4.4 Slowly Changing Motions

Although Stokes law applies to a small sphere moving steadily in a fluid, or to a small sphere held fixed in a slow, viscous flow, such motions are not common. The main example of the first occurs when a small sphere falls steadily at Re ?1. Another steady motion occurs, of course, if a particle is completely entrained by a steadily translating fluid. However, in this case, the fluid exerts no force on the particle because there is no relative motion.

In fact, in most situations, the motions of free particles are unsteady and, in the absence of external forces, they are induced by changes in the velocity of the fluid around them. We will consider the effects of these changes on the particle force later. Here, we proceed as we did for the heat transfer calculation and first assume that the force acting on a sphere at rest in a steadily moving fluid would, for Re ?1, be given by F _p=6 ? ? _f au _f, where u _f is the fluid velocity far from the sphere. Now suppose the particle is not at rest but is moving...