Dynamical Equations of Aspherical Bubbles Based on Lagrangian Formalism

在拉格朗日形式框架下，从理论上推导了气泡边界多极展开的非球形气泡相互作用的动力学方程。用球面谐波扩展气泡边界，从而可以处理气泡的变形和平移。为了构成系统的拉格朗日，选择相应的球面谐波幅值作为广义坐标。导出的方程显示了观察到的气泡变形和喷射开始等行为。研究主要考虑到三阶振型（八极振型）的球谐波阶数，因为八极振型是导致气泡不对称变形的最低振型振荡，这对气泡的行为影响最大。

The dynamical equations for interacting non-spherical bubbles are derived theoretically in the framework of Lagrangian formalism with multipole expansion of the bubble boundaries. The bubble boundaries are expanded with spherical harmonics so that deformation and translation of the bubbles can be treated. In order to compose the Lagrangian of the system, corresponding amplitudes of the spherical harmonics are chosen as the generalized coordinate. The derived equations show the observed behaviour such as bubble deformation and onset of jetting. The study mainly takes account of the order of the spherical harmonics up to 3rd mode (octupole mode) because octupole mode is the lowest mode oscillation contributing asymmetrical deformation of the bubble and this has the most significant effect on the behaviour of the bubbles.