The motion of a dipolar spherical particle in homogeneous shear and time-periodic fields

Abstract

The effects of the intensity and frequency of a time-periodic external field on the rotary motion of a dipolar spherical particle suspended in homogeneous shear are studied with the goal of providing insight into problems concerning the motion of swimming microorganisms and the macroscopic behaviour of ferrofluids. The analysis reveals two modes of motion: convergence of the particle to a global time-periodic attractor, and quasi-periodic motion. The former mode of particle rotation generally appears for sufficiently strong fields. However, asymptotic analysis clarifies that it may occur even for very weak fields as a cumulative result of appropriate resonance interactions.

A sufficient condition for the occurrence of a global time-periodic attractor is established for an external field acting in the plane of shear. Asymptotic results together with numerical evidence indicate that this condition is in fact a necessary condition as well. Making use of this condition we obtain the division of the plane of parameters into domains respectively corresponding to quasi-periodic motion and global time-periodic attractors. The latter domain has the structure of non-intersecting Arnold's tongues. Throughout each, the average frequency of dipole rotation about the vorticity vector is a constant (integral) multiple of the forcing frequency (frequency locking). In the case of quasi-periodic motion, there simultaneously coexist separate domains in orientation space where the rotary motion is locally characterized by different constant rotation numbers. These may assume both rational and irrational values. Potential implications of the distinction between these modes of rotary motion on the characterization of effective (macroscale) ferrofluid properties are briefly discussed.