(209e) Role of Rotating Magnetic Field of a Multipole Stator Winding In a Cylindrical Ferrofluid Flow

The mechanism driving flow of a ferrofluid in a cylindrical geometry and with rotating magnetic fields is a source of controversy in the theoretical description of ferrofluid phenomena. The spin diffusion theory of Shliomis predicts that ferrofluid flow is driven by diffusion of internal angular momentum. However, Glazov proposed that non-uniformities in the magnetic field generated by non-ideal stator windings drive the flow, and this hypothesis is the principal objection to the spin diffusion theory being applicable in experimental situations. Motivated by this situation, we have revisited the problem of the effect of field inhomogeneity in ferrofluid flow by obtaining a self-consistent analytical solution for the ferrohydrodynamic problem where the ferrofluid fills the gap between two resting coaxial cylinders, and is subjected to a rotating magnetic field generated by a multipole stator winding distribution, for which the non-uniformity in the magnetic field increases as the number of pole pairs increases. The solution includes the special case for which the diameter of the internal cylinder is zero, which corresponds to the “spin-up flow” geometry. Velocity profiles are obtained for both geometries, where results show that only for non-zero spin viscosity flow is predicted for any number of pole pair in the stator winding. In the spin-up flow geometry, for any number of poles in the stator, the bulk ferrofluid flow co-rotates with the magnetic field rotation. For a two pole stator winding (m = 1) ferrohydrodynamic equations predict a counter-rotation of the ferrofluid flow with the magnetic field close to the internal cylinder, and co-rotating ferrofluid flow with the magnetic field close to the external cylinder, which is in qualitative agreement with previous experimental measurements made with oil based ferrofluid. However, for a multipole stator winding (m > 2) the solution predicts that ferrofluid co-rotates with the magnetic field in the annular gap.  Experimental measurements, using a four pole stator winding (m = 2) show that ferrofluid co-rotates with the magnetic field in spin-up flow and annular geometries, which is in qualitative agreement with the theory, providing further evidence of the existence of the couple stress and the importance of the spin viscosity in describing ferrofluid flows.