An asymptotic solution is obtained for flow of a ferrofluid between two stationary coaxial cylinders of infinite extent, including the effects of spin viscosity. This solution takes into account the nonuniformity of the magnetic field in the annular space that arises as a consequence of the demagnetizing field of the inner cylinder. In the limit of zero spin viscosity the analysis predicts there is no flow in the annular gap. For nonzero spin viscosity the analysis predicts flow in the annulus with corotation of field and fluid close to the outer cylinder wall and counter-rotation close to the inner cylinder wall. The asymptotic predictions for the translational velocity are compared to experimental measurements obtained using the ultrasound velocity profile method for ferrofluid in an annular gap. The observed experimental velocity profiles are in qualitative agreement with the predictions of the asymptotic model, and transition between corotation to counter-rotation of fluid and field was observed at an intermediate radial position. These observations provide further evidence of the existence of couple stresses in ferrofluids and the importance of spin viscosity in describing some ferrofluid flows.