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.
On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities
