On the Settling Speed of Free and Fixed Suspensions

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Abstract
Dilute suspensions of small spheres in a viscous fluid are examined for the three cases of: I, a regular periodic array; II, a random array of freely moving particles; and III a random array in which the particles are held rigidly. It is known that the difference between average force and average velocity for a particle is a different function of concentration for the three cases. A unified analytical treatment is given for the point particle approximation, which explains the different qualitative dependence on concentration. It is shown that the differences between I and II are kinematical in nature, depending entirely on the difference between the geometries of regular and random arrays. The different result for case III is due to the shielding effect of a dilute suspension when the velocities, but not the forces, of the particles are fixed.