Perturbed breakup of gas bubbles in water: Memory, gas flow, and coalescence

The pinch-off of an air bubble from an underwater nozzle ends in a singularity with a remarkable sensitivity to a variety of perturbations. I report on experiments that break both the axial (i.e., vertical) and azimuthal symmetry of the singularity formation. The density of the inner gas influences the axial asymmetry of the neck near pinch-off. For denser gases, flow through the neck late in collapse changes the pinch-off dynamics. Gas density is also implicated in the formation of satellite bubbles. The azimuthal shape oscillations described by Schmidt et al. can be initiated by anisotropic boundary conditions in the liquid as well as with an asymmetric nozzle shape. I measure the $n=3$ oscillatory mode and observe the nonlinear, highly three-dimensional outcomes of pinch-off with large azimuthal perturbations. These are consistent with prior theory.

Figure 1

Air bubble pinch-off. The bubble is blown from a 6-mm-diameter circular hole cut in a plate that was carefully leveled. The plate reflects the image of the bubble. The bright spot in the dark bubble is due to back-lighting.

Figure 2

(a) Schematic top view of the experimental apparatus. Bubbles are blown slowly from a nozzle, back-lit, and imaged synchronously with two high-speed cameras. Several nozzles of different shapes and sizes may be fitted to the apparatus. (b) Variant of the gas supply of (a) for generating bursts. Gas is slowly released through a needle valve until the bubble reaches a preset state; a pressure sensor then triggers the release of the gas held in the reservoir.

Figure 3

Minimum neck radius $R_{\min }$ as a function of time until break-up $\tau$ for axisymmetric pinch-offs of ${\mathrm{N}}_2$ ($ˆ$, density 1.14 g/L, 3 events), low-pressure He ($\times$, 0.0411 g/L, 6 events), and SF${}_6$ ($+$, 5.63 g/L, 7 events). $t_{*}$, and thus $\tau$, is determined for each movie by fitting with Eq. (1) with $\alpha =0.56$ (dashed line). Only points below $R_{\min }~120$ $\mu$m were used for fitting. The consistency of these data show that the changing gas density does not strongly change the character of the singular dynamics over most of the collapse; however, SF${}_6$ points below 60 $\mu$m could not be fitted with an exponent of 0.56.

Figure 4

A pressure drop in the upper portion of the bubble is responsible for the upward gas flow. The pressure in the gas supply line is plotted as a function of the time $\tau$ before pinch-off is seen on video, for He at 0.17 g/L as it disconnects from a round, level, 6-mm-diameter nozzle. The pressure drops by 9.0 Pa in the final 10 ms of pinch-off. Pressure is relative to the value measured 400 ms after pinch-off, once vibrations of the residual interface have died down. The data for SF${}_6$ are not significantly different, except for a lag due to the slower speed of sound.

Figure 5

Influence of gas density on the vertical asymmetry of the neck. (a) Measurement of cone angles ${\theta }_u$ and ${\theta }_l$. For consistency and repeatability, cone angles are computed by fitting a straight line to the region of the profile between $7R_{\min }$ and $12R_{\min }$ away from the minimum. (b) Lower cone angle ${\theta }_l$ as a function of neck minimum radius $R_{\min }$, for three gas densities: lowest vacuum SF${}_6$ (5.5 g/L, circles), highest vacuum SF${}_6$ (1.5 g/L, squares), and highest vacuum He (0.039 g/L, triangles). Time proceeds from right to left; thus the neck grows more slender during collapse. (c) Vertical asymmetry ${\theta }_u-{\theta }_l$ as a function of gas density ${\rho }_g$, sampled at $R_{\min }=10$ $\mu$m (circles), 20 $\mu$m (diamonds), and 60 $\mu$m (triangles). The data suggest a small residual asymmetry that is not due to gas inertia. (d) Vertical asymmetry as a function of neck minimum radius $R_{\min }$ for the three gas densities shown in (b). Higher gas density gives rise to a growth in asymmetry late in collapse. Earlier, asymmetry appears to evolve from an initial condition at large $R_{\min }$ that is determined by the gas density, but with a trend that is independent of the gas.

Figure 6

Upward motion of the neck minimum is driven by a vertical gas flow as a bubble disconnects from a round, level, 6 mm-diameter nozzle. (a) Height of the neck minimum above the nozzle as a function of minimum radius $R_{\min }$ for He at 0.029 g/L ($•$) and SF${}_6$ at 5.7 g/L ($ˆ$). Increasing gas density causes an upward motion of the neck minimum, which for SF${}_6$ is more pronounced and begins at much larger $R_{\min }$. $\tau$ ranges from 12 to 0.1 ms over the course of this plot. (b) Height of the minimum as a function of $R_{\min }$ closer to pinch-off, showing a singular motion at late times that increases with gas density. $z_{\min }$ is compared with its extrapolated value at $R_{\min }=0$, $z_{\mathrm{pinch}}$, to make the curves’ different slopes more apparent. From top to bottom, the five curves correspond to He at 0.041 g/L, He at 0.16 g/L, SF${}_6$ at 1.6 g/L, ${\mathrm{N}}_2$ at 1.1 g/L, and SF${}_6$ at 5.6 g/L. The range of $R_{\min }$ plotted represents the final 100 $\mu$s of pinch-off.

Figure 7

(a) Horizontal neck cross sections showing azimuthal perturbations [7], and how they are measured. The length $(A-B)/2$, measured with two cameras, is the $n=2$ asymmetry $S_2$, and the average radius $\mathrm{R̄}=(A+B)/4$. The length $(C-D)/2$ is $S_3$, where $C$ and $D$ are measured from the geometric center of the neck, marked by a closed dot; here $\mathrm{R̄}=(C+D)/2$. In practice, $C$ and $D$ are determined by assuming that the geometric center is stationary during the pinch-off, which is valid for the leveled nozzle used here [6]. (b) Cross sections at a later time, showing the nature of the shape oscillations (not to scale). The midpoint of the $n=3$ shape, as seen by a camera looking toward the top of the page, is shifted to the right, even though its geometric center has not moved. The intermediate shape for each mode is a circle. (c) Nozzle orifice shapes used to generate these perturbations. (d) The two-plate nozzle, used to excite an $n=2$ oscillation by placing two transparent walls near a circular orifice.

Figure 8

Vibrational memory of $n=2$ azimuthal perturbations. The independent variable ${\mathrm{R̄}}_{\min }$ is the average radius of the neck at its minimum; time proceeds from right to left. Theoretical curves (continuous lines) are computed according to Schmidt et al. [7, 25], taking the initial condition at ${\mathrm{R̄}}_{\min }=128$ and 156 $\mu$m for (a) and (b), respectively. Representative error bars are shown on a fraction of the points for clarity. (a) The $n=2$ asymmetry $S_2$ oscillates due to a slot-shaped nozzle (see Fig. 7). $S_2>0$ corresponds to elongation perpendicular to that of the nozzle. 108 separate pinch-off events are plotted. (b) $S_2$ oscillates due to parallel vertical plates on each side of the nozzle. $S_2>0$ corresponds to elongation parallel to the walls. 34 events are plotted. The frequency is the same in both experimental configurations, but the phase is slightly shifted and the transient behavior is significantly different.

Figure 9

Minimum neck radius $R_{\min }$ as a function of time until breakup $\tau$ for pinch-off from a 6-mm-diameter circular nozzle ($ˆ$, 3 events); and minimum average radius ${\mathrm{R̄}}_{\min }$ of the $n=2$ oscillatory pinch-offs of Figs. 8a ($\times$, 12 events) and 8(b) ($•$, 4 events). All plotted events used ${\mathrm{N}}_2$ gas. $\tau$ was determined for each movie by fitting with the power law $R_{\min }\propto {\tau }^{0.56}$ (solid line). Only points below $R_{\min }~120$ $\mu$m were used for fitting.

Figure 10

Vibrational memory of an $n=3$ azimuthal perturbation due to a three-armed nozzle (see Fig. 7). (a) Vibration signal $S_3$ as a function of neck minimum radius ${\mathrm{R̄}}_{\min }$. $S_3$ is subject to an arbitrary offset (as in Fig. 7). To better show the consistent phase and amplitude of the 21 events plotted, a different offset is fitted to each event. For ${\mathrm{R̄}}_{\min }\gtrsim 50$ $\mu$m (shaded vertical line) the neck is sufficiently large compared to the perturbation that the cross section remains convex (see insets), and so $S_3$ may be accurately measured. To the left of this line, the cross section is concave and the measurement no longer corresponds to $S_3$. This change in geometry is because the perturbation amplitude remains nearly constant as the neck size decreases. (b) The valid portion of the $S_3$ data, with a theoretical curve as in Fig. 8. The initial condition was taken at ${\mathrm{R̄}}_{\min }=280$ $\mu$m. (c) Frames from one of the movies plotted in this figure. Due to the $n=3$ vibration, the apparent center of the neck oscillates from side to side in time. In the final frame, with ${\mathrm{R̄}}_{\min }=20$ $\mu$m, the neck shape above and below the minimum recapitulates this prior side-to-side motion.

Figure 11

(a) Horizontal translation of the neck minimum as a function of neck size $R_{\min }$ for a 6-mm-diameter nozzle leveled to within $0.{02}^{°}$ ($+$), and tilted by 0.1${}^{ˆ}$ ($▵$) and 0.75${}^{ˆ}$ ($ˆ$) (four movies each). Motion is away from the direction of tilt. Unlike that from oscillation, translation from tilting is monotonic. (b) By contrast, the signal from placing a single vertical wall 8 mm from the nozzle center ($•$, 14 movies) is weakly nonmonotonic, changing at the onset of inertial collapse ($~$400 $\mu$m). Here motion is initially toward the wall.

Figure 12

Varying burst pressure varies the oscillation parameters. Oscillation signal $S_2$ is plotted vs average neck radius at the minimum ${\mathrm{R̄}}_{\min }$. Each curve represents a burst pressure, labeled in units of kPa. Here ${\mathrm{R̄}}_{\min }$ has a linear scale because of the importance of behavior before the inertially dominated power-law collapse. At lower pressures each increase in the pressure advances the oscillation phase and increases its amplitude; this effect becomes more complicated and oscillations less sinusoidal at higher pressures. As in Fig. 8, $S_2>0$ corresponds to elongation perpendicular to that of the nozzle. In the lower left, the oscillation for quasistatic inflation (i.e., zero burst pressure) from Fig. 8a is plotted with its $S_2$ multiplied by $-10$ and offset for comparison. Increasing the burst pressure gradually changes the phase of the oscillation, and generally increases its amplitude. For each pressure, data points are plotted with a smoothed curve to aid the eye. The points for 0.35 kPa are from 15 pinch-off events; each other curve is from two.

Figure 13

Outcome of a small burst of ${\mathrm{N}}_2$ (0.35 kPa) from a slot-shaped nozzle, as seen simultaneously from a direction perpendicular to the slot (left) and parallel to it (right). In the left image there is a satellite bubble in the center of the hole. Scale bar is 0.5 mm.

Figure 14

(a) The morphology of Fig. 13 is explained by observing a much stronger burst (5.5 kPa) with the same orthogonal cameras, producing simultaneous images “1” and “2.” A similar geometry is formed but on a much larger scale, Note that the geometry has changed orientation. Scale bar is 0.5 mm. (b) Enlarged and contrast-enhanced detail from (a) shows that the “hole” is actually a $~$1-$\mu$m-thick sheet of gas, as indicated by the presence of Newton’s Rings. Right: 87 $\mu$s later, breakup of the thin sheet proceeds from left to right, leaving hundreds of satellite bubbles. (c) Horizontal cross section of the experiment, showing placement of the cameras used in part (a), and a neck cross section predicted by Turitsyn et al. [8] that exhibits smooth contact between opposite sides of the neck, consistent with formation of the thin sheet.

Figure 15

300 ns arc-flash exposures taken at different times in the course of two similar bursts from a slot nozzle. Scale bar is 0.5 mm. Left: Newton’s Rings show a thin sheet as in Fig. 14. Right: coalescence proceeds around the perimeter of the sheet, showing a process for producing a single, central satellite bubble as in Fig. 13, rather than many bubbles, as in Fig. 14.