Breakup to non-breakup transition of air entrained into viscous liquid by a disk: analogy of the self-similar dynamics with critical phenomena

TL;DR

The study investigates self-similar pre-detachment dynamics of air entrainment by a disk into a viscous liquid in confinement. Using high-speed imaging, the authors identify three detachment regimes and reveal a breakup-to-non-breakup transition within the corn-forming regime, controlled by the lubricating film thickness $e$. They demonstrate that the breakup dynamics exhibit self-similarity with scaling exponents $\beta$, $\\Delta$, and $\\delta$ that depend on the continuous parameter $e$, and that the interface can be collapsed onto a master curve $h = h_m\\Gamma(z/z_m)$ with $\\Gamma(\\xi) \\sim \\xi^{1/\\delta}$ for large $\\xi$. This leads to the intriguing conclusion that an uncountably infinite set of universality classes may describe confined hydrodynamic singularities, inviting renormalization-group analyses and broader exploration of self-similar dynamics in confined geometries.

Abstract

Self-similarity in partial differential equations has been widely exploited to study many phenomena in physical sciences. We have studied the interfacial dynamics when air is entrained into viscous liquid by a disk in a confined geometry. In a previous study using an original experimental system, we found the sheet- and corn-forming regimes, in which a sheet and cone of air are respectively formed before air detaches from the disk. The sheet eventually breaks up but the corn, which appears when a bit more confined, does not. Here, we find a third regime, in which a corn eventually breaks up, by investigating different ranges of confining parameters: the transition from breakup to non-breakup can occur within the corn regime. Furthermore, with the data obtained in the third regime we deeply explore analogy with critical phenomena to find out that the counterpart of the critical exponents dependent on a length scale. Since the scale is a number not discrete but continuous, the present hydrodynamic analog suggests the existence of an uncountably infinite number of universality classes. The rich physics revealed in our study suggests a promising direction of the study of the self-similar dynamics: exploring analogy with critical phenomena, focusing on confined geometries in many natural and industrial phenomena.

Figures (6)

• Figure 1: (a) Experimental setup. A metal disk of thickness $D_{0}$ ($=1$ mm) and radius $R$ ($10~$to $12.5$ mm) falls in the cell of thickness $D$ ($2$ to $6$ mm) filled with a viscous liquid of kinematic viscosity $\nu$ (100 to 1000 cS). 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 $e$. (b) Snapshots just before and at breakup illustrating the setting of axes, in the case with topology change for $(R,D_{0},e,\nu)=(10,3,0.5,100)$ in mm or cS.
• Figure 2: Snapshots of entrainment of air by a disk into liquid, leading to detachment of air from the disk for $R=10$ mm, $D_{0}=1$ mm, $\nu=100$ cS. The liquid film thickness $e$ are 1, 1.5, and 2 mm, respectively, in (a) to (c). The time label 0 ms corresponds to $t=t_{d}$ defined in the text. In the three rightmost front-view photos reveal that a small bubble remains on the solid surface in (b) and (c) while such a bubble cannot be seen in (a): the breakup transition point lies between $e=$$1$ and 1.5 mm. Side-view shots near the detachment (0 ms) are separated by 6 ms, but are not synchronized with the front-view shots (a set of side-view snapshots are obtained from an experiment performed on a day different from the day on which the corresponding front-view snapshots but conducted for the same parameters).
• Figure 3: (a) Plots of $h_{m}$, $z_{m}$, and $z_{G}$ as a function of $t-t_{c}$ for $e=0.5$, 1, 1.5 and 2 mm at $D_{0}=1$ mm and $R=10$ mm, where $t_{c}$ is the critical time precisely defined in the text. In (a4), $h_{m}$ is comparable to $z_{m}$. (b) Plots in (a), regrouped and plotted on a log-log scale. Solid lines are obtained by fitting (using the data between the two crosses) with a function with a function $y=ax$ for $z_{G}$ and $y=ax^{b}$ for 2$h_{m}$ and $z_{m}$ (see the text for details). In (b3), the quantities $2z_{m}(t)$ and $4z_{m}(t)$ instead of $z_{m}(t)$ are plotted for $e=1.5$ and 2.0 mm, to avoid overlap with $z_{m}(t)$ for $e=1.0$ mm. (c) Renormalized plot of $z_{G}(t)$, where the dotted line represents Eq. (\ref{['eq1a']}) with $k=0.254$. As discussed below, the slope in (b2) [(b3)] determines the exponent $\beta$ [$\Delta$].
• Figure 4: Shape change for $e=1$, 1.5, 2.0, and 0.5 mm for $(R,D_{0},\nu)=(10,1,100)$ in mm or cS. Since the right and left interfaces are the mirror image of the other, only the right interface after averaging is shown. The error bars in capturing the interface are less than the size of markers. The time development of interfacial shape before [after] rescaling is respectively shown in (a1) [(a2) and (a3)] for $e=1$ mm. We see a better collapse for times not too close to $t=t_{c}$ (before $-9$ ms) [see (a3)]. The corresponding plots for $e=$1.5, 2.0, and 0.5 mm are shown respectively in (b) to (d). The data labeled $R^{\ast}$ in (a3) are obtained for different parameter set $(R,D_{0},\nu)=(12.5,1,100)$, which collapse well on the master curve [the good collapse seems limited for the upper branch (of our focus), i.e., the shape above the constriction point].
• Figure 5: Translated shape functions for $e=1.0,1.5,2.0$, and 0.5 mm on linear scales in the period of good collapse [(a1) to (d1)] and on log-log scales for $-21$ to $-30$ ms with a fitting line obtained by fitting the data in the region between the two black crosses for $-21$ ms [(a2) to (d2)]. For all cases, $(R,D_{0},\nu)=(10,1,100)$ in mm or cS.