J. Fluid Mech. zyxwvutsrq (1!386), vol. 167, pp. 415426 Printed in Great Britain 415 zy Viscous interactions of many neutrally buoyant spheres in Poiseuille flow By JEFFREY A. SCHONBERG, DONALD A. DREW AND GEORGES BELFORT Department of Chemical Engineering and Environmental Engineering zyx and Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA (Received 25 June 1984 and in revised form 12 November 1985) zyx A theory is developed to predict the motion of Nneutrally buoyant spheres suspended in laminar flow between parallel plates. The spheres are at large separation yet nearer each other than the duct walls, and the Reynolds number is small. In this parameter range, viscous interactions are larger than inertial effects, and can be represented in terms of a superposition of ‘strainlets’. Several examples are given to show this viscous interaction effect. Near the leading edge of a front of spheres or near the trailing edge significant lateral migration velocities can occur, being a t least one order of magnitude larger than inertially induced migration velocities. This phenomenon may have a negative effect on ‘chromatographic zyxw ’ separation schemes, affecting particle concentration, recovery and resolution. 1. Introduction The motion of small particles in a viscous medium in which the particle number per unit volume (concentration) varies from dilute to concentrated suspensions is found in a wide variety of processes. Examples include (0 ’Neil1 1981 ) : suspension and polymer rheology (Russel 1981), continuum mechanics (Drew 1983), and diffusional transport processes in suspensions of Brownian particles, erythrocyte motion in capillary blood flow, gel permeation chromatography, field flow fractionation, flow of fibre suspensions in papermaking processes and of latex particles in emulsion based paints, cross-flow filtration of concentrated suspensions, ferro-fluid rheology, shear- gradient coagulation in hydrosols, emulsification mechanisms in colloid mills and the motion of suspended rock crystals in molten rocks. Since Stokes’ celebrated work (Stokes 1851) on the motion of particles in viscous flow at small Reynolds numbers, a plethora of studies has appeared in the literature. New developments and extensions have accounted for such effects as particle rotation, non-spherical geometry, inertial effects, wall effects, particle-particle interactions, density effects, liquid droplets and gas bubbles, and the inclusion of electrical and magnetic forces acting on the particles (Batchelor 1976). Of specific interest here is the motion and behaviour of neutrally buoyant solid spherical particles suspended in laminar flow moving in a non-porous duct under near-creeping-flow conditions. Although the Reynolds number of the bulk flow can never be exactly zero, the relative importance of viscous versus inertial effects for low-Reynolds-number hydrodynamics is of interest in the creeping-flow limit. Various researchers (Ho & Leal 1974; Vasseur & Cox 1976; Ishii & Hasimoto 1980; and Altena & Belfort 1984) have solved the integral equations (Cox & Brenner 1968) for the inertially induced lateral migration of single neutrally buoyant particles