  Ferrofluids are a synthetic magnetic colloid consisting of magnetized
nanoparticles surrounded by a repulsive surfactant layer. When subjected to an
external magnetic field the ferrofluid acquires a macroscopic magnetization
density which leads to magnetic behavior that is intricately coupled to the
ambient fluid dynamics. Ferrofluids share several features with the chiral
active fluids composed of unidirectionally spinning hematite cubes, which have
been shown to possess a 2D non-dissipative odd viscosity term (Nature Physics,
15, 1188-1194(2019)). In standard ferrofluid dynamics, 3D versions of parity
breaking terms are not commonly observed, partly because of the small size of
the magnetic particles. In this work, we investigate if there are unique
mechanisms in ferrofluids that can lead to a 3D odd viscosity term. Our results
show that coupling the fluid vorticity ($\vec{\omega}$) to the magnetization
($\vec{M}$) with a term proportional to $\vec{\omega}\cdot\vec{M}$ leads to
parity breaking terms in ferrofluid hydrodynamics, and results in a three
dimensional odd viscosity term when the magnetization is relaxed to the
direction of a uniform and static applied field. Hele-Shaw cells are commonly
used devices to investigate ferrofluids and we demonstrate that this coupling
reproduces the parity odd generalization of Darcy's Law discussed in a recent
work (Phys. Rev. Fluids 7, 114201 (2022)). A potential experimental setup is
discussed which may reveal the presence of this coupling in a ferrofluid
confined to a Hele-Shaw cell.
