Odd Viscosity
J. E. Avron
TL;DR
This work analyzes the concept of odd viscosity, the time-reversal-odd part of the viscosity tensor that can exist in two-dimensional isotropic media when time-reversal symmetry is broken. It develops the theoretical framework by decomposing the viscosity tensor into symmetric and antisymmetric parts, derives the 2D stress relations with a single odd coefficient $\eta^a$, and shows that in 3D isotropy forbids the odd part. The paper then explores unusual wave phenomena that arise when odd viscosity dominates, including chiral, quadratic-dispersion viscosity waves (and their absence in incompressible isotropic media), the energy flux carried by odd viscosity, and boundary scattering where odd viscosity acts as a polarizer. Finally, it extends to Navier–Stokes dynamics, establishing a Bernoulli-type relation modified by $\eta^a$, and solves a rotating-disc problem to reveal how odd viscosity induces a normal pressure proportional to rotation, contrasting with the torque produced by dissipative viscosity. The results provide a coherent, non-dissipative, chirality-rich counterpart to conventional fluid dynamics with potential implications for 2D quantum Hall-like systems and other time-reversal-broken fluids.
Abstract
When time reversal is broken the viscosity tensor can have a non vanishing odd part. In two dimensions, and only then, such odd viscosity is compatible with isotropy. Elementary and basic features of odd viscosity are examined by considering solutions of the wave and Navier-Stokes equations for hypothetical fluids where the stress is dominated by odd viscosity.