On retraction of a ruptured film and the viscous effects

Abstract

A numerical study is conducted on film retraction (or hole expanding) after the film is ruptured. A finite element method coupled with phase field model (governed by the Cahn-Hilliard equation) is applied. It is found that our method can successfully reproduce the film retraction process. Two modes of the retraction are found depending on viscous effect. When the viscous effect is small (mode II), a “ring-like” ridge appears at the end of the film. Between the ridge and the undisturbed part of the film, there is a necking region. When the viscous effect is strong, there are no necking regions (mode I), and the film thickness decreases gradually from the ridge. The two modes can be characterized by Ohnesorge number, with the critical value around 0.1. Our results show that the neck region thinners linearly in mode II in the early age. A very rough estimation of the breaking time can be obtained by extrapolating the measured data at around t B = 16 ρ l e 3 / γ . In the expression, ρ l , e , γ denote the liquid density, half of the film thickness, and the interfacial tension, respectively. In a later stage, the concave interface leads to a capillary pressure pointing outward, and further decelerates the thinning process. During this stage, the capillary number, denoting the ratio between the viscous stress and capillary force, is greatly smaller than unity. On the other hand, we also note that the previous theoretical model, which predicts the expanding speed of the hole, mainly concerns the later retraction stage. In the stage, the entrained liquid volume from the film into the ridge can be neglected. We modified the theory with considering of the ridge growth in early stage. The prediction is consistent with the numerical results very well. The analyses show that the viscous effects play subtle roles during the film retraction process, even when Ohnesorge number is very small: half of the released surface energy is dissipated.