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 initial length-to-thickness ratio <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>l</a:mi> <a:mn>0</a:mn> </a:msub> <a:mo>/</a:mo> <a:msub> <a:mi>h</a:mi> <a:mn>0</a:mn> </a:msub> </a:mrow> </a:math> and a characteristic Ohnesorge number, Oh, which is inversely related to the Reynolds number of the flow, 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. We find that 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 governing equations to a one-dimensional free-boundary problem in which the sheet thickness satisfies the heat equation with time-dependent boundary conditions. The reduced problem depends on a single dimensionless parameter <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi mathvariant="script">L</b:mi> <b:mo>=</b:mo> <b:msub> <b:mi>l</b:mi> <b:mn>0</b:mn> </b:msub> <b:mo>/</b:mo> <b:mrow> <b:mo>(</b:mo> <b:mn>4</b:mn> <b:msub> <b:mi>h</b:mi> <b:mn>0</b:mn> </b:msub> <b:mtext>Oh</b:mtext> <b:mo>)</b:mo> </b:mrow> </b:mrow> </b:math> . Numerical solutions of the reduced model agree well with previous studies based on different formulations, including full Navier-Stokes simulations, and reveal that the sheet undergoes distinct retraction regimes depending on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi mathvariant="script">L</d:mi> </d:math> and a dimensionless time after rupture <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>T</f:mi> </f:math> . 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msup> <g:mi>T</g:mi> <g:mrow> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mn>2</g:mn> </g:mrow> </g:msup> </g:math> , characteristic of diffusive processes, while at late times it decays as <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mn>1</h:mn> <h:mo>/</h:mo> <h:msup> <h:mi>T</h:mi> <h:mn>2</h:mn> </h:msup> </h:mrow> </h:math> due to the finiteness of the sheet. Both results are consistent with previous findings. We also analyze an intermediate regime, not previously studied in detail, which arises for very long sheets ( <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi mathvariant="script">L</i:mi> <i:mo>≫</i:mo> <i:mn>1</i:mn> </i:mrow> </i:math> ). During this phase, the retraction speed approaches the classical Taylor-Culick value, despite the sheet not exhibiting a large circular rim as in classical studies of film rupture. When <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>T</k:mi> <k:mo>≈</k:mo> <k:mi mathvariant="script">L</k:mi> </k:mrow> </k:math> , the speed undergoes a sudden deceleration, which we characterize numerically, going from the Taylor-Culick speed to that predicted by the late-time asymptotics.