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.