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In hydrodynamic topological transitions, one mass of fluid breaks into two ortwo merge into one. For example, in the honey-drop formation when honeydripping from a spoon, honey is extended to separate into two as the liquidneck bridging them thins down to micron scales. At the moment when topologychanges due to the breakup, physical observables such as surface curvaturelocally diverges. Such singular dynamics have widely attracted physicists,revealing universality in their self-similar dynamics, which share much incommon with critical phenomena in thermodynamics. Many experimental exampleshave been found, which include electric spout and vibration-induced jeteruption. However, only a few cases have been physically understood on thebasis of equations that govern the singular dynamics and even in such a casethe physical understanding is mathematically complicated inevitably involvingdelicate numerical calculations. Here, we study breakup of air film entrainedby a solid disk into viscous liquid in a confined space, which leads toformation, thinning and breakup of the neck of air. As a result, weunexpectedly find that equations governing the neck dynamics can be solvedanalytically by virtue of two remarkable experimental features: only a singlelength scale linearly dependent on time remains near the singularity anduniversal scaling functions describing singular neck shape and velocity fieldare both analytic. The present solvable case would be essential for our betterunderstanding of the singular dynamics and will help unveil the physics ofunresolved examples intimately related to daily-life phenomena and diversepractical applications.

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