On the Stokes resistance of multiparticle systems in a linear shear field

Abstract

A general theory is presented for determining the quasistatic Stokes resistance of a particle or an aggregate of particles of arbitrary shape which move with arbitrary translational and rotational velocities in an infinite viscous fluid subjected to a general linear shear flow in the absence of the particles. It is shown that the hydrodynamic resistance of the particles may be characterized by two intrinsic matrices-the grand resistance matrix and the shear resistance matrix, the elements of each matrix being dependent only upon the instantaneous geometrical configuration of the multiparticle system. Explicit expressions for these matrix elements are given for a single ellipsoidal particle and for two spheres. Analytical, closed-form expressions are given for the case where the spheres are widely spaced compared with their radii. When the separation between the spheres is arbitrary, the matrix elements are only determinate numerically for specified ratios of sphere radii and center-to-center separation distances. These numerical values are derived from available bipolar-coordinate system solutions for two spheres. Besides providing a compact and systematic representation of the intrinsic Stokes resistance of multiparticle systems subjected to shear, which does not depend upon the precise velocities of the particles or on the nature of the shear flow, it is demonstrated (by way of example) that the matrix representation permits an immediate determination of the translational and rotational velocities of the individual particles in a neutrally buoyant multiparticle system.