A multipole-accelerated algorithm for close interaction of slightly deformable drops

An efficient multipole-accelerated boundary-integral algorithm is developed to study close-contact, three-dimensional interaction of two drops at zero Reynolds numbers and very small, but non-zero, capillary numbers Ca, when the drops are nearly spherical. The numerical difficulties (compared to the case of larger deformations) include severe stability limitations on the time step and a singular perturbation for Ca ≪ 1, requiring very high surface resolution both in the small gap and in the “outer” region. The mesh triangle vertices are grouped into a large number of non-overlapping “patches,” with economical, rotation-based multipole reexpansions to handle patch-to-patch interactions and limit the use of expensive direct summations. A novel concept of a “dynamical projective mesh” is developed, to maintain fixed-topology, gap-adaptive surface triangulations. For O(105) boundary elements per drop in close contact, the algorithm has, at least, an order-of-magnitude advantage over the standard boundary-integral method, making such dynamical calculations feasible. In gravity-induced and shear-induced motion, exact results are obtained for the dynamics of the surface clearance hmin (which attains values less than 0.001 of the drop radii) and for the “separation angle” βsep (determining the configuration when two drops in apparent contact start to separate). The shear flow problem is studied in the wide range of drop-to-medium viscosity ratios 0.25 ⩽ λ ⩽ 10. Comparisons are made with prior and extended asymptotic theories of coalescence (based on matching the thin-film solution with the outer solution for spherical drops) to determine their range of validity and assess the “pumping flow” effect neglected in the theories. Pumping flow is most important for small λ and/or nearly head-on collisions; otherwise, the drops move past each other with too little time for the pumping flow to have a strong effect. The asymptotic techniques are extended to λ ≫ 1, and shown to be very accurate for λ = 4 and 10 in the wide range of Ca ≪ 1. Scaling laws for hmin and βsep are found, both numerically and analytically; in particular, for λ = O(1), βsep approaches its limiting value π/2 (corresponding to zero “driving force” in the asymptotic theory) extremely slowly, with a difference of O(Ca1/3), as Ca → 0.