Theory of ferromagnetic microswimmers

This paper considers the dynamics of a microscale swimmer based on two magnetic beads that are elastically coupled together. A time-varying external magnetic field is imposed that has two principal effects: one is to exert a torque on the magnetic beads. The second is to change the orientation of the magnetic field dipoles in one or both beads, depending on their ferromagnetic properties. This then creates an attraction or repulsion between the two dipoles. The combination of dipole attraction/repulsion, moderated by the elastic coupling, and torque gives motions that are not generally time reversible and can lead to unidirectional swimming, that is persistent motion in one direction, in a Stokes flow regime. The equations of motion for the swimmer are set up using a Lagrangian formulation and supplemented by equations giving the dipole orientation of the magnetic fields of the beads in the external field. The equations are non-dimensionalized and key parameters determined. Numerical simulations reveal a number of regimes that are studied using simplified models and multiple scale analysis. Approximate thresholds are obtained above which the swimmer moves in a closed path and below which the orientation is `trapped' giving unidirectional motion. Three mechanisms for such trapping are isolated and discussed.