Dynamics of nonspherical microbubble oscillations above instability threshold

Time-resolved dynamics of nonspherical oscillations of micrometer-sized bubbles are captured and analyzed using high-speed imaging. The axisymmetry of the bubble shape is ensured with certainty for the first time from the recordings of two synchronous high-speed cameras located at ${90}^{\circ }$. The temporal dynamics of finite-amplitude nonspherical oscillations are then analyzed for various acoustic pressures above the instability threshold. The experimental results are compared with recent theories accounting for nonlinearities and mode coupling, highlighting particular effects inherent to these mechanisms (saturation of the instability, triggering of nonparametric shape modes). Finally, the amplitude of the nonspherical oscillations is given as function of the driving pressure both for quadrupolar and octupolar bubbles.

Figure 1

Schematic view of the optical setup. The axis of symmetry of the nonspherical bubble is identified by the unit vector ${\mathbf{e}}_{\mathbf{b}}$.

Figure 2

Temporal evolution of the amplitude of the spherical mode ($a_0$) and the first three surface modes, for (a) a bubble of radius $48\mu \text{m}$ exhibiting a quadrupolar deformation, and (b) a bubble of radius $70\mu \text{m}$ exhibiting a octupolar deformation. All the modal coefficients are dimensionless. The time length corresponds to one period of modulation $f_m^{-1}=12.8\text{ms}$. (a) Quadrupolar mode. $R_0=48\mu \text{m}$, $P_a=18.4\text{kPa}$, $\eta =0.38$. (b) Octupolar mode. $R_0=70\mu \text{m}$, $P_a=8.6\text{kPa}$, $\eta =0.33$.

Figure 3

Bubble orientation and associated side views on both cameras. The axis of symmetry of the nonspherical bubble is identified by the unit vector ${\mathbf{e}}_{\mathbf{b}}$.

Figure 4

Snapshot series on two acoustic periods corresponding to case (a) in Fig. 3. Time between two successive snapshots is $5.55\mu \text{s}$.

Figure 5

Temporal evolution of the modal coefficients on ten acoustic periods corresponding to case (a) in Fig. 3, obtained from camera 1 (blue curves with circles) and camera 2 (green curves with squares).

Figure 6

Snapshot series on two acoustic periods corresponding to case (b) in Fig. 3. Time between two successive snapshots is $5.55\mu \text{s}$.

Figure 7

Temporal evolution of the modal coefficients on ten acoustic periods corresponding to case (b) in Fig. 3, obtained from camera 1 (blue curves with circles) and camera 2 (green curves with squares). The red solid curve on the top graph corresponds to the apparent spherical oscillations computed from camera 2 using Eq. (4).

Figure 8

Temporal evolution of the modal coefficients for a bubble of radius $R_0=44\mu \text{m}$ driven at mean acoustic pressure $P_a=22.5\text{kPa}$ and modulation amplitude $\eta =0.23$. The time length corresponds to one period of modulation $f_m^{-1}=12.8\text{ms}$. (a) Experiments. (b) Theory.

Figure 9

Comparison between theory (black curves) and experiments (blue curves with circles) at the acoustic time scale.

Figure 10

Shape modes amplitude as function of the acoustic pressure. The experimental results are represented by symbols for $n=2$ (blue circles), $n=3$ (green triangles), and $n=4$ (red squares). The error bars are only provided for the main modes for clarity; however, similar uncertainties are obtained for the other shape modes. The theoretical results are represented by curves with colors corresponding to modal coefficients and line styles corresponding to three values of bubble radius. (a) Quadrupolar bubble. $R_0\in [46–49]\mu \text{m}$. (b) Octupolar bubble. $R_0\in [70–73]\mu \text{m}$.