The Unsteady Motion of Solid Bodies in Creeping Flows

J. Feng and D. D. Joseph

*J. Fluid Mech*. **303**, 83-102 (1995)

**Abstract** In treating unsteady particle motions in creeping
flows, a quasi-steady approximation is often used, which assumes that the
particle's motion is so slow that it is comprised of a series of steady states.
In each of these states, the fluid is in a steady Stokes flow and the total
force and torque on the particle are zero. This paper examines the validity
of the quasi-steady method. For simple cases of sedimenting spheres, previous
work has shown that neglecting the unsteady forces causes a cumulative error
in the trajectory of the spheres. Here we will study the unsteady motion
of solid bodies in several more complex flows: the rotation of an ellipsoid
in a simple shear flow, the sedimentation of two elliptic cylinders and four
circular cylinders in a quiescent fluid and the motion of an elliptic cylinder
in a Poiseuille flow in a two-dimensional channel. The motion of the fluid
is obtained by direct numerical simulation and the motion of the particles
is determined by solving their equations of motion with solid inertia taken
into account. Solutions with the unsteady inertia of the fluid included or
neglected are compared with the quasi-steady solutions. For some flows, the
effects of the solid inertia and the unsteady inertia of the fluid are important
quantitatively but not qualitatively. In other cases, the character of the
particles' motion is changed. In particular, the unsteady effects tend to
suppress the periodic oscillations generated by the quasi-steady approximation.
Thus, the results of quasi-steady calculations are never uniformly valid
and can be completely misleading. The conditions under which the unsteady
effects at small Reynolds numbers are important are explored and the implications
for modeling of suspension flows are addressed.