The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on May 22, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(3):289-309; doi:10.1093/qjmam/hbm006
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Stability of an imploding spherical shock wave in a van der Waals gas II
( Department of Mathematics, University of California, Los Angeles, CA 90095, USA )
1 Corresponding author: 
roberts{at}math.ucla.edu
Received 11 May 2006.
Revise 17 October 2006.
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Abstract |
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The emission of light from a sonoluminescing bubble may depend on whether shock waves launched each acoustic cycle by the imploding surface of the bubble focus on to a sufficiently small volume at the bubble centre. This in turn may depend on whether the shock maintains its stability as it travels inwards. With this application in mind, the linear stability of an imploding spherical shock was studied in Part I, using a van der Waals equation of state for the gas. Conditions for instability were determined, but the subsequent fate of the perturbations of the bubble surface was unknown. Would the instabilities grow and persist at finite amplitude or would they disappear during implosion? The answers to such questions are sought here by integrating the gas dynamics equations using the finite-difference essentially non-oscillatory method of Shu and Osher. The shock is initiated by a nearly spherical ‘piston’ and its subsequent evolution, including its finite-amplitude deviations from sphericity, is determined. Two types of behaviour are found depending on the parameter 
, where b is the van 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 focuses and, although it is impossible to continue the integrations up to the moment of implosion, it appears that it will focus on a small volume at the centre of the bubble. This is in sharp contrast 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 cylindrical imploding shocks as well as with earlier theoretical investigations of imploding cylindrical and spherical shocks in an ideal gas (
) that used the Chisnell, Chester and Witham (CCW) approximation or the geometrical shock approximation of Whitham. Plausibly, the polygonal distortions reduce the volume on to which the imploding shock in a sonoluminescing bubble focuses.