Modified Darcy's Law for parity odd three-dimensional fluids

Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap. The governing equation of Hele-Shaw flow is Darcy's Law, which also describes ideal inviscid flow, and flow through a porous medium. In this work, we derive a modification to Darcy’s Law due to a longitudinal component of the three-dimensional parity-odd viscosity tensor, dubbed 'longitudinal odd viscosity’, and show that for incompressible fluids this is the only modification that is possible within the Hele-Shaw geometry. The modified Darcy’s Law acquires an off-diagonal contribution, similar in form to that seen in a rotating Hele-Shaw cell, or a ferrofluid in an external magnetic field. We discuss the observable effects of the modified Darcy’s law in various experiments relevant to the Hele-Shaw setup. In particular, we show that when confined to a channel, a single fluid exerts a transverse force on the walls. These results can be directly applied to active matter fluids, or other parity-odd three-dimensional fluids confined to a Hele-Shaw cell.