Dynamical Equations of Aspherical Bubbles Based on Lagrangian Formalism

在拉格朗日形式的框架下，结合气泡边界的多极展开，从理论上推导了相互作用的非球形气泡的动力学方程。气泡边界用球谐函数展开，以便处理气泡的变形和平移。为了构成系统的拉格朗日量，选择相应的球谐振幅作为广义坐标。推导出的方程显示了所观察到的行为，如气泡变形和发病的喷射。该研究主要考虑到球谐函数的阶数到第三模式（八极模式），因为八极模式是最低模式振荡，导致气泡的不对称变形，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.