Buoyancy-driven interactions between two deformable viscous drops

Abstract

Time-dependent interactions between two buoyancy-driven deformable drops are studied in the low Reynolds number flow limit for sufficiently large Bond numbers that the drops become significantly deformed. The first part of this paper considers the interaction and deformation of drops in axisymmetric configurations. Boundary integral calculations are presented for Bond numbers ℬ = Δρga2/σ in the range 0.25 ≤ ℬ < ∞ and viscosity ratios λ in the range 0.2 ≤ λ ≤ 20. Specifically, the case of a large drop following a smaller drop is considered, which typically leads to the smaller drop coating the larger drop for ℬ [Gt ] 1. Three distinct drainage modes of the thin film of fluid between the drops characterize axisymmetric two-drop interactions: (i) rapid drainage for which the thinnest region of the film is on the axis of symmetry, (ii) uniform drainage for which the film has a nearly constant thickness, and (iii) dimple formation. The initial mode of film drainage is always rapid drainage. As the separation distance decreases, film flow may change to uniform drainage and eventually to dimpled drainage. Moderate Bond numbers, typically ℬ = O(10) for λ = O(1), enhance dimple formation compared to either much larger or smaller Bond numbers. The numerical calculations also illustrate the extent to which lubrication theory and analytical solutions in bipolar coordinates (which assume spherical drop shapes) are applicable to deformable drops.The second part of this investigation considers the 'stability’ of axisymmetric drop configurations. Laboratory experiments and two-dimensional boundary integral simulations are used to study the interactions between two horizontally offset drops. For sufficiently deformable drops, alignment occurs so that the small drop may still coat the large drop, whereas for large enough drop viscosities or high enough interfacial tension, the small drop will be swept around the larger drop. If the large drop is sufficiently deformable, the small drop may then be ‘sucked’ into the larger drop as it is being swept around the larger drop. In order to explain the alignment process, the shape and translation velocities of widely separated, nearly spherical drops are calculated using the method of reflections and a perturbation analysis for the deformed shapes. The perturbation analysis demonstrates explicitly that drops will tend to be aligned for ℬ > O(d/a) where d is the separation distance between the drops.