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

Abstract

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, λ, 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, λ ∗ , enters these equations. In the constant-force case, the transformed drainage rate increases monotonically with λ ∗ 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 λ ∗ -values outside the range 10<λ ∗<10 3 . In the constant-velocity case the behavior is more subtle, drainage at the periphery of the film being strongly affected by the plug contribution in the adjoining outer region. This provides an explanation for the much higher final drainage rates predicted numerically under constant-velocity conditions in the partially-mobile case. From a practical point of view the most important case to model is that dividing coalescing from non-coalescing drop collisions. While the constant-force approximation is probably closest to the final interaction in this case, the sensitivity of the drainage behavior to the outer boundary conditions suggests that more realistic simulations are required which take account of the actual, time-dependent interaction force/velocity.