Free-Surface Variational Principle for an Incompressible Fluid with Odd Viscosity

We present variational and Hamiltonian formulations of incompressible fluid dynamics with a free surface and nonvanishing odd viscosity. We show that within the variational principle the odd viscosity contribution corresponds to geometric boundary terms. These boundary terms modify Zakharov’s Poisson brackets and lead to a new type of boundary dynamics. The modified boundary conditions have a natural geometric interpretation describing an additional pressure at the free surface proportional to the angular velocity of the surface itself. These boundary conditions are believed to be universal since the proposed hydrodynamic action is fully determined by the symmetries of the system.

Figure 1

The choice of normal $\widehat{n}$ and tangent $\widehat{s}$ unit vectors, and the angle $\alpha$ for the case of half-plane geometry.