A strong-interaction theory for the motion of arbitrary three-dimensional clusters of spherical particles at low Reynolds number

Abstract

This paper contains an ‘exact’ solution for the hydrodynamic interaction of a three-dimensional finite cluster at arbitrarily sized spherical particles at low Reynolds number. The theory developed is the most general solution to the problem of an assemblage of spheres in a three-dimensional unbounded media. The boundary-collocation truncated-series solution technique of Ganatos, Pfeffer & Weinbaum (1978) for treating planar symmetric Stokes flow problems has been extensively modified to treat the non-symmetric multibody problem. The orthogonality properties of the eigenfunctions in the azimuthal direction are used to satisfy the no-slip boundary conditions exactly on entire rings on the surface of each particle rather than just at discrete points.Detailed comparisons with the exact bipolar solutions for two spheres show the present theory to be accurate to five significant figures in predicting the translational and angular velocity components of the particles at all orientations for interparticle gap widths as close as 0.1 particle diameter. Convergence of the results to the exact solution is rapid and systematic even for unequal-sized spheres (a1/a2 = 2). Solutions are presented for several interesting and intriguing configurations involving three or more spherical particles settling freely under gravity in an unbounded fluid or in the presence of other rigidly held particles. Advantage of symmetry about the origin is taken for symmetric configurations to reduce the collocation matrix size by a factor of 64. Solutions for the force and torque on three-dimensional clusters of up to 64 particles have been obtained, demonstrating the multiparticle interaction effects that arise which would not be present if only pair interactions of the particles were considered. The method has the advantage of yielding a rather simple expression for the fluid velocity field which is of significance in the treatment of convective heat and mass transport problems in multiparticle systems.