A solution is obtained for the ferrohydrodynamic problem of ferrofluid flow in the annular gap between two coaxial cylinders under the influence of the magnetic field generated by a multipole stator winding. The solution is applicable to small values of the applied magnetic field amplitude, takes into account the spatial harmonics produced by the multipole stator winding, and includes the effect of spin viscosity. The special case for which the internal diameter goes to zero, corresponding to the cylindrical “spin-up flow” geometry, is also obtained. Results show that for zero spin viscosity (η'=0), no flow is predicted for any number of poles in the winding. In the case of nonzero spin viscosity, for a two-pole stator winding (m = 1) the solution predicts counter-rotating flow with respect to the magnetic field close to the inner cylinder and co-rotating flow with respect to the magnetic field near the outer cylinder, in agreement with previously reported experimental observations. For stator windings with larger number of poles (m>1), the solution predicts co-rotation of fluid and field in the annular gap. Using a four-pole stator winding (m=2), experimental measurements show that a ferrofluid co-rotates with the magnetic field in both the cylindrical and annular cases, with qualitative agreement between the theory with nonzero spin viscosity and the experiments. Non-idealities in the stator winding distribution, e.g., due to slot effects, are considered through linear superposition of higher order harmonics, again predicting zero flow in the absence of spin viscosity and counter-rotating flow with respect to the magnetic field close to the inner cylinder wall only for the case where the fundamental harmonic (m=1) is dominant.