Breakup of a thin air film surrounded by viscous liquid: derivation of self-similar solutions

Recently, we studied a viscous breakup of a fluid sheet and demonstrated an example of self-similar dynamics through agreement between experiment and theory [1]. A simple feature of this example is remaining of a single length scale near the breakup singularity. However, the governing equations have yet to be derived from the first principle. In this study, we successfully derive the self-similar solution from the Navier-Stokes equations, extending a method developed for the axisymmetric breakup [2]. Our case is clearly different from the axisymmetric case, since, as already noted, only a single length scale remains near the critical point. However, both cases are governed by the same principle: the scaling forms are determined in a way to avoid the divergence of dissipation near the critical point.
[1] Nakazato, Hana, Yuki Yamagishi, and Ko Okumura. Phys. Rev. Fluids (2018)
[2] Eggers, Jens. Rev. Mod. Phys. 69.3 (1997)