The simultaneous effect of small deformation and short-range van der Waals attraction on the coalescence efficiency of two different-sized slowly sedimenting drops is considered. For spherical drops, it has been shown previously that the tangential mobility of drop surfaces makes collision possible even without van der Waals attraction; on the other hand, even a small amount of deformation precludes drops from coming into contact unless van der Waals attraction is accounted for. In the present work, the conditions are delineated when these two small-scale factors, acting in opposite directions, have a considerable combined effect on the coalescence efficiency. The problem is solved by matched asymptotic expansions valid for small capillary numbers (Ca). The outer solution, for two spherical drops moving in apparent contact without van der Waals attraction, determines the contact force as a function of time. This force is used as the driving force for the inner solution of the relevant integro-differential thin-film equations (coupling the flow in the small-gap region to that inside the drops) to determine whether coalescence occurs during the apparent contact motion. The initial gap profile for the inner solution is provided by matching with the outer trajectory for spherical drops approaching contact.
The analysis shows that, for Ca[Lt ]1, the near-contact deformation is mainly axisymmetric, greatly simplifying the inner solution; nevertheless, determination of the critical horizontal offsets leading to coalescence and the parametric analysis are computationally very intensive. To facilitate these tasks, a substantially new, highly efficient, and absolutely stable numerical method for solving stiff thin-film equations is developed. Unlike for spherical drops, when the upstream intersection area is a circle, the existence of a second coalescence zone for deformable drops is found over much of the parameter space. Results are mapped out for a range of four dimensionless parameters (capillary number, size and drop-to-medium viscosity ratios, dimensionless Hamaker parameter). As a physical application, predicted coalescence efficiencies are shown for a system of ethyl salicylate drops in diethylene glycol.
The present solution extends the range of drop sizes where the coalescence efficiencies are known theoretically and can be used in drop population dynamics. Comparison with full three-dimensional boundary-integral calculations for deformable drops without van der Waals attraction is also made to demonstrate that, when the drop-to-medium viscosity ratio is of the order of unity, the present asymptotic approach is valid in a wide range of small and moderately small capillary numbers.