A scaling relation for the capillary-pressure driven drainage of thin films

Hydrodynamic interactions between deformable particles such as drops or vesicles are an integral part of the rheology of emulsions and suspensions. In addition, the drainage of the thin film separating two colliding drops or vesicles is crucial for understanding the dynamics of coalescence or adhesion, which can lead to phase separation. However, despite several decades of study, this phenomenon is still not well understood and existing analytical theories do not agree quantitatively with experimental and numerical results. In this article, new scaling arguments are presented to analyze the drainage process, once the film becomes sufficiently thin. In particular, it is shown that the length over which the pressure varies in the film changes as the film drains, and follows a specific scaling relation. The mass balance in the film is then revisited in light of the new scaling for the pressure gradient. Numerical simulations are conducted to test the new scaling arguments and evaluate the revised mass balance. In the case of vesicles, they exhibit an excellent fit with the new scaling theory. The theory is also found to apply well to drops, but only when the flow inside the drops is determined predominantly by the flow in the thin film rather than by the ambient flow.