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 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.
Highlights
1) All equations we wrote so far are legitimate in any spatial dimension except for the odd viscosity term of (5) which is specific to two-dimensional hydrodynamics
As the frequency Ω of the surface waves remains finite in the limit νe → 0 we find that the Bcomponent of the solution (16,17) and vorticity of the fluid (18), in particular, is localized near the surface of the fluid within the dynamic boundary layer of thickness δ
In this work we considered the motion of a free surface of an incompressible fluid with odd viscosity
Summary
There has been much interest in the role of parity violating effects in two dimensional incompressible hydrodynamics. We begin by introducing the general hydrodynamic equations in Sec 2 and derive the odd viscosity generalization of Lamb’s solutions in Sec 4 We analyze these solutions in the limit νe → 0 and obtain the scaling of the boundary layer thickness and velocity and vorticity profile with respect to νe (Sec 4.2). We discuss a one parameter family of non-linear equations (Eq 149), which contains the chiral Burgers equation and the Benjamin-Davis-Ono (BDO) equation as limiting cases
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