In-phase synchronization between two auto-oscillating bubbles

Acoustically driven bubbles are nonlinear oscillators showing a wide range of behaviors such as period-doubling bifurcations, deterministic chaos, and synchronization to an external signal. Here we demonstrate that bubbles driven with a constant heat source can couple with each other and yield in-phase synchronization, even in the absence of an external sound field. Besides perfect synchronization, we resolve the parameter space where independent oscillations and bubble merging occur. The main finding that only weakly driven oscillating bubbles synchronize provides a path for experimental scale-up to study spatiotemporal synchronization. In the wider scope the findings are relevant for heat transfer applications from structured heaters where complex multibubble oscillations are expected.

Figure 1

Sketch of the experiment to generate and monitor two auto-oscillating bubbles on two closely placed electrical microheaters.

Figure 2

Full experimental setup that studies the system of two oscillate-boiling bubbles. The black arrows represent signal transmission and the red lines represent power supply connections. The scale bar in the heater's top view represents $20\mu \mathrm{m}$.

Figure 3

Two synchronized oscillate-boiling bubbles. Here the distance between heaters is $40\mu \mathrm{m}$. The power of the left and the right heater is 34.9 and 36.0 mW, respectively. (a) Captured images before and during the time when two bubbles synchronize. A digital filter was applied to remove dirt in the background. Refer to video 2 in [24] for the raw data. The time between two consecutive frames is $3.33\mu \mathrm{s}$. The scale bar shows $20\mu \mathrm{m}$. (b) Spectrogram of the acoustic signal emitted from two synchronized oscillate-boiling bubbles. (c) Distance between the bubbles' centers of mass and their effective radii retrieved from the recording. (d) Correlation coefficient between the effective radii of the two bubbles, calculated with a window time of $100\mu \mathrm{s}$, i.e., 30 frames of the recordings.

Figure 4

Two stable oscillate-boiling bubbles (see video 1 in [24]). Here the distance between two heaters is $50\mu \mathrm{m}$. The power of the left and the right heater is 25.4 and 25.1 mW, respectively. (a) Captured images of the bubble dynamics. The sizes of the pinch-off bubbles are close to the optical resolution, so they are barely noticeable. The time between two consecutive frames is $3.33\mu \mathrm{s}$. The scale bar represents $20\mu \mathrm{m}$. (b) Spectrogram of the acoustic signal emitted from the two bubbles. (c) Distance of the bubbles' centers of mass and their effective radii retrieved from the recording during the period marked in (b). The inset shows a close-up of the data in the black rectangle.

Figure 5

Two bubbles grow and merge. Here the distance between two heaters is $40\mu \mathrm{m}$. The power of the left and the right heater is 48.0 and 47.7 mW, respectively. (a) Captured images when two bubbles grow and merge (see video 3 in [24]). The time between two consecutive frames is $33.3\mu \mathrm{s}$. The scale bar shows $20\mu \mathrm{m}$. (b) Spectrogram of the acoustic signal emitted from two growing bubbles. (c) Distance of the bubbles' centers of mass and their effective radii retrieved from the recording.

Figure 6

Parameter space of two nearby oscillate-boiling bubbles as a function of different heating powers and distances between heaters. All data points share the same error bar, which is plotted separately. The classification region is constructed using the $k$-nearest-neighbor algorithm [25] with $k=5$.

Figure 7

Numerical calculation of two closely placed bubbles, both operating in (a) stable oscillate-boiling mode with liquid jet impact and (b) unstable oscillate-boiling mode without liquid jet impact. Here the power supplied to bubble 1 is $34\mathrm{mW}$ and to bubble 2 is $30\mathrm{mW}$ and their distance is $40\mu \mathrm{m}$.

Figure 8

Temperature distribution of the substrate at 1 ms after the current is triggered. Here each heater is operated at 35 mW and placed at four different distances following the experiment. (a) and (b) Top view of the thermal profile of the substrate with the longest and shortest distances. (c)–(f) Temperature distribution along the white line with different distances of the heaters.

Figure 9

Spectrograms of the (a) emitted acoustic signal, (b) temperature of the left heater, and (c) temperature of the right heater when two bubbles oscillate independently. These results are from the same experiment as in Fig. 4.

Figure 10

Spectrograms of the (a) emitted acoustic signal, (b) temperature of the left heater, and (c) temperature of the right heater when two bubbles synchronize. These results are from the same experiment as in Fig. 3.

Figure 11

(a) Power profile used in two cases: with liquid jet impact (stable oscillate boiling) and without liquid jet impact (unstable oscillate boiling). (b) Corresponding calculated bubble dynamics.

Figure 12

Geometry of the simulation for the thermal profile of the heating substrates.