Approach to universality in axisymmetric bubble pinch-off

The pinch-off of an axisymmetric air bubble surrounded by an inviscid fluid is compared in four physical realizations: (i) cavity collapse in the wake of an impacting disk, (ii) gas bubbles injected through a small orifice, (iii) bubble rupture in a straining flow, and (iv) a bubble with an initially necked shape. Our boundary-integral simulations suggest that all systems eventually follow the universal behavior characterized by slowly varying exponents predicted by J. Eggers et al. [Phys. Rev. Lett. 98, 094502 (2007)]. However, the time scale for the onset of this final regime is found to vary by orders of magnitude depending on the system in question. While for the impacting disk it is well in the millisecond range, for the gas injection needle universal behavior sets in only a few microseconds before pinch-off. These findings reconcile the different views expressed in recent literature about the universal nature of bubble pinch-off.

Figure 1

(Color online) Illustration of the bubble collapse in the four different systems: (i) impacting disk, (ii) gas injection through a needle orifice, (iii) bubble in a straining flow, and (iv) initially necked bubble. Solid blue lines correspond to the free surface at pinch-off, while dashed and dotted black lines represent earlier bubble shapes. The disk and the needle are depicted in red (light gray).

Figure 2

(Color online) “Classical” plot of the neck radius versus the time to pinch-off (a) for system (i) with $\text{Fr}=5.1$ ($R_0=2\text{cm}$, $V_0=1\text{m}/\text{s}$), (b) for system (ii) in setup A, and (c) for systems (iii) and (iv) shown in dark gray (lower line) and magenta (upper line), respectively. The dashed line represents a slope of 1/2.

Figure 3

(Color online) Illustration of the cavity surface, the minimal neck radius $r_0$, and the local radius of curvature $r_c$.

Figure 4

(Color online) The aspect ratio $\gamma$ plotted as a function of the time to pinch-off $\tau$ for system (i) with $\text{Fr}=5.1$ (blue lower line) as well as system (ii) in setup A (red upper line). This shows that one can use $\gamma$ instead of $\tau$ as a measure for the approach to pinch-off.

Figure 5

(Color online) (a) The local exponent $\alpha$ as a function of the aspect ratio $\gamma$ for system (i) with $\text{Fr}=3.4$ (cyan, rightmost curve), $\text{Fr}=5.1$ (blue curve), $\text{Fr}=46$ (brown curve), $\text{Fr}=500$ (green curve), and $\text{Fr}=4000$ (black, leftmost curve). After an initial transient all curves follow the same universal regime. The dashed line is Eq. (2). The local maxima correspond roughly to the start of the universal regime. (b) The local exponent for system (ii) in the three configurations: A (red, dark gray curve), B (gray curve), and C (yellow, very light gray curve). All curves A–C lie practically on top of each other. (c) The local exponent for system (iii) in dark gray (upper curve) and system (iv) in magenta (light gray, lower curve) follows the same universal behavior close to pinch-off. Small jumps in the data are due to the crossover between different node positioning algorithms employed in the initial and the final stages of the simulation (see EPAPS document [25]), while the deviation of the numerical data away from the universal curve at the very end stems from the uncertainty in determining the exact time of closure.

Figure 6

(Color online) The local Froude number for system (i) and an impact Froude number $\text{Fr}=3.4$ (cyan, rightmost curve) and $\text{Fr}=4000$ (black) and the local Weber number for system (ii) with configuration A (red, dark gray) and B (gray, leftmost curve). The onset of the universal regime can be located roughly after the respective nondimensional quantities have become larger than order unity (horizontal dashed line).

Figure 7

(Color online) (a) Illustration of the characteristic aspect ratio of the cavity. (b) The averaged exponent measured in [4] (black diamonds) is reproduced very well by our model with the constant $K=0.46$ (red line).