4.8 Unsteady Effects at Re ?1

In Sections 4.4 and 4.5, we studied some unsteady motions of small spheres under the assumption that all changes occurred slowly. This allowed us to use Stokes law, in which the fluid force is proportional to the instantaneous relative velocity between particle and fluid. But, in reality, that force also depends on the acceleration, as we saw for a sphere that is executing translational oscillations at a single frequency, in which case

(4.8.1)

In this section, we examine the effects of acceleration of both fluid and particle by considering the force on a particle that is executing translational oscillations in response to a monochromatic oscillation of the fluid, as may occur when a container filled with an incompressible fluid is made oscillate as a whole. But, before we can do that, we must evaluate the force on a small spherical particle in the oscillatory fluid. This can be obtained from the above result, provided we take into account the fluid motion. Thus, if the fluid is also executing translational oscillations having a single frequency, the force on the sphere could be obtained by solving the unsteady Stokes equations, referred to axes moving with the fluid, instead of the fixed system used to obtain (4.8.1). The analysis would be similar to that used in Section 4.2, except for two changes, which can be incorporated directly into the final result. Firstly, instead of u _p, we must use the particle velocity relative...