cylindricalimploding shocks as well as with earlier theoretical investigationsof imploding cylindrical and spherical shocks in an ideal gas() that used the Chisnell, Chester and Witham (CCW) approximation or the geometrical shockapproximation of Whitham. Plausibly, the polygonal distortionsreduce the volume on to which the imploding shock in a sonoluminescingbubble focuses.