Surface-wave instabilities, period doubling, and an approximate universal boundary of bubble stability at the upper threshold of sonoluminescence

Numerical results are presented for hydrodynamic instabilities surrounding the parameter space of sonoluminescing bubbles. The Rayleigh-Taylor instability is shown to limit bubble oscillations only in a small region near a border at small radii in noisy environments. Also, two different noise-induced instabilities by secondary collapses are identified. The reason for period-doubled, anisotropic emission is found to be the coupling of radial and surface oscillations. Furthermore, an approximate universal border is shown to exist above which the bubble is destroyed by the parametric instability. The results are compared to experimental results from several publications using different experimental setups.

Figure 1

Rayleigh-Taylor instability: The driving pressure is 1.55 bars, the mean ambient bubble radius $0.903\mu \text{m}$, and the driving frequency 20 kHz. Shown is the ratio of the amplitude of surface distortion to mean radius. Maxima and minima are seen around the main bubble collapse and subsequent afterbounces. The $n=2$ (upper graph) surface harmonic shows a smaller distortion at the main collapse which is amplified by the afterbounces. The $n=6$ (lower graph) surface harmonic shows a large distortion at the main collapse and hardly any reaction to the afterbounces.

Figure 2

(Color online) Rayleigh-Taylor instability of an argon-water vapor bubble: In the parameter space spanned by driving pressure and equilibrium radius, the average maximum of the ratio of instantaneous surface deformation to mean radius is color coded. The maximum deformation ratio of modes $n=2$–6 is taken around the main collapse and averaged over ten consecutive driving oscillation cycles. Within the contour lines the value is larger than unity, implying splitting of the bubble. The equilibrium radius is taken as the value of radius with no periodic forcing amplitude applied.

Figure 3

Two types of afterbounce instabilities: Upper: Driving pressure is 1.4 bars, ambient radius $5.23\mu \text{m}$. Lower: Driving pressure is 1.7 bars, ambient radius $2.51\mu \text{m}$. Driving frequency is 20 kHz. The times of the main collapses are indicated by arrows.

Figure 4

(Color online) Amplification of surface mode oscillations of an argon-water vapor bubble by afterbounces: In the parameter space spanned by driving pressure and equilibrium radius the average maximum of the ratio of surface deformation $\left(n=2\right)$ amplitude to mean radius is color coded. The average maximum occurring during a whole driving oscillation period of 6 different cycles is taken. The individual cycles are nonconsecutive and randomly disturbed. Within the contour lines the value is larger than unity, implying splitting of the bubble. Two main regions of this instability can be seen to exist.

Figure 5

Parametric instability of an argon-water vapor bubble. The maximal amplitude of surface deformation increases during several periods of the driving sound to a value larger than the radius, thus leading to bubble breakup. Upper graph: The upper trace shows bubble radius, while the lower trace shows surface oscillation amplitude $\left(n=2\right)$. Equilibrium radius is $R_0=5.74\mu \text{m}$, 1.4 bars is the sound pressure. Lower graph: Excerpt of upper graph around point of splitoff. $f=20\text{kHz}$ is the driving frequency for all calculations.

Figure 6

(Color online) Parametric stability of a argon-water vapor bubble: In the parameter space spanned by driving pressure and equilibrium radius the growth rate of surface deformation per period $\left(n=2\right)$ is color coded. Within the white contour lines the value is larger than unity, implying parametric growth of the surface deformation.

Figure 7

Parametric growth of surface deformation of an argon-water vapor bubble: The time dependence of the $\left(n=2\right)$ surface oscillation per radius ratio is shown for four consecutive time sections around the main collapse (marked by arrows). Data from two different parameter settings have been taken. First row: equilibrium radius $R_0=5.839\mu \text{m}$ at 1.35 bars; second row: $R_0=6.0\mu \text{m}$ at 1.35 bars. Note the alternating symmetry in the first row.

Figure 8

Maximal bubble radius times driving frequency at parametric instability threshold as a function of ratio of driving pressure over ambient pressure. Bubble oscillations below the line are parametrically stable. Calculations are shown for ambient pressures of $+/-1.4\text{kPa}$ Ar (squares, triangles) around 1 atm (circles) with a driving frequency of 20 (filled symbols), 14 (hollow squares), 24 (stars), and 30 kHz (hollow circles) at 1 atm. The boundary layer is $\delta =0$. The line shows a linear regression over all data points.

Figure 9

Maximal bubble radius times driving frequency as a function of normalized pressure for various data sets obtained from literature: crosses [60, 61], stars [62], filled circles [63], down triangles [16], up triangles [64], filled squares, own data, hollow circles [65]. Where indicated the ambient pressure (set to 1 atm if unknown) has been augmented by hydrostatic pressure guessed from published cell dimensions. The line is determined from Fig. 8.

Figure 10

Maximal bubble radius times driving frequency at parametric instability threshold as a function of normalized pressure. Data from various models and approximations are shown. Circles are calculated from the Gilmore bubble model using all equations specified earlier. Crosses denote the Keller-Miksis model (KM). The modified Rayleigh-Plesset bubble model (RPm) is shown with several changes applied to demonstrate the robustness of the approximation of a universal boundary. The line is determined from Fig. 8.

Figure 11

(Color online) Phase space spanned by driving pressure and maximal radius showing regions of Rayleigh-Taylor (RT), afterbounce (AB), and parametric (PM) instabilities. A region for high forcing amplitudes exists between AB1 and AB2 where none of the above instabilities exist.