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Three Dimensional Odd Viscosity in Ferrofluids with Vorticity-Magnetization Coupling

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 ($\vecω$) to the magnetization ($\vec{M}$) with a term proportional to $\vecω\cdot\vec{M}$ 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.