An analysis of the far-field response to external forcing of a suspension in Stokes flow in a parallel-wall channel

TL;DR

This paper analyzes the far-field flow response of particles in a parallel-wall channel under creeping flow, deriving macroscopic transport equations and evaluating the influence of particle arrangements on suspension flow.

Contribution

It introduces a macroscopic framework for suspension flow in parallel-wall channels, linking microscopic pressure dipoles to macroscopic transport properties and providing explicit Green's functions.

Findings

Transport coefficients satisfy Onsager reciprocal relations.

Derived explicit Green's functions for flow between parallel walls.

Proposed a Clausius-Mossotti approximation for permeability.

Abstract

The leading-order far-field scattered flow produced by a particle in a parallel-wall channel under creeping flow conditions has a form of the parabolic velocity field driven by a 2D dipolar pressure distribution. We show that in a system of hydrodynamically interacting particles, the pressure dipoles contribute to the macroscopic suspension flow in a similar way as the induced electric dipoles contribute to the electrostatic displacement field. Using this result we derive macroscopic equations governing suspension transport under the action of a lateral force, a lateral torque or a macroscopic pressure gradient in the channel. The matrix of linear transport coefficients in the constitutive relations linking the external forcing to the particle and fluid fluxes satisfies the Onsager reciprocal relation. The transport coefficients are evaluated for square and hexagonal periodic arrays of fixed and freely suspended particles, and a simple approximation in a Clausius-Mossotti form is proposed for the channel permeability coefficient. We also find explicit expressions for evaluating the periodic Green's functions for Stokes flow between two parallel walls.