Motion of a rigid particle in a rotating viscous flow: an integral equation approach

Abstract

A boundary integral method is presented for analysing particle motion in a rotating fluid for flows where the Taylor number ${\cal T}$ is arbitrary and the Reynolds number is small. The method determines the surface traction and drag on a particle, and also the velocity field at any location in the fluid.Numerical results show that the dimensionless drag on a spherical particle translating along the rotation axis of an unbounded fluid is determined by the empirical formula $D/6\pi = 1+(4/7){\cal T}^{1/2}+(8/9\pi){\cal T}$, which incorporates known results for the low and high Taylor number limits. Streamline portraits show that a critical Taylor number ${\cal T}_c\ap 50$ exists at which the character of the flow changes. For ${\cal T} < {\cal T}_c$ the flow field appears as a perturbation of a Stokes flow with a superimposed swirling motion. For ${\cal T} > {\cal T}_c$ the flow field develops two detached recirculating regions of trapped fluid located fore and aft of the particle. The recirculating regions grow in size and move farther from the particle with increasing Taylor number. This recirculation functions to deflect fluid away from the translating particle, thereby generating a columnar flow structure. The flow between the recirculating regions and the particle has a plug-like velocity profile, moving slightly slower than the particle and undergoing a uniform swirling motion. The flow in this region is matched to the particle velocity in a thin Ekman layer adjacent to the particle surface.A further study examines the translation of spheroidal particles. For large Taylor numbers, the drag is determined by the equatorial radius; details of the body shape are less important.