# 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 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 themore »

- Authors:

- Stony Brook University
- The Graduate Center, CUNY
- City College of New York, Stony Brook University

- Publication Date:

- Research Org.:
- State Univ. of New York (SUNY), Albany, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division

- OSTI Identifier:
- 1461900

- Alternate Identifier(s):
- OSTI ID: 1596115

- Grant/Contract Number:
- [SC0017662]

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- [Journal Name: SciPost Physics Proceedings Journal Volume: 5 Journal Issue: 1]; Journal ID: ISSN 2542-4653

- Publisher:
- SciPost Foundation

- Country of Publication:
- Netherlands

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Burgers equation; Hydrodynamics; Incompressible fluids; Parity symmetry; Shear viscosity; Viscosity

### Citation Formats

```
Abanov, Alexander, Can, Tankut, and Ganeshan, Sriram. Odd surface waves in two-dimensional incompressible fluids. Netherlands: N. p., 2018.
Web. doi:10.21468/SciPostPhys.5.1.010.
```

```
Abanov, Alexander, Can, Tankut, & Ganeshan, Sriram. Odd surface waves in two-dimensional incompressible fluids. Netherlands. doi:10.21468/SciPostPhys.5.1.010.
```

```
Abanov, Alexander, Can, Tankut, and Ganeshan, Sriram. Fri .
"Odd surface waves in two-dimensional incompressible fluids". Netherlands. doi:10.21468/SciPostPhys.5.1.010.
```

```
@article{osti_1461900,
```

title = {Odd surface waves in two-dimensional incompressible fluids},

author = {Abanov, Alexander and Can, Tankut and Ganeshan, Sriram},

abstractNote = {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.},

doi = {10.21468/SciPostPhys.5.1.010},

journal = {SciPost Physics Proceedings},

number = [1],

volume = [5],

place = {Netherlands},

year = {2018},

month = {7}

}

DOI: 10.21468/SciPostPhys.5.1.010

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