A new three-dimensional boundary-integral algorithm for deformable drops moving in a viscous medium at low Reynolds numbers is developed, which overcomes some familiar difficulties with boundary-integral calculations. The algorithm is used to simulate different modes of interaction between drops or bubbles, primarily for buoyancy-driven motion. The present iterative method for mean curvature calculation is found to be more robust and accurate than contour integration schemes. A novel iterative strategy based on combining biconjugate gradient and simple iterations overcomes the poor convergence of “successive substitutions” for drops in very close approach with extreme viscosity ratio. A substantially new variational method of global mesh stabilization solves the problem of mesh degradation with advantageous, soft stability constraints. A curvatureless boundary-integral formulation is also derived and shown to provide, in principle, a more accurate description of the drop breakup than the conventional formulation. The efficiency of these techniques is demonstrated by numerical examples for two drops in gravity-induced motion with high surface resolutions. The present code successfully simulates mutual approach of slightly deformable drops to extremely small separations, as well as their rotation when in “apparent contact,” thus bridging the gap between finite deformation calculations and a recent asymptotic theory for small capillary numbers. Also provided is a 3D simulation of the experimental phenomenon of enhanced bubble coalescence, discovered by Manga and Stone [J. Fluid Mech. 256, 647 (1993); 300, 231 (1995)]. For drops of viscosity comparable to that of the surrounding fluid, it is shown in contrast that breakup is a typical result of hydrodynamic interaction in gravity-induced motion for large and even moderate capillary numbers. The code is readily applicable to any type of an ambient flow and may be adapted to more than two drops.

A trajectory analysis is used to determine the effect of small deformations and van der Waals attractions on the collision efficiency of two non-Brownian drops freely suspended in a linear flow at small Reynolds number. Simple shear flow and uniaxial compressional and extensional flow are considered. Treating the capillary number (Ca) as a small parameter permits an approach similar to matched asymptotic expansions. For Ca≪1, the analysis shows that the deformation is mainly axisymmetric and that the tangential motion of the drops in apparent contact is unaffected to leading order by the small deformation. A comparison with full three-dimensional boundary-integral calculations confirms the accuracy of the asymptotic approach. In the dimensionless parameter space, results for the collision efficiency are mapped for four parameters: Ca, size ratio, drop-to-medium viscosity ratio, and a dimensionless Hamaker parameter. For spherical drops in uniaxial compression and extension, the collision efficiencies are identical due to the reversibility of Stokes flow. When small deformation is introduced, however, the collision efficiencies are lower for compression than for extension. For slightly deformable drops in simple shear flow, the critical capture cross section upstream is no longer a circle, in contrast to the behavior of spherical drops in the absence of van der Waals forces. For all flow types, a key result is that the collision efficiency decreases rapidly from the corresponding value for spherical drops, as the capillary number increases beyond a critical value, due to small deformations. Consequently, droplet growth by coalescence will be arrested when the drops reach a prescribed size, as shown by population dynamics simulations for a model physical system.

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.

A three-dimensional boundary-integral algorithm for interacting deformable drops in Stokes flow is developed. The algorithm is applicable to very large deformations and extreme cases, including cusped interfaces and drops closely approaching breakup. A new, curvatureless boundary-integral formulation is used, containing only the normal vectors, which are usually much less sensitive than is the curvature to discretization errors. A proper regularization makes the method applicable to small surface separations and arbitrary λ, where λ is the ratio of the viscosities of the drop and medium. The curvatureless form eliminates the difficulty with the concentrated capillary force inherent in two-dimensional cusps and allows simulation of three-dimensional drop/bubble motions with point and line singularities, while the conventional form can only handle point singularities. A combination of the curvatureless form and a special, passive technique for adaptive mesh stabilization allows three-dimensional simulations for high aspect ratio drops closely approaching breakup, using highly stretched triangulations with fixed topology. The code is applied to study relative motion of two bubbles or drops under gravity for moderately high Bond numbers [Bscr ], when cusping and breakup are typical. The deformation-induced capture efficiency of bubbles and low-viscosity drops is calculated and found to be in reasonable agreement with available experiments of Manga & Stone (1993, 1995b). Three-dimensional breakup of the smaller drop due to the interaction with a larger one for λ=O(1) is also considered, and the algorithm is shown to accurately simulate both the primary breakup moment and the volume partition by extrapolation for moderately supercritical conditions. Calculations of the breakup efficiency suggest that breakup due to interactions is significant in a sedimenting emulsion with narrow size distribution at λ=O(1) and [Bscr ][ges ]5–10. A combined capture and breakup phenomenon, when the smaller drop starts breaking without being released from the dimple formed on the larger one, is also observed in the simulations. A general classification of possible modes of two-drop interactions for λ=O(1) is made.

The evolution of the drop-size distribution in immiscible fluid mixtures following well-specified shear histories is investigated by in situ microscopy, allowing determination of the shear-induced coalescence efficiency epsilon. At small capillary number Ca, epsilon is constant, whereas at larger values of Ca, epsilon decreases, in agreement with theory accounting for slight deformation of the drops in close approach. Coalescence causes the drop-size distribution to broaden in general, but greater deformation of the larger drops at high shear rates causes the drop-size distribution to remain narrow.