Retraction Dynamics of a Highly Viscous Liquid Sheet

Abstract

We study the one-dimensional capillary-driven retraction of a finite, planar liquid sheet in the asymptotic regime where both the Ohnesorge number $\mathrm{Oh}$ and the initial length-to-thickness ratio $l_0/h_0$ are large. In this regime, the fluid domain decomposes into two regions: a thin-film region governed by one-dimensional mass and momentum equations, and a small tip region near the free edge described by a self-similar Stokes flow. Asymptotic matching between these regions yields an effective boundary condition for the thin-film region, representing a balance between viscous and capillary forces at the free edge. Surface tension drives the thin-film flow only through this boundary condition, while the local momentum balance is dominated by viscous and inertial stresses. We show that the thin-film flow possesses a conserved quantity, reducing the equation of thickness to heat equation with time-dependent boundary conditions. The reduced problem depends on a single dimensionless parameter $\mathcal{L} = l_0 / (4 h_0 \mathrm{Oh})$. Numerical solutions of the reduced model agree well with previous studies and reveal that the sheet undergoes distinct retraction regimes depending on $\mathcal{L}$ and a dimensionless time after rupture $T$. We derive asymptotic approximations for the thickness profile, velocity profile, and retraction speed during the early and late stages of retraction. At early times, the retraction speed grows as $T^{1/2}$, while at late times it decays as $1/T^2$. An intermediate regime arises for very long sheets ($\mathcal{L} \gg 1$). During this phase, the retraction speed approaches the Taylor-Culick value. When $T \approx \mathcal{L}$, the speed undergoes fast deceleration from the Taylor-Culick speed to late-time asymptotics.