Three-dimensional stability of a thin film between two approaching drops

Abstract

In most simulations of the coalescence process, the rupture of the thin film between two drops is assumed to be axisymmetric. In this paper, we examine the possibility of a nonaxisymmetric rupture by carrying out a three-dimensional linear stability analysis of an axisymmetric thin film region. First, the effect of tangential interfacial velocity on the stability of a flat film is analyzed and a scaling analysis is provided to predict the dependence of the critical film thickness on the dimensionless parameters of the problem: the capillary number Ca, the dimensionless Hamaker constant A∗, and the viscosity ratio λ. Multigrid integration and implicit eigensystem solution techniques are used to simultaneously solve the boundary integral and film evolution equations for the disturbance shape and growth rate. The calculations show that a fixed-end disturbance, which decays to zero at the edge of the film, exhibits maximum instability in the axisymmetric mode. On the other hand, the first nonaxisymmetric mode is the most unstable for the class of free-end disturbances that approach the edge of the film with zero slope. Next, the stability calculation is interfaced with the full axisymmetric simulation of two drops approaching in a biaxial extensional flow in order to obtain the correct base-state film shape at each time interval. In this case, the thin film first becomes unstable to a nonaxisymmetric disturbance for both kinds of boundary conditions. The critical thickness from the stability calculation is compared with the critical film thickness obtained earlier from numerical calculations of the collision and interaction of a pair of fully axisymmetric drops and the effect of base-state curvature on film stability is investigated.