Surface modes with controlled axisymmetry triggered by bubble coalescence in a high-amplitude acoustic field

When two bubbles encounter each other in a moderate ultrasound field they coalesce into a single bubble that shows purely spherical oscillations. For a sufficiently large acoustic field, this coalescence can lead to sustained nonspherical oscillations of the resulting bubble. We experimentally capture the time-resolved dynamics of the coalesced bubble, starting with the moment of film rupture. This allows the transient and steady-state regimes of the oscillating bubble to be studied. The amplitude dynamics for both of these regimes are successfully compared to numerical modeling, taking into account the coupling of volume and surface modes. Initial conditions for each surface mode are taken from the Legendre polynomial projection of the experimentally obtained bubble shape immediately after film rupture. We also observe that the symmetry axis of the zonal spherical harmonics is defined by the coalescence. The axis is identical to the rectilinear translational motion of the two approaching bubbles before impact. This high-amplitude coalescence technique provides a unique opportunity to study axisymmetric and sustained nonspherical bubble oscillations under controlled initial conditions.

Figure 1

(a) Schematic presentation of the experimental setup. (b) Visualization of the bubble trajectories before coalescence: a single bubble is trapped at a stable position in the acoustic field (top half); once a second bubble is approaching from the left, the two of them will be attracted towards each other due to secondary Bjerknes forces (bottom half, superposition of several photos). The moment of coalescence is not shown here but will be discussed in the following sections. Once the bubbles have coalesced, the new bubble will return to a stable position.

Figure 2

(a) Definition of the bubble shape $r_s\left(\theta \right)$. (b) The associated modal decomposition. This example corresponds to a bubble of $R_0=68.1\mu \mathrm{m}$ [one snapshot from the series in Fig. 3, frame size $220\mu \mathrm{m}\times 220\mu \mathrm{m}]$. The modal decomposition gives one single measurement point of the complete graph that will be shown in Fig. 7.

Figure 3

Extracts of two coalescing bubbles that result in a single bubble with different modes: (a) spherical mode, (b) mode 2, (c) mode 3, and (d) mode 4. The reference time of $0\mu \mathrm{s}$ corresponds to the first image after the film rupture, indicated by the red box. The frames presented here have a height and width of $220\mu \mathrm{m}$. The bubble sizes are summarized in Table  1. The corresponding videos can be found in the Supplemental Material [30].

Figure 4

Pressure-radius diagram showing the different modes of the coalesced bubbles in the established regime. $\circ$, spherical mode; $\times$, mode 2; $▵$, mode 3; $\square$, mode 4. The background colors correspond to the analytical solutions: white, spherical mode; blue, mode 2; green, mode 3; red, mode 4.

Figure 5

Temporal evolution of the radius $R\left(t\right)$ and the surface mode coefficients $a_1\left(t\right)$ to $a_4\left(t\right)$ extracted from the modal decomposition. For this bubble (for parameters, see Table 1, row a) only spherical oscillations remain in the steady-state regime, once the surface modes have decayed. Yellow line, simulation; blue line or points, experimental data. (a) The first $1.5\mathrm{ms}$ of the experiment, starting at the moment of film rupture $t=0\mathrm{ms}$. (b) During the steady-state regime (in the range $10\mathrm{ms}$ to $15\mathrm{ms}$), with 200 experimental points presented with respect to two acoustic periods $2T=2/f=0.064\mathrm{ms}$.

Figure 6

As for Fig. 5. This bubble (for parameters, see Table 1, row b) shows mode 2 oscillations in the steady-state regime.

Figure 7

As for Fig. 5. This bubble (for parameters, see Table 1, row c) shows mode 3 oscillations in the steady-state regime.

Figure 8

As for Fig. 5. This bubble (for parameters, see Table 1, row d) shows mode 4 oscillations in the steady-state regime.