Retraction Dynamics of a Highly Viscous Liquid Sheet

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.

💡 Research Summary

The paper investigates the capillary‑driven retraction of a finite planar liquid sheet in the asymptotic limit where both the Ohnesorge number (Oh) and the initial length‑to‑thickness ratio (l₀/h₀) are large. In this regime the flow naturally separates into two distinct regions. The bulk of the sheet is described by a one‑dimensional thin‑film model that retains inertia, viscosity and, in principle, capillarity. Near the free edge a small “tip” region of order the sheet thickness is governed by the full two‑dimensional Stokes equations and exhibits a self‑similar solution. By performing matched asymptotic expansions the authors derive an effective boundary condition for the thin‑film region:
4 μ h ∂v/∂x = −γ at the tip.
This condition represents a balance between the viscous stress exerted by the bulk flow and the capillary force associated with the highly curved tip. It is valid only when Oh ≫ 1, i.e., when viscous forces dominate over inertia in the tip.

A scaling analysis shows that, within the thin‑film region, capillary stresses associated with thickness gradients are negligible compared with viscous stresses. Consequently the capillary term in the thin‑film momentum equation can be dropped, leaving only inertial and viscous contributions. Balancing these yields the visco‑inertial length and time scales
l_I = 4 h₀ Oh, t_I = 4 μ h₀/γ,
which are directly related to the classic Taylor‑Culick speed u_TC = γ/(ρ h₀). The only remaining dimensionless group is
ℒ = l₀/(4 h₀ Oh) = l₀/l_I,
which measures the sheet length in units of the visco‑inertial length.

The thin‑film equations, together with the kinematic condition at the moving edge and the no‑flow condition at the fixed end, possess a conserved quantity Q = Hv − ∂H/∂X (in nondimensional variables). Using the initial condition (uniform thickness, zero velocity) one finds Q = 0, which implies a linear relationship between velocity and the slope of the thickness profile. Substituting this relation reduces the coupled continuity and momentum equations to a single diffusion (heat) equation for the thickness:
∂H/∂T = ∂²H/∂X²,
with time‑dependent boundary conditions at the moving tip (derived from the viscous‑capillary balance) and at the fixed end (no slip, zero slope). An equivalent formulation in terms of velocity yields Burgers’ equation with similar boundary conditions.

Numerical integration of the reduced heat equation reproduces the full dynamics of the sheet for a wide range of ℒ. The results agree quantitatively with earlier full Navier–Stokes simulations (Brenner & Gueyffier 2000) and with other thin‑film studies (Savva & Bush 2013; Deka & Pierson 2020). The authors identify three distinct temporal regimes for the retraction speed u(T):

1. Early‑time diffusion‑dominated regime (T ≪ 1): The tip is driven by the viscous‑capillary balance, but the bulk flow spreads diffusively. The retraction speed grows as u ∝ T^{1/2}.

2. Intermediate regime for long sheets (ℒ ≫ 1, 1 ≪ T ≪ ℒ): The bulk flow adjusts quickly to the tip condition, and the speed approaches the constant Taylor‑Culick value u_TC, even though no pronounced rim forms. This demonstrates that the classic Taylor‑Culick speed can arise purely from the viscous‑capillary coupling at the tip, without a massive rim.

3. Late‑time deceleration (T ≈ ℒ): As the retracting edge nears the fixed end, the finite length of the sheet becomes important. The speed undergoes a rapid drop from u_TC to a much slower decay, following u ∝ T^{−2}. This deceleration reflects the global mass conservation constraint and the imposed zero‑velocity condition at the far end.

The paper also provides asymptotic approximations for the thickness and velocity profiles in each regime, confirming that the conserved quantity leads to simple analytic forms (e.g., Gaussian‑like thickness profiles in the diffusion stage). For ℒ ≫ 1 the intermediate regime is shown to be a long‑time “quasi‑steady” state, while for ℒ ≪ 1 the sheet never attains the Taylor‑Culick speed and decelerates almost immediately after rupture.

Overall, the work clarifies the physical roles of viscosity, inertia and surface tension in highly viscous sheet retraction. By isolating the tip region and rigorously matching it to the thin‑film bulk, the authors eliminate the need for ad‑hoc curvature regularizations that have plagued earlier thin‑film models. The reduction to a single diffusion equation with a single governing parameter ℒ offers a compact yet accurate framework for predicting sheet retraction in applications ranging from bubble bursting, ink‑jet printing, to pathogen spread in thin liquid films. The methodology—identifying a conserved quantity, performing asymptotic matching, and reducing to a simpler PDE—has potential relevance for other free‑surface problems involving thin films and moving contact lines.