Trapping and exclusion zones in complex streaming patterns around a large assembly of microfluidic bubbles under ultrasound

Pulsating bubbles have proved to be a versatile tool for trapping and sorting particles. In this article, we investigate the different streaming patterns that can be obtained with a group of bubbles in a confined geometry under ultrasound. In the presence of an external flow strong enough to oppose the streaming velocities but not drag the trapped bubbles, we observe either the appearance of exclusion zones near the bubbles or asymmetric streaming patterns that we interpret as the superposition of a two-dimensional (2D) streaming function and of a potential flow. When studying a lattice of several bubbles, we show that the streaming pattern can be accurately predicted by superimposing the contributions of every pair of bubbles present in the lattice, thus allowing one to predict the sizes and the shapes of exclusion zones created by a group of bubbles under acoustic excitation. We suggest that such systems could be used to enhance mixing at a small scale or to catch and release chemical species initially trapped in vortices created around bubble pairs.

Figure 1

(a) Scheme of the experimental setup showing the water tank, the microfluidic device where the bubbles are created and studied, the external acoustic excitation, and the microscope used for visualization. (b) Side-view detail showing the PDMS microchannel, with two bubbles trapped on micropits imprinted in the upper wall. (c) Streaming fountain pattern around two vibrating bubbles without external flow. In this experiment, the distance between the two bubbles is $150\mu \text{m}$, the scale bar is $100\mu \text{m}$, and the excitation frequency is $91\text{kHz}$. The flow is fed with $2\text{−}\mu \text{m}$ tracers in order to visualize the trajectories. Arrows indicate the direction of motion of the particles.

Figure 2

Modification of the streaming pattern produced by two excited bubbles under an external flow. Flow is applied from the left with the following velocities: (a) $U_{\infty }/U_{\mathrm{streaming}}=0.0874$, (b) $U_{\infty }/U_{\mathrm{streaming}}=0.186$, and (c) $U_{\infty }/U_{\mathrm{streaming}}=0.364$. The total flow is visualized using tracers $2\mu \text{m}$ in diameter. Their positions are superimposed to emphasize the streamlines. The same bubble pair was used over all three experiments, with constant acoustical pressure and frequency $\left(90\text{kHz}\right)$. The scale bar corresponds to $100\mu \text{m}$ and the red arrows materialize the stagnation points.

Figure 3

Scheme of notations used in the theory. For a group of several bubbles (two bubbles here), the streaming is always calculated with respect to the generating bubble (here bubble 1). The notations are those used in this scheme. When ${\theta }_{\mathrm{dip}}=\pi$, a fountain-type dipolar field is created.

Figure 4

Influence of an external flow on the streaming pattern produced by two bubbles. Plain hollow arrows represent the equivalent hydrodynamic dipoles induced by the streaming and dashed arrows represent the hydrodynamic dipoles induced on the bubbles by the external flow (arrows not at the same scale). Flow is coming from the left with the same ratio of external flow to streaming velocity than Fig. 2: (a) $U_{\infty }/U_{\text{streaming}}=0.0874$, (b) $U_{\infty }/U_{\text{streaming}}=0.186$, and (c) $U_{\infty }/U_{\text{streaming}}=0.364$. These values were chosen with the external flow values of Fig. 2 and assuming $U_{\text{streaming}}=6.18$ mm/s. The scale bar is $100\mu \text{m}$ and the color bar represents values of isolines of the stream function in ${\text{m}}^2{\text{s}}^{-1}.$

Figure 5

(a) Experimental exclusion zone superimposed with theoretical prediction from Ref. [9]. The excluded zone is the zone within the red curve. The external flow is coming from the left. (b) Experimental distance of the “stop point” as a function of the external flow. Curve is a fit using Eq. (7), giving $U_{\text{streaming}}=6.18\text{mm}/\text{s}$. Errors bars for $U_{\infty }$ are the standard deviation of the speed of particles outside the streaming flow. Errors bars for $r/R_0$ take origin in the drift of the bubbles' radii during the experiment. All points were acquired during the same experiment, varying the external flow $U_{\infty }$ from 10 to $0.5\mathrm{mm}/\mathrm{s}$. The excitation frequency is $90\text{kHz}$. (c) Numerical excluded surface, normalized by the bubble surface: Black dots are the results of numerical simulations whereas the red line is the excluded surface in the case of a circular excluded zone with the radius given by Eq. (7).

Figure 6

Influence of an increasing external flow on the streaming pattern produced by two bubbles oriented at 45 and ${90}^{\circ }$. Same ratios of external flow to streaming velocity in Figs. 2 and 4: (a) $U_{\infty }/U_{\text{streaming}}=0.0874$, (b) 0.186, and (c) 0.364. Dashed arrows represent the dipole induced for each bubble by the external flow whereas plain arrows represent dipoles induced by the acoustic excitation. The scale bar is $100\mu \text{m}$ and the color bar represents values of isolines of the stream function in ${\text{m}}^2{\text{s}}^{-1}$.

Figure 7

Comparison of the model with experiments for more than two bubbles in the absence of external flow. The top images are experimental results and the bottom images are numerical results according to the theory developed at the beginning of this section. (a) Three bubbles under ultrasonic excitation without external flow (excitation frequency is 91 kHz). The bold plain arrow represents the total streaming dipole for one bubble, which can be decomposed in the two dipoles induced by each neighbors represented by the thin plain arrows. (b) Seven bubbles under ultrasonic excitation (excitation frequency is 53 kHz).

Figure 8

Streaming around a group of three bubbles undergoing an external flow: comparison between experiments (top) and the model represented in this section. The external flow is applied from the left with the following velocities: (a) $U_{\infty }=0.295\mathrm{mm}/\mathrm{s}$ and (b) $U_{\infty }=1.474\mathrm{mm}/\mathrm{s}$. The simulation was fitted to the experiments by adjusting the streaming velocities: for (a) $U_{\text{streaming}}=8.4\mathrm{mm}/\mathrm{s}$ and for (c) $U_{\text{streaming}}=5.6\mathrm{mm}/\mathrm{s}$. In addition to the dipolar flows already present in Fig. 7, there is a dipolar flow produced by the interaction between the external flow and the bubbles represented by the dashed arrow. Excitation frequency is 89 kHz.

Figure 9

Left: A “pinball” of microfluidic bubbles. The flow is coming from the right with a velocity $U_{\infty }=304\mu \mathrm{m}/\mathrm{s}$. Right: Model described in Sec. 4a, using the acoustic pressure amplitude $P_{ac}$ as the only adjusting parameter, showing qualitative agreement with the experiment.