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

  • Authors:

  • Alexander Z. Zinchenko;Robert H. Davis

  • Affiliations:

  • Engineering Center, Department of Chemical and Biological Engineering, University of Colorado, ECCH 111, Campus Box, Boulder, CO 80309-0424, USA;Engineering Center, Department of Chemical and Biological Engineering, University of Colorado, ECCH 111, Campus Box, Boulder, CO 80309-0424, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2005

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Abstract

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(10^5) 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 h"m"i"n (which attains values less than 0.001 of the drop radii) and for the ''separation angle'' @b"s"e"p (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=0.