Far-field approximation for hydrodynamic interactions in parallel-wall geometry

TL;DR

This paper derives a far-field approximation for hydrodynamic interactions in a fluid between parallel walls, simplifying calculations by using Hele-Shaw flow assumptions and improving computational efficiency in particle suspension simulations.

Contribution

It introduces an asymptotic far-field flow model in parallel-wall geometry, enabling more efficient evaluation of hydrodynamic interactions in suspensions.

Findings

Flow assumes Hele-Shaw form at large distances

Derived asymptotic Green's matrix expressions

Enhanced efficiency of hydrodynamic simulation algorithms

Abstract

A complete analysis is presented for the far-field creeping flow produced by a multipolar force distribution in a fluid confined between two parallel planar walls. We show that at distances larger than several wall separations the flow field assumes the Hele-Shaw form, i.e., it is parallel to the walls and varies quadratically in the transverse direction. The associated pressure field is a two-dimensional harmonic function that is characterized by the same multipolar number m as the original force multipole. Using these results we derive asymptotic expressions for the Green's matrix that represents Stokes flow in the wall-bounded fluid in terms of a multipolar spherical basis. This Green's matrix plays a central role in our recently proposed algorithm [Physica A xx, {\bf xxx} (2005)] for evaluating many-body hydrodynamic interactions in a suspension of spherical particles in the parallel-wall geometry. Implementation of our asymptotic expressions in this algorithm increases its efficiency substantially because the numerically expensive evaluation of the exact matrix elements is needed only for the neighboring particles. Our asymptotic analysis will also be useful in developing hydrodynamic algorithms for wall-bounded periodic systems and implementing acceleration methods by using corresponding results for the two-dimensional scalar potential.