Retraction Dynamics of a Highly Viscous Liquid Sheet

TL;DR

This work analyzes the capillary-driven retraction of a highly viscous, long planar liquid sheet by introducing a two-region asymptotic framework: a thin-film region where capillary stresses are negligible and a small tip region governed by Stokes flow. An effective edge boundary condition derived from matching the two regions reduces the governing equations to a one-dimensional heat equation for the film thickness, with a single dimensionless parameter $\mathcal{L} = \frac{l_0}{4 h_0 \mathrm{Oh}}$ controlling the dynamics. The thickness-velocity relation yields a conserved quantity that further reduces the problem, enabling closed-form asymptotics for early times ($dX_c/dT \sim 2\sqrt{T/\pi}$), late times ($dX_c/dT \sim \mathcal{L}/(1+T)^2$ for finite sheets, and $dX_c/dT \to 1$ for infinitely long sheets), and an intermediate regime where the speed approaches the Taylor–Culick value. Numerical solutions agree with prior studies and Navier–Stokes simulations, providing a unified picture of retraction regimes and a route to apply the Stefan-like reduced model to viscous-sheet rupture problems in various geometries.

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.

Figures (5)

• Figure 1: Schematic of a ruptured planar thin sheet. A flat surface across the $x$–$z$ plane is fixed at $x = l_0$ and ruptured along the line $x = 0$. As time $t$ progresses, the free edge moves to $x = x_c(t)$. The half-thickness along the $y$-axis is given by $h(x, t)$ and shown as a continuous blue curve.
• Figure 2: Numerically computed profiles of the thickness $H(X,T)$ and velocity $V(X,T)$ are shown for $\mathcal{L} = 0.1$, $\mathcal{L} = 1$, and $\mathcal{L} = 20$ at various times. In panels (a–d), the circles indicate the asymptotic predictions for $H$ and $V$ given by Eqs. \ref{['eq: H short expansion']} and \ref{['eq: V asympt short']}. In panels (e) and (f), the circles correspond to the asymptotic formulae in Eqs. \ref{['eq: wkb text']} and \ref{['eq: velocity profile']}, while the squares in the insets show the results from Eqs. \ref{['eq: H short expansion']} and \ref{['eq: V asympt short']}.
• Figure 3: Time evolution (blue line) of the retraction speed $d X_c/dT$ for (a) $\mathcal{L}=0.1$, (b) $\mathcal{L}= 1$, and (c) $\mathcal{L}=20$. The dashed red line is the early-time asymptotic Eq. \ref{['eq: xc asympt small t']}. The yellow dotted-dashed line is the late-time asymptotics Eq. \ref{['eq: u asymp short']}. The purple dotted line in (c) is the infinite-sheet asymptotics Eq. \ref{['eq: infinite asymptotics']}.
• Figure 4: The maximum value of $d X_c/dT$ reached by the thin film during its retraction for different $\mathcal{L}$. The dashed red line is the maximum of Eq. \ref{['eq: u asymp short']}, which is simply given by $\mathcal{L}$, and the dotted yellow line is the Taylor-Culick velocity \ref{['eq: taylor culick speed']}.
• Figure 5: For large $\mathcal{L}$, the sheet undergoes a rapid deceleration during retraction, crossing over from the Taylor--Culick speed to the regime described by Eq. \ref{['eq: u asymp short']}. (a) Crossover time $T_c$ when this deceleration occurs, and (b) duration $\Delta T_c$ of this transition, shown for various values of $\mathcal{L}$. The dashed red line represents $T_c = \mathcal{L}$.