Self-similar dynamics of air film entrained by a solid disk in confined space: A simple prototype of topological transitions

Abstract

In hydrodynamic topological transitions, one mass of fluid breaks into two or two merge into one. For example, in the honey-drop formation when honey dripping from a spoon, honey is extended to separate into two as the liquid neck bridging them thins down to micron scales. At the moment when topology changes due to the breakup, physical observables such as surface curvature locally diverges. Such singular dynamics have widely attracted physicists, revealing universality in their self-similar dynamics, which share much in common with critical phenomena in thermodynamics. Many experimental examples have been found, which include electric spout and vibration-induced jet eruption. However, only a few cases have been physically understood on the basis of equations that govern the singular dynamics and even in such a case the physical understanding is mathematically complicated inevitably involving delicate numerical calculations.Here, we study breakup of air film entrained by a solid disk into viscous liquid in a confined space, which leads to formation, thinning and breakup of the neck of air. As a result, we unexpectedly find that equations governing the neck dynamics can be solved analytically by virtue of two remarkable experimental features: only a single length scale linearly dependent on time remains near the singularity and universal scaling functions describing singular neck shape and velocity field are both analytic. The present solvable case would be essential for our better understanding of the singular dynamics and will help unveil the physics of unresolved examples intimately related to daily-life phenomena and diverse practical applications.The self-similar dynamics in hydrodynamics was already on focus in 1982, when the dynamics of viscous instability of a moving front was studied [1] and the renormalization group theory in statistical physics, which elucidates universality appearing in critical phenomena in thermodynamics, was recognized worldwide beyond fields [2]. In 1993, the self-similar dynamics was first studied for breakup of droplets (capillary pinch-off) [3], one typical example of topological transitions. In the seminal paper, the scaling ansatz, one of the key ideas leading to universality in critical phenomena, was shown to be useful. This paper ignited a surge of publication on the self-similarity in fluid breakup phenomena in 1993 and 1994 [4-7] In 1999, analytical solution for the dynamics of droplet coalescence, another typical example of hydrodynamic topological transitions, was published [8]. This is followed by many studies on the dynamics of droplets and bubbles, such as coalescence [9][10][11][12][13], non-coalescence [14], pinch-off [15,16], and droplet impact onto substrates [17][18][19][20][21][22], with a rapid technological advance in high-speed imaging and numerical computing. However, since then, the field has rather remotely been developed from the self-similar dynamics, while one of the main foci has been on simple scaling laws [23], which is another key concept in u...

A framework for crossover of scaling law : dynamical impact of viscoelastic board

“…I call this class similarity of the second class [43]. The variables of the second class normalizes the difference of the variables of the first class to integrate the single lines, which corresponds to the data collapse [44,45]. In the second class, self-similar variables represents the dynamics of energy involved in the process.…”