The effect of the dispersed to continuous-phase viscosity ratio on film drainage between interacting drops

The deformation and drainage of the film between colliding drops is studied numerically at small capillary numbers, small Reynolds numbers and a range of dispersed to continuous-phase viscosity ratios, ¿, covering the transition from partially-mobile to immobile interfaces. Two types of collision are considered: constant approach velocity and constant interaction force. The problem is solved numerically by means of a finite difference method for the equations in the continuous phase and a boundary integral method or finite-element method in the drops. The velocity profile in the gap between the drops is the sum of a uniform and a parabolic contribution, governed respectively by viscous forces within the dispersed and the continuous phases. Solutions to date concern the limiting cases of partially-mobile or immobile interfaces, in which either the parabolic or plug contribution is negligible. A transformation of variables then results in a universal set of governing equations. In the intermediate regime a transformed viscosity ratio, ¿*, enters these equations. In the constant-force case, the transformed drainage rate increases monotonically with ¿* and the final (rate-determining) stage of drainage is well described by a power-law dependence of the minimum film thickness on time, enabling compact analytical approximations to be developed for the drainage time. These expressions reduce to those in the partially-mobile and immobile limits for ¿*-values outside the range 1