Fig. 3.

Self-similarity of the neck profile. (A) The evolution of the bubble neck profile in time (data corresponding to $d=750\hspace{0.17em}\mu \mathrm{m}$, $\mu =1.4\hspace{0.17em}\mathrm{P}\mathrm{a}.\mathrm{s}$, and $\mathrm{C}\mathrm{a}=0.008$); blue and green symbols represent the data corresponding to the early and late-time self-similar regimes, and red symbols represent the transition between the two. (B) Scaling the neck profile with the minimum neck diameter collapses the data corresponding to the early-time self-similar regime, where $\tilde{R}\left(\xi \right)=\tilde{r}\left(\tilde{z},\tilde{\tau }\right)/{\tilde{\tau }}^{1/5}$ and $\xi =\left(\tilde{z}-{\tilde{z}}_0\right)/{\tilde{\tau }}^{1/5}$. The dashed line, overlaying the blue symbols corresponding to the early-time self-similar regime, represents the self-similar solution of the long-wave model (SI Appendix, section 2). The data corresponding to the late-time self-similar regime, however, deviate from the predictions of the long-wave model. (C) The definition of parameters used to characterize the bubble neck profile. (D) The evolution of the axial length scale defined as $\tilde{\zeta }=\sqrt{{\tilde{r}}_0{\tilde{r}}_c}$ versus time to pinch-off. In the early-time regime, $\tilde{\zeta }=\sqrt{{\tilde{r}}_c{\tilde{r}}_0}\sim {\tilde{\tau }}^{1/5}$, consistent with the predictions of the long-wave model. In the late-time regime, however, ${\tilde{r}}_0\sim \tilde{\tau }$ and $\tilde{\zeta }\sim {\tilde{\tau }}^{1/2}$, which indicates that the axial radius of curvature becomes constant, i.e., the neck profile becomes a parabola that simply translates in time (13, 42). (E) Scaling the axial length scale with the expressions obtained in D leads to the collapse of all bubble neck profiles during the entire pinch-off process (shown in A) onto a single parabolic curve: $r/r_0=1+{\left[\left(z-z_0\right)/\zeta \right]}^2$ (dashed line).