Odd surface waves in two-dimensional incompressible fluids

Journal Article · 26 July 2018 · SciPost Physics

DOI: https://doi.org/10.21468/SciPostPhys.5.1.010 · OSTI ID:1461900

Abanov, Alexander ^[1]; Can, Tankut ^[2]; Ganeshan, Sriram ^[3]

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1. Stony Brook University
2. The Graduate Center, CUNY
3. City College of New York, Stony Brook University

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 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.

• Research Organization: State Univ. of New York (SUNY), Albany, NY (United States); Stony Brook Univ., NY (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division
• Grant/Contract Number: SC0017662
• OSTI ID: 1461900
• Journal Information: SciPost Physics, Journal Name: SciPost Physics Journal Issue: 1 Vol. 5; ISSN 2542-4653
• Publisher: SciPost FoundationCopyright Statement
• Country of Publication: Netherlands
• Language: English