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.