Odd viscosity in three dimensional flows

A fluid that breaks microscopic time-reversal symmetry, for example, by being composed of spinning units, can acquire a so-called “odd-viscosity”, an antisymmetric contribution to the viscosity tensor. In isotropic two-dimensional fluids, it has been shown that the odd viscosity does not modify the flow in the incompressible limit. In this work, we consider the extension of odd viscosity to three dimensional flows. We find a host of additional odd viscosity coefficients that give rise to dramatic changes in the flow even in incompressible fluids. By revisiting classic hydrodynamic problems in the Stokes limit, we elucidate the effects of these additional coefficients, and discuss the modifications in the presence of odd stress. Our work provides guiding principles for experimental and numerical investigations of complex fluids in suitable biological and active matter systems.