Asymptotic Dynamics of Magnetic Micro-Swimmers

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.