Trajectory and flow properties for a rod spinning in a viscous fluid. Part 1. An exact solution

Abstract

An exact mathematical solution for the low-Reynolds-number quasi-steady hydrodynamic motion induced by a rod in the form of a prolate spheroid sweeping a symmetric double cone is developed, and the influence of the ensuing fluid motion upon passive particles is studied. The resulting fluid motion is fully three-dimensional and time varying. The advected particles are observed to admit slow orbits around the rotating rods and a fast epicyclic motion roughly commensurate with the rod rotation rate. The epicycle amplitudes, vertical fluctuations, arclengths and angle travelled per rotation are mapped as functions of their initial coordinates and rod geometry. These trajectories exhibit a rich spatial structure with greatly varying trajectory properties. Laboratory frame asymmetries of these properties are explored using integer time Poincaré sections and far-field asymptotic analysis. This includes finding a small cone angle invariant in the limit of large spherical radius whereas an invariant for arbitrary cone angles is obtained in the limit of large cylindrical radius. The Eulerian and Lagrangian flow properties of the fluid flow are studied and shown to exhibit complex structures in both space and time. In particular, spatial regions of high speed and Lagrangian velocities possessing multiple extrema per rod rotation are observed. We establish the origin of these complexities via an auxiliary flow in a rotating frame, which provides a generator that defines the epicycles. Finally, an additional spin around the major spheroidal axis is included in the exact hydrodynamic solution resulting in enhanced vertical spatial fluctuation as compared to the spinless counterpart. The connection and relevance of these observations with recent developments in nano-scale fluidics is discussed, where similar epicycle behaviour has been observed. The present study is of direct use to nano-scale actuated fluidics.