Partial persistence of memory in bubble breakup: incomplete universality acquired by broken symmetry

Abstract

When a water drop falls from a faucet, the drop is created with the formation of an axisymmetric constriction region, which thins down to breakup. Such formation of a fluid drop has been extensively studied as a representative of the singular dynamics widely observed in nature. The singular dynamics is often self-similar, i.e., shapes at different times collapsing onto a master curve after rescaling, and the self-similar dynamics has been categorized as either universal or non-universal: the master curve is independent of or dependent on the length scales that set the initial boundary conditions, as if memory is erased or retained. Here, we focus on the post-breakup dynamics and confine the system to break the axisymmetry, introducing three length scales, which leads to a third category of incomplete universality, where memory is partially retained: the master curve could be dependent on the smallest scale but independent of the other two scales. Affecting of only the smallest length scale on the master curve underscores the importance of scale separation for the emergence of universality. The present study suggests a promising direction for the study on the singular dynamics by exploring the symmetry.

Figures (6)

• Figure 1: (a) Experimental setup. A metal disk of thickness $D_{0}$ ($=2.0-3.5$ mm) and radius $R$ ($10~$to $15$ mm) falls in the cell of thickness $D$ ($3$ to $4.5$ mm) filled with a viscous liquid of kinematic viscosity $\nu$ (1 to 50 St). The disk entrains air into the liquid, which finally detaches from the disk. The difference between $D$ and $D_{0}$ defines the liquid film thickness $e$, indicated in the side view from the right edge of the front view. (b) Snapshots at breakup and after breakup illustrating the setting of axes for $(e,D_{0},R,\nu)=(0.5,3,10,1)$ in mm or St. $z=z_{c}$ is set to the origin of the $z$ coordinate.
• Figure 2: Snapshots of entrainment of air by a disk into liquid, leading to breakup of a sheet of air for $(R,D_{0},e,\nu)=(10,3,0.5,10)$ in mm or St. In the top panel, overall time development is shown. In the bottom magnified snapshots, the sharp tip at short times become rounded with time, with the thick vertical line in the middle indicates the border. The time label 0 ms corresponds to $t=t_{c}$ defined in the text.
• Figure 3: (a) $z_{m}$ vs. $t^{\prime}=t-t_{c}$, where $t_{c}$ is the critical time defined in the text for various parameters $(e,D_{0},R,\nu)$ in mm or St. The data marked with a star (*) in the legend are those obtained for a different $\Delta\rho$ by using a brass disk instead of a disk of stainless steel. (b) Distinct data collapse by Eq. (\ref{['eq1c']}). All the data in (a) are plotted on rescaled axes, based on Eq. (\ref{['eq1c']}), demonstrating a clear data collapse with a quasi scaling-crossover between the regimes reasonably well characterized by the exponent $\Delta^{\prime}\simeq0.8$ and 1/2, as indicated by the results of fitting where $y=z_{m}(t)/D$ and $x=\Delta\rho gRt^{\prime}/\eta$.
• Figure 4: (a1) to (a3) Temporal change of the interface $h(z,t)$ at $e=0.5$ mm for the parameter sets $(D_{0},R,\nu)=(3,10.0,5)$, $(1.0,10.0,10)$ and $(2.0,12.5,10)$ in mm or St. (b1) to (b3) The space-time dependent collapse of the shape by Eq. (\ref{['eq18a']}). The collapse near the tip persists for long time, while that away from the tip is observed only for short time.
• Figure 5: (a) The master curves universal (and highly reproducible) for change in $D_{0}$, $R$, and $\nu =\eta/\rho$ at $e=0.5$ mm (but non-universal for change in $e$ and $\Delta \rho$, as shown below in Fig. \ref{['Fig6']} (a3)). Shapes at earlier times in Fig. \ref{['Fig4']} are compared. The data marked as (N2) are basically obtained on a different day. (b) Right branches in (a) are shown with the axes interchanged, together with the results of fitting, where $y=h(z,t)R/z_{m}^{2}$ and $x=(z-z_{m})/z_{m}$.