Updated formulation of magnetic body force in ferrofluids

The magnetic body force is critical for modelling of convection in ferrofluids. Despite a long history in the development of the theories for ferro-hydrodynamics, literature from the last five years shows that a universal consensus has not been reached concerning the formulation of this term for ferrofluids. We present an updated derivation of the body force directly from the Lorentz force and Maxwell's equations. The derivation requires that the differential volume experiencing the body force only contains complete dipole current loops. This has the effect that an additional surface integral term to account for bound surface current is not needed when modelling situations where the ferrofluid has interfaces with other materials. We compare results from our derived body force with five other formulations from the literature for the case of a single conductor in ferrofluid under static and convection conditions. Most formulations become similar in the limit of small magnetic susceptibility. For a susceptibility of the order of 1, as is typical for ferrofluids, the calculated body force from the formulations differed by a factor of about four, greatly affecting thermomagnetic convection predictions for a heated microwire.