  Micro-swimmers put into motion by a rotating magnetic field have provided
interesting challenges both in engineering and in mathematical modelling. We
study here the dynamics of a permanent-magnetic rigid body submitted to a
spatially-uniform steadily-rotating magnetic field in Stokes flow. This system
depends on two external parameters: the Ma- son number, which is proportional
to the angular speed of the magnetic field and inversely proportional to the
magnitude of the field, and the conical angle between the magnetic field and
its axis of rotation. This work focuses on asymptotic dynamics in the limits of
low and high Mason number, and in the limit of low conical angle. Analytical
solutions are provided in these three regimes. In the limit of low Mason
number, the dynamical system admits a periodic solution in which the magnetic
moment of the swimmer tends to align with the magnetic field. In the limit of
large Mason number, the magnetic moment tends to align with the average
magnetic field, which is parallel to the axis of rotation. Asymptotic dynamics
in the limit of low conical angle allow to bridge these two regimes. Finally,
we use numerical methods to compare these analytical predictions with numerical
solutions.
