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.