Stress in a suspension near rigid boundaries

Abstract

The stress system near a rigid boundary in a suspension of neutrally buoyant spheres is considered under the assumption of small Reynolds number. The suspending fluid is assumed to be Newtonian and incompressible. An ergodic principle is formulated for parallel mean flows, and the bulk stress is expressed as a surface average. Only dilute suspensions are considered and particle interactions are neglected. A uniform shear flow past a plane wall with a single spherical particle is studied first. A series solution is developed and the mean velocity and stress fields are computed for a force-free and couple-free sphere and also for spheres with couples applied by external means. The translational and angular velocities of the particle and the stress distribution on the surface of the particle are calculated. Properties of dilute suspensions of spheres are found by appropriate surface averages over the solutions for a single particle. The mean stress on a plane parallel to the wall is shown to reduce to the Einstein value when the distance from the boundary is sufficiently large. Mean velocity profiles of the suspension for Couette flow and Poiseuille flow are developed. It is shown that in an average sense particles rotate more slowly than the ambient fluid in a region approximately three sphere radii thick adjacent to the plane wall. But for the suspension as a whole, an apparent slip velocity always develops in this region. This results in an apparent viscosity which is less than the infinite-suspension value of Einstein.