  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.
