Hydrodynamics of two-dimensional compressible fluid with broken parity: Variational principle and free surface dynamics in the absence of dissipation
Hydrodynamics of two-dimensional compressible fluid with broken parity: Variational principle and free surface dynamics in the absence of dissipation
复制标题
破缺宇称的二维可压缩流体的流体动力学：无耗散情况下的变分原理和自由表面动力学
DOI：
10.1103/physrevfluids.5.104802
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
G. Monteiro
中科院分区：
文献类型：
--
作者：
A. Abanov;T. Can;S. Ganeshan;G. Monteiro
本文研究了具有破缺宇称的各向同性可压缩非耗散流体在二维空间中的自由面边界条件。描述流体的体动力学的流体动力学方程以及自由表面边界条件明确地依赖于宇称破缺非耗散奇粘性项。我们构造了一个变分原理的有效作用的形式，它给出了体流体动力学方程和自由表面边界条件。自由表面边界条件需要一个额外的边界项的行动，类似于1 +1D手征玻色子场耦合到背景几何。我们解决了深水情况下的线性化流体动力学方程，并推导出手征表面波的色散。我们表明，在长波长限制的流动剖面表现出振荡的旋涡边界层附近的自由表面。层的厚度由奇粘度与声速之比$\delta \sim \nu_o/c_s$给出的长度标度控制。在不可压缩极限下，$c_s\to \infty$涡边界层变得奇异，层内涡量发散为$\omega \sim c_s$。边界层是由奇粘性将速度$\boldsymbol\nabla \cdot \boldsymbol{v}$的散度耦合到涡量$\boldsymbol\nabla \times \boldsymbol{v}$而形成的。它的结果在非平凡的手征自由表面动力学，即使在没有外力。奇粘性诱导边界层的结构与传统的耗散剪切粘性自由表面边界层有很大的不同。
We consider an isotropic compressible non-dissipative fluid with broken parity subject to free surface boundary conditions in two spatial dimensions. The hydrodynamic equations describing the bulk dynamics of the fluid as well as the free surface boundary conditions depend explicitly on the parity breaking non-dissipative odd viscosity term. We construct a variational principle in the form of an effective action which gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resembles a $1+1D$ chiral boson field coupled to the background geometry. We solve the linearized hydrodynamic equations for the deep water case and derive the dispersion of chiral surface waves. We show that in the long wavelength limit the flow profile exhibits an oscillating vortical boundary layer near the free surface. The thickness of the layer is controlled by the length scale given by the ratio of odd viscosity to the sound velocity $\delta \sim \nu_o/c_s$. In the incompressible limit, $c_s\to \infty$ the vortical boundary layer becomes singular with the vorticity within the layer diverging as $\omega \sim c_s$. The boundary layer is formed by odd viscosity coupling the divergence of velocity $\boldsymbol\nabla \cdot \boldsymbol{v}$ to vorticity $\boldsymbol\nabla \times \boldsymbol{v}$. It results in non-trivial chiral free surface dynamics even in the absence of external forces. The structure of the odd viscosity induced boundary layer is very different from the conventional free surface boundary layer associated with dissipative shear viscosity.