4.2 Translational Oscillations

We first consider translational monochromatic oscillations, without rotation, of a rigid sphere in a fluid that is otherwise at rest. The velocity of the particle and its diameter are such that the Reynolds number is very small, so that the Stokes equations apply. This particular motion is important on its own, but also its results can be used to study other motions. For example, the force that acts on more general unsteady translational motions may then be obtained from the monochromatic result by means of Fourier transform methods.

We thus take the sphere to have a translational velocity u _p (t) along a straight line, say one of the axes of a Cartesian system of coordinates. For purely monochromatic motions, the magnitude of u _p (t) is u _p (t)=U _{p 0} cos ( ?t). Now consider the fluid velocity u _f. After all transient motions have died, this is also a monochromatic function of time, having the same frequency as that of the particle. Thus, we may express both velocities as

(4.2.1a,b)

where the vector U _f (x) is, in general, a complex function of position that is to be determined. The vector x appearing here represents the position of a field point with respect to a fixed system of coordinates, which we place at the mean position of the sphere s center. We note in passing that, in this system, the time derivative has the usual meaning...