Odd surface waves in two-dimensional incompressible fluids

TL;DR

This paper explores the unique surface wave phenomena in two-dimensional incompressible fluids with odd viscosity, deriving linear and nonlinear wave equations, including a novel chiral Burgers equation, and analyzing their solutions and singularity formation.

Contribution

It introduces the chiral Burgers equation as a new nonlinear model for surface dynamics in fluids with odd viscosity, extending linear wave solutions to the nonlinear regime.

Findings

Surface waves are chiral and exist without gravity or shear viscosity.

Derived the nonlinear chiral Burgers equation governing surface dynamics.

Found exact solutions and singularity formation in the nonlinear regime.

Abstract

We consider free surface dynamics of a two-dimensional incompressible fluid with odd viscosity. The odd viscosity is a peculiar part of the viscosity tensor which does not result in dissipation and is allowed when parity symmetry is broken. For the case of incompressible fluids, the odd viscosity manifests itself through the free surface (no stress) boundary conditions. We first find the free surface wave solutions of hydrodynamics in the linear approximation and study the dispersion of such waves. As expected, the surface waves are chiral and even exist in the absence of gravity and vanishing shear viscosity. In this limit, we derive effective nonlinear Hamiltonian equations for the surface dynamics, generalizing the linear solutions to the weakly nonlinear case. Within the small surface angle approximation, the equation of motion leads to a new class of non-linear chiral dynamics governed by what we dub the {\it chiral} Burgers equation. The chiral Burgers equation is identical to the complex Burgers equation with imaginary viscosity and an additional analyticity requirement that enforces chirality. We present several exact solutions of the chiral Burgers equation. For generic multiple pole initial conditions, the system evolves to the formation of singularities in a finite time similar to the case of an ideal fluid without odd viscosity. We also obtain a periodic solution to the chiral Burgers corresponding to the non-linear generalization of small amplitude linear waves.