Does the magnetic field of a multipole stator winding drive flow of a ferrofluid in a cylindrical container?

The flow of a ferrofluid in a stationary cylindrical container driven by a rotating magnetic field has received considerable attention since the inception of the field of ferrohydrodynamics. Much controversy has resulted regarding the existence, or lack thereof, of bulk flow under conditions of a rotating uniform magnetic field, which can be generated for example, using a two-pole stator winding. The original observations of flow at the interface showed counter-rotation of field and fluid, whereas recent observations of bulk flow using the ultrasound technique have shown co-rotation of field and fluid. Various theories have been advanced over the years to explain the observed phenomena, including the spin diffusion theory of Shliomis and the hypothesis that it is field non-uniformity, generated by non-ideal stator winding distributions, that actually drives the flow, as first proposed by Glazov. We have revisited this problem from an analytical perspective by solving the ferrohydrodynamic and magnetoquasistatic equations self-consistently for the case of ferrofluid in a cylindrical container, with and without an internal co-axial cylinder, and driven by the field generated by a multipole stator winding distribution. In such a winding increasing the number of poles results in increasingly non-uniform fields. It is shown that regardless of the number of poles in the stator winding the ferrohydrodynamic equations do not predict any flow in either geometry as long as the spin viscosity parameter is assumed to be zero. Velocity profiles are obtained for both geometries and arbitrary number of poles for the case of non-zero spin viscosity. It is shown that only for the case of a two-pole stator winding and ferrofluid constrained to the annular space between an inner and outer cylinder do the ferrohydrodynamic equations predict co-rotation of fluid and field close to the outer cylinder and counter-rotation of fluid and field close to the inner cylinder, in qualitative agreement with experimental measurements made using an oil based ferrofluid.