Odd surface waves in two-dimensional incompressible fluids

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.

34 pages, 7 figures, revised version: added discussion on the Hamiltonian structure and conservation laws, new references added, minor typos corrected. Close to published version

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• Edge wave and boundary layer of vortex matter
• Free surface variational principle for an incompressible fluid with odd viscosity
• Hydrodynamics of two-dimensional compressible fluid with broken parity: variational principle and free surface dynamics in the absence of dissipation
• Probe particles in odd active viscoelastic fluids: how activity and dissipation determine linear stability
• Non-linear shallow water dynamics with odd viscosity
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• Resolving Hall and dissipative viscosity ambiguities via boundary effects
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• Observable signatures of Hall viscosity in lowest Landau level superfluids
• Slip-induced odd viscous flow past a cylinder
• Taylor columns and inertial-like waves in a three-dimensional odd viscous liquid
• Nonlinear shallow-water waves with vertical odd viscosity
• Odd elasticity in disordered chiral active materials