The accelerated motion of rigid bodies in non-steady stokes flow

Abstract

The force and torque acting on an accelerating rigid body of arbitrary shape, moving at low Reynolds number through a fluid at rest in infinity, are considered. The expressions found for the force due to pure translation and the torque due to pure rotation of the body each include three tensors which relate the acceleration and velocity to the force and the torque. In the case of combined translation and rotation, three “coupling tensors” are added to each of the above expressions. These expressions are extended for the case of a particle, immersed in a quiet fluid and acted upon by an impulse. Generalized Faxen's theorems are derived for non-steady flows which do not vanish in infinity. Finally, the effect of non-zero initial velocity of the fluid and the body is considered. The stop distance is shown to depend linearly on the initial velocity of the body through a displacement tensor which consists of the traditional quasi-stationary term and an additional tensor. This additional tensor depends on the geometry of the body and on the initial velocity field of the fluid. It is infinite if the kinetic energy of the initial field is infinite. Likewise, the expression for the force acting on the body contains an additional term which depends on time, on the geometry of the problem and on the initial velocity field.