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

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 two-dimensional 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 the Stokes flow between two parallel walls.