Ferrofluid flow in a spherical cavity under an imposed uniform rotating magnetic field: Spherical spin-up flow

Abstract

An analytical solution is obtained for the ferrohydrodynamic problem of flow of a ferrofluid in a spherical cavity under the influence of a uniform rotating magnetic field produced by a fluxball winding. The ferrohydrodynamics equations are decoupled for low values of the applied magnetic field amplitude to allow analytical solution. A governing equation for the divergence of spin velocity is obtained and used to demonstrate that the divergence of the spin velocity is zero under conditions of small uniform magnetic field. The condition of zero spin viscosity results in no flow being predicted in the cavity. For the case of non-zero spin viscosity the condition of zero divergence of spin results in the inability of the ferrohydrodynamics equations to satisfy boundary conditions for both the normal and tangential components of the spin velocity at the sphere wall. Solutions are obtained for cases where boundary conditions are imposed on either the normal or tangential component of the spin velocity. In both cases, the solutions predict that the ferrofluid rotates rigid-body-like about the axis of field rotation in the core and there is a region near the outer wall where flow reduces to satisfy the no slip boundary condition on the translational velocity. However, the two solutions differ in their asymptotic behavior when the spin viscosity vanishes. It is found that when the boundary condition is applied to the tangential component of the spin velocity, the zero spin viscosity solution is recovered as η′ → 0, whereas when the boundary condition is applied to the normal component of the spin velocity the solution diverges as η′ → 0. The results indicate that studying the flow of ferrofluids in a spherical cavity could yield important tests of long-held, arguably ad hoc assumptions surrounding the governing equations and boundary conditions of ferrofluid flow.