二层流体中波动问题的Hamilton正则方程

研究了两种常密度不可压缩理想流体组成的垂直分层的二流体系统的无旋等熵流动，考虑了上层流体与空气及两层流体间的表面张力。流动区域在水平方向无限伸展，上层流体有限深度，下层流体无限深。利用自由面及分界面相对于静止时平衡位置的偏移以及两层流体的速度势构造了Hamilton函数。为导出Hamilton正则方程引用了Euler描述下的流体运动的变分原理。自由面的位移是Hamilton意义下的正则变量，其对偶变量是上层流体在自由面上取值的速度势与密度的乘积。另一个正则变量是分界面的位移，其对偶变量是下层流体的密度与下层流体速度势在分界面上所取值的乘积减去上层流体密度与上层流体速度势在分界面上所取值的相应乘积。导出的Hamilton结构对分析分层流动中表面波与内波的相互作用是重要的。

Hamiltonian Canonical Equations of Wave Motion in Two Layers of Fluid

The irrotational and isoentropic flow of a vertically stratified two-fluid system consisting of incompressible ideal fluids of constant density was studied. The surface tension between upper fluid and air and the interfacial tension between two fluids were taken into account. The flow domain has unbounded horizontal extension. The upper fluid has finite depth, the lower fluid is infinitely deep. Hamiltonian function was constructed by using the velocity potentials of two fluids and the displacements of free surface and interfacace relative to the equilibrium positions when the fluids are at rest. Variational principle for Eulerian formulation of the fluid motion was applied to derive Hamiltonian canonical equations. The displacement of free surface is canonical variable in Hamilton's sense, and its conjugate variable is the product of the velocity potential evaluated at the free surface and the density of the upper fluid. The displacement of the interface is another canonical variable, and its conjugate variable is the product of the density of lower fluid and the velocity potential of lower fluid evaluated at the interface minus the corresponding product of the density of upper fiuid and the velocity potential of upper fluid evaluated at the interface. The Hamiltonian structure derived is important in analyzing the interaction between surface-waves and internal waves in stratified flows.