Viscous Flow Past a Random Array of Spheres

Abstract

The steady, slow motion of a viscous fluid past a random, uniform array of identical nonoverlapping finite spheres is studied. The expected force F acting on any given sphere is obtained by averaging over the ensemble of possible positions of all other spheres. It is shown that, if the Stokes equations hold in the fluid, then for small particle volume concentration c, F has an expansion of the form $$\text{F=[ 1 + (3 /}\sqrt{2}{\mathrm{)\; c}}^{\text{1/2}}\text{+ (135/64)c\hspace{0.17em}}\log {\mathrm{c\; ]\; F}}_0{\mathrm{+\; cD\; ·\; F}}_0{\text{+ 0 (c}}^{\text{3/2}}\log \mathrm{c),}$$ where F₀ is the Stokes force on a single isolated sphere in the same flow and D is a tensor which is given in terms of the two‐sphere distribution function for the array and the Stokes solution for two spheres oriented arbitrarily in an unbounded flow. A physical model for the solution is presented and compared with the model of Brinkman.