Axisymmetric Bubble Pinch-Off at High Reynolds Numbers

Analytical considerations and potential-flow numerical simulations of the pinch-off of bubbles at high Reynolds numbers reveal that the bubble minimum radius, $r_n$, decreases as $\tau \propto r_n^2\sqrt{-\ln r_n^2}$, where $\tau$ is the time to break up, when the local shape of the bubble near the singularity is symmetric. However, if the gas convective terms in the momentum equation become of the order of those of the liquid, the bubble shape is no longer symmetric and the evolution of the neck changes to a $r_n\propto {\tau }^{1/3}$ power law. These findings are verified experimentally.

Figure 1

Time evolution of the pinch-off region of a bubble, $\Lambda =1.2\times {10}^{-3}$, at $We={\widehat{\rho }}_l{\widehat{U}}_l^2\widehat{R}/\widehat{\sigma }=12$. In both cases, the number of discrete points along the interface (513) were rearranged at every time step in order for them to be equispaced. (a) Symmetric breakup ($z_0=0$) and (b) asymmetric case ($z_0=0.25$). Note that, while in (a) the radial scale is stretched more rapidly than the axial one, (b) suggests that a self-similar solution could be reached near the pinch-off time if the Kelvin-Helmholtz (shear) instabilities were prevented.

Figure 2

Time evolution of the radius of the neck, $r_n(\tau )$, for different values of $z_0$ and $\Lambda$ (otherwise stated, $\Lambda =1.2\times {10}^{-3}$ at $We={\widehat{\rho }}_l{\widehat{U}}_l^2\widehat{R}/\widehat{\sigma }=12$). Notice that the transition from the approximately $1/2$ to the $1/3$ power law is delayed as $\Lambda$ and/or $Q_g$ decrease. The results given for $z_0=0.25$, $\Lambda =1.2\times {10}^{-4}$ ($Q_g\simeq 0.59$, $q\simeq 0.197$, $r_{\mathrm{tran}}\simeq 3.3\times {10}^{-2}$) and those given for $z_0=0.1$, $\Lambda =1.2\times {10}^{-3}$ ($Q_g\simeq 0.26$, $q\simeq 0.21$, $r_{\mathrm{tran}}\simeq 4.3\times {10}^{-2}$) are almost identical since the values of $r_{\mathrm{tran}}$ are nearly the same in both cases. The dashed line represents $\tau =2.2\times r_n^2\sqrt{-\ln r_n^2}$.

Figure 3

Time evolution of $Q_g$ (dashed lines) and $v_g$ (solid lines) for $z_0=0.25,0.1$, $\Lambda =1.2\times {10}^{-3}$, and $We=12$. It can be observed that, whereas $v_g$ increases almost two decades, $Q_g$ remains nearly constant during the breakup process.

Figure 4

Close-up views showing two different sequences of the bubble pinch-off corresponding to the experimental conditions of Fig. 5. The air and water flow rates are $Q_a=610\mathrm{ml}/\min$, $Q_w=4\mathrm{l}/\min$ in photographs (a)–(d), and $Q_a=850\mathrm{ml}/\min$, $Q_w=6\mathrm{l}/\min$ in (e)–(h). The time interval between the shown frames is $\Delta t=100\mu \mathrm{s}$ in (a)–(d), and $\Delta t=100/3\mu \mathrm{s}$ in (e)–(h). The spatial resolution is, in all cases, $18.6\mu \mathrm{m}/\mathrm{\text{pixel}}$, and the movies were recorded with a shutter time of $1/30000{\mathrm{s}}^{-1}$. Gas and liquid Reynolds numbers based on the needle diameter are, for both experimental conditions, of the order of $\sim O({10}^3)$ and $\sim O({10}^4)$, respectively.

Figure 5

Experimental results for the evolution of ${\widehat{r}}_n$ with $\widehat{\tau }$ in a coflowing air-water jet. The gas is injected through a needle (inner and outer radii 0.419 and 0.635 mm, respectively) placed coaxially within a liquid jet discharging from a 4 mm radius circular nozzle. $Q_a$ and $Q_w$ denote the air and water flow rates, respectively. A detailed description of the experimental facility is given in 10.