The effect of viscosity on the spherical stability of oscillating gas bubbles

Abstract

Gas bubbles driven in radial oscillations are subject to an instability of the spherical shape that is opposed by surface tension and viscosity. An exact linear formulation for the study of the phenomenon has been available for many years, but its complexity has discouraged a detailed investigation. With the recent theory of sonoluminescence of Lohse and co-workers [Hilgenfeldt et al., Phys. Fluids, 8, 2808 (1996)], there has been a renewed interest in the problem and new data have become available. This paper presents a numerical method for the solution of the pertinent equations and compares the theory with these new data. The coupling of the strong nonlinearity of the bubble radial oscillations with the parametric mechanism of the surface instability results in a very complex structure for the stability boundary. Nevertheless, a good agreement between theory and data is found. A comparison with earlier approximate models is also made.

Single-bubble sonoluminescence

“…These experimental and theoretical studies examine larger bubble sizes with smaller forcing pressures for which sonoluminescence cannot occur. Applying the above shape stability analysis in the regime P a ϳ0.5-1 atm Brenner et al, 1999;Hao and Prosperetti, 1999b;Augsdö rfer et al, 2000) gives similar thresholds to those found in experiment by Eller and Crum (1970), Horsburgh (1990), and Gaitan and Holt (1998). For P a Շ0.9 atm, the nϭ3 mode takes over as the most unstable surface mode from the nϭ2 mode Brenner et al, 1999;Augsdö rfer et al, 2000), just as was found in the experiments by Gaitan and Holt (1998).…”

“…Holt and Gaitan (1996;Gaitan and Holt, 1999) also measured shape instabilities in the sonoluminescence regime and close to it. Their experimental results were compared with theory in the work of Brenner, Dupont, et al (1998) and Hao and Prosperetti (1999b), with good agreement in evidence [see Fig. 35(a)].…”

“…It is therefore important to ask how the shape stability results change when the boundary layer approximation is not made, and the full integrodifferential equations are solved. Such a comparison was carried out by Wu and Roberts (1998) and Hao and Prosperetti (1999b). A typical result for forcing pressure around P a ϳ1 atm is shown in Figs.…”

“…This interaction is nonlocal in time, and so the problem requires solving integrodiffer- ential equations for the vorticity in the liquid, coupled with the shape oscillations. Details can be found in the literature (Prosperetti, 1977d;Hilgenfeldt et al, 1996;Hao and Prosperetti, 1999b). Here we simply summarize the results of a simple ''boundary layer'' approximation, which assumes that the vorticity is localized in a thin region around the bubble.…”

“…A typical result for forcing pressure around P a ϳ1 atm is shown in Figs. 35(a) and 35(b), taken from Hao and Prosperetti (1999b): Hardly any difference between the exact result and the boundary layer approximation can be seen. Wu and Roberts (1998) found similar agreement even for larger P a in the regime of SBSL.…”