Dynamical Equations of Aspherical Bubbles Based on Lagrangian Formalism

在拉格朗日形式的框架下，考虑气泡边界的多极展开，从理论上推导了非球形气泡相互作用的动力学方程。气泡边界用球谐函数展开，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量，选取相应的球谐函数的幅值作为广义坐标。所导出的方程显示了观察到的气泡变形和喷流开始等行为。由于八极模是导致气泡不对称变形的最低模振荡，对气泡的行为有最大的影响，所以本文主要考虑球谐振级到三阶(八极模)。

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.