An O(N) algorithm for Stokes and Laplace interactions of particles

Abstract

A method for computing Laplace and Stokes interactions among N spherical particles arbitrarily placed in a unit cell of a periodic array is described. The method is based on an algorithm by Greengard and Rokhlin [J. Comput. Phys. 73, 325 (1987)] for rapidly summing the Laplace interactions among particles by organizing the particles into a number of different groups of varying sizes. The far-field induced by each group of particles is expressed by a multipole expansion technique into an equivalent field with its singularities at the center of the group. The resulting computational effort increases only linearly with N. The method is applied to a number of problems in suspension mechanics with the goal of assessing the efficiency and the potential usefulness of the method in studying dynamics of large systems. It is shown that reasonably accurate results for the interaction forces are obtained in most cases even with relatively low-order multipole expansions.

A method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall and a gas-liquid interface

“…This will allow an easy integration into an O͑N͒ algorithm described earlier by Sangani and Mo. 5 The method is applied to a number of problems including chains of particles, drops, and porous particles.…”

“…The present study is intended to give expressions that are necessary so that the method of Sangani and Mo 5 can be extended to account for the presence of a rigid plane wall, a nondeformable gas-liquid interface, or both. The method is based on determining an image system for Lamb's multipoles.…”

“…Transport phenomena-flow and mass transfer-play a role equally important as the cell biology in determining the morphology of such microcolonies, and this complex interplay of transport and biological factors can be studied by numerical simulations of a flow involving many particles near a wall. Mo and Sangani 4 and Sangani and Mo 5 described an efficient method for computing low Reynolds number hydrodynamic interactions in a system consisting of many particles. The velocity induced by a spherical particle is expressed in terms of Lamb's multipoles, i.e., the coefficients appearing in Lamb's spherical harmonic solution of the Stokes equations of motion.…”

Effective-medium theories for predicting hydrodynamic transport properties of bidisperse suspensions

“…The lubrication effects require even greater values of N s , and hence multipoles, to determine hydrodynamic transport properties accurately. Since including the lubrication effects explicitly according to the scheme outlined by Sangani and Mo 5 slows down the convergence rate of the iterative method used in the algorithm, and since modifying the method for monodisperse suspensions outlined in Sangani and Mo 5 to account for the lubrication effects in bidisperse suspensions requires considerably more effort, it was chosen to carry out calculations with greater N s for one configuration and apply the correction obtained from the single configuration to the results obtained by N s ϭ3.…”

“…The simulation results are obtained by modifying the method described in Sangani and Mo. 5 It is shown that modified effective-medium theories yield reasonably accurate estimates of the hydrodynamic transport coefficients and the conditionally averaged velocities.…”