Time-resolved velocity and pressure field quantification in a flow-focusing device for ultrafast microbubble production

Flow-focusing devices have gained great interest in the past decade, due to their capability to produce monodisperse microbubbles for diagnostic and therapeutic medical ultrasound applications. However, up-scaling production to industrial scale requires a paradigm shift from single chip operation to highly parallelized systems. Parallelization gives rise to fluidic interactions between nozzles that, in turn, may lead to a decreased monodispersity. Here we study the velocity and pressure field fluctuations in a single flow-focusing nozzle during bubble production. We experimentally quantify the velocity field inside the nozzle at $100\mathrm{n}\mathrm{s}$ time resolution, and a numerical model provides insight into both the oscillatory velocity and pressure fields. Our results demonstrate that, at the length scale of the flow-focusing channel, the velocity oscillations propagate at fluid dynamical timescale (order of $\mu \mathrm{s}$) whereas the dominant pressure oscillations are linked to the bubble pinch-off and propagate at a much faster timescale (order of $\mathrm{n}\mathrm{s}$).

Figure 1

(a) Schematic of the flow-focusing region with typical snapshot from a high speed recording and explanation of the geometry of the flow-focusing channel and its cross section. (b) Schematic of the experimental setup.

Figure 2

Visualization of the postprocessing. (a) More than 130 000 data points for the tracer particles presented along the channel profile. The total particle velocity $V_{xy}$ is indicated by the color code. The gray line indicates the border of the shadow region of the channel wall, where no particles can be detected. (b) Total velocity $V_{xy}$ and flow direction $\alpha$ for the two sections marked by the rectangles in (a). The colored bold lines correspond to the velocities and angles obtained through postprocessing for the 2D imaging plane. The thin, magenta line indicates a Poiseuille flow for comparison. (c) Reconstruction of the velocity in the cross section $x\approx 25\mu \mathrm{m}$, based on the assumption that the velocity is maximum in the channel center and that detected tracer particles are equally spaced in the $z$-direction for a given position $y$.

Figure 3

Comparison between numerical model and experimental results. (a) Snapshot for reference. (b) Average flow profiles for experiments and simulations without bubbles. (c) Average flow profiles for experiments and simulations with bubbles. The center region where bubbles pass has been left on purpose in order not to induce any bias through averaging. (d) Temporal evolution of the jet tip and detached bubble in the flow-focusing channel, experiments (upper halves) compared to the numerically obtained gas-liquid interface (black line in the lower half plots). The thin gray lines that are added at the rear of the numerically obtained bubbles indicate the shape that would result from a projection of the bubble for comparison to the experimental shape.

Figure 4

(a) Experimental bubble velocity $V_b$ at the end of the flow-focusing channel as a function of the corresponding mean flow velocity $\overline{V}=(Q_g+Q_l)/A$ for a large number of experimental bubble sizes and liquid flow rates. The color code shows the different liquid flow rates used during bubble production. (b) Flow rate ratio $V_b/\overline{V}$ as a function of the bubble radius normalized by the channel radius and corresponding model from Eqs. (1) to (5). No bubbles smaller than $R_{b,\min }$ could be formed in the present flow-focusing device and within the parameter space explored.

Figure 5

(a, b) Average velocity (upper half) and pressure fields (lower half) for the case (A) without bubbles and the case (B) with bubbles, respectively. Note that for (b) the area where bubbles pass has intentionally been left blank in order not to induce any ambiguities in the definition of the mean velocity value. (c) Average pressure along three different lines parallel to the $x$-axis without bubbles (black line) and with bubbles (red line), both at a total flow rate $Q_t=Q_l+Q_g=200\mu \mathrm{L}{\min }^{-1}$. Note that for the radial position $y=4\mu \mathrm{m}$ for the case with bubbles the time-averaged pressure includes moments where the gas phase is present, which leads to a slightly increased average pressure. As a reference, the minimum pressure appearing at about the channel end at $x\approx 30\mu \mathrm{m}$ was set to zero. (d, e) Pressure drop in the inlet region I as a function of liquid flow rate and gas flow rate, respectively [approx. $\mathrm{\Delta }{p}_{\mathrm{I}}$: Eq. (6)]. (f) Pressure drop the region II as a function of total flow rate [parabolic: Eq. (7); approx. $\mathrm{\Delta }{p}_{\mathrm{II}}$: Eq. (8)].

Figure 6

Numerical oscillatory velocity in the gas phase for $y=0$ and different positions $x$ in the flow-focusing channel. (a) Gas flow rate as a function of time. (b) Gas velocity for different positions inside the gas thread (position of pinch-off is $x\approx 5\mu \mathrm{m}$). Limitations of the incompressible simulations are discussed in the text.

Figure 7

Velocity and pressure oscillations in the flow-focusing channel. (a, b) Typical experimental and numerical particle trajectories highlight the time-dependent behavior at a given location. The time stamps correspond to the moment when the respective particles were passing the position indicated by the dots. (c) Velocity oscillations $V^{\prime }=V-\overline{V}$ for positions $y=5\mu \mathrm{m}$ and $x$ given in the figure. The colored lines present the numerical results and the dashed black line the corresponding experimental data. (d, e) Velocity and pressure map for two instants during the bubbling period $T$. (f, g) Velocity $V$ and pressure $p$, respectively, for a fixed radial position $y=6\mu \mathrm{m}$, different positions $x$, and different moments in time. The hatched areas correspond to the presence of a bubble in the respective channel sections $x$. The colored arrows in (f) indicate the positions in (c).