Displacement flows under elastic membranes. Part 2. Analysis of interfacial effects

Abstract

We investigate the injection of inviscid gas into the narrow liquid-filled gap between a rigid base plate and an overlying elastic sheet. After an early-time transient in which the gas deflects the sheet into a large blister, the viscous liquid displaced by the expanding bubble starts to accumulate in a wedge which advances as the elastic sheet peels away from the base. We analyse theoretically the subsequent interaction between viscous forces, elastic (bending or tension) forces and capillary forces. Asymptotic expressions are derived for the speed of spreading of the bubble, which reveal that the effect of the capillary pressure drop at the bubble tip is to suck down the sheet over the liquid wedge and thereby reduce the speed. We show that the system passes through three different asymptotic regimes in sequence. At early times, capillary effects are weak and hence the spreading of the bubble is controlled dominantly by the viscous-peeling process at the wedge tip. The capillary forces grow in importance with time, and at late times they dominate viscous effects and balance with elastic forces, leading to quasi-static spreading. Finally, at very late times, the capillary suction generates a narrow bottleneck at the wedge tip, which pushes a large ridge of liquid ahead of it. These results hold in the framework of standard lubrication theory as well as with an improved lubrication model, which takes into account films of wetting liquid deposited behind the advancing bubble tip. The predictions of the model are shown to be in excellent agreement with the Navier–Stokes simulations and experimental results from Part 1 of this work.