The stability of a bubble in a weakly viscous liquid subject to an acoustic traveling wave

The volume oscillations, translation, and axisymmetric deformation of a bubble in an acoustic traveling wave are considered. Assuming the bubble translation and deformation is small, but placing no restriction on the volume oscillations, a combination of the Rayleigh dissipation function and perturbation analysis is employed to account for the effects of viscosity in the absence of vorticity to third order in the small interaction terms. Contributions from the acoustic field are also determined to this order, while the free oscillation terms are drawn from a previously derived model correct to the same order of analysis. To permit the study of large amplitude acoustic forcing, appropriate compressibility terms are phenomenologically added to the volume pulsation equation. Stability maps of driving pressure versus driving frequency and driving pressure versus the equilibrium bubble radius are presented. A predominant number of results are for micron-sized bubbles driven in the ultrasonic regime, but the behavior of larger bubbles driven at frequencies in the kilohertz range is also considered. In all cases, bubbles driven above the natural frequency of their respective volume oscillations are markedly more stable with regard to the acoustic driving amplitude, consistent with previous observations. Below these respective natural frequency values, the stability/instability fronts display a much more complex structure. Accounting for shape mode viscous damping causes a general increase in bubble stability, together with a reduction in the stability/instability front complexity. In the case of micron-sized bubbles this stabilization is markedly more significant for bubbles driven above the natural frequency of the respective volume mode oscillations; for larger bubbles driven in the kilohertz range, the influence of shape mode damping is less significant.