The retraction of the edge of a planar liquid sheet

Abstract

Our objective is to investigate the motion of a free two-dimensional edge of a liquid sheet, which recedes and accumulates fluid as it is pulled back toward the bulk of the sheet due to surface tension. In the long time limit, the velocity of this edge reaches a constant value given by Taylor [Proc. R. Soc. London, Ser. A 253, 313 (1959)] and Culick [J. Appl. Phys. 31, 1128 (1960)], independent of the fluid viscosity. The way this value is reached, however, depends on the viscosity. In order to follow quantitatively the acceleration process, time-dependent numerical simulations of the two-dimensional Navier–Stokes equations have been performed. For inertia-dominated flow situations, the results show that the motion of the rim is relatively well characterized by a global momentum balance. However, if viscous effects are dominant, the rim is shown to extend far out into the film. In this case, the decrease in film thickness toward the undisturbed film can be described quite precisely by another one-dimensional model. By comparison of the individual contributions to the energy balances, following from numerical computation and from the global momentum balance, a closer look has also been taken on the inner mechanism of the rim formation. Finally, on the basis of the performed simulations, a stability criterion for a moving liquid curtain of finite length is derived, which determines whether a newly formed free rim leads to a break-up of the curtain.

Transverse instabilities of ascending planar jets formed by wave impacts on vertical walls

“…While the axial stretching of the rim, due to the radial crown expansion, decreases the amplitude of the perturbation [13], no such assumption can be made here as the planar sheet is initially ejected from the two-dimensional wave crest without any initial transverse stretching (Section 3 (f)). The longitudinal oscillation of the sheet thickness, whose amplitude decreases with the distance to the rim, have also been observed during retraction for low Ohnesorge number flows [30,31]. As yet the presence of any longitudinal oscillation has not been confirmed in the present experiments, however, the generation mechanism for the observed transverse variations in sheet thickness may be interpreted as a re-orientation of capillary waves on a thin stretched sheet [20].…”

The Elasticity of Soap Bubbles Containing Wormlike Micelles

“…However, for highly viscous films, an alternative explanation for the absence of a rim was obtained from a hydrodynamic analysis of the flow near the edge of purely viscous liquids. 25 It was shown that the rim formation is governed by a single parameter,…”

Viscous sheet retraction

“…With this simple calculation, we see the role of viscosity in the dynamics of retraction: it affects how the momentum is distributed through the film, but does not affect its terminal speed u c . As also pointed out in the two-dimensional numerical calculations of Sünderhauf et al (2002), in the long time limit, half of the surface energy is converted to kinetic energy, while the other half is ultimately dissipated through the action of viscosity.…”

“…Their results however were somewhat limited due to the short extent of the fluid sheet, and no conclusion could be drawn concerning the dynamics in the long time limit. More recently, Sünderhauf, Raszillier & Durst (2002) performed two-dimensional simulations of the Navier-Stokes equations, but neglected the ambient fluid on the basis of the prior work of Song & Tryggvason (1999). They focused primarily on exploring the acceleration phase of the film edge towards the terminal Taylor-Culick speed and provided some insights into the stability of falling liquid sheets.…”