Translation and oscillation of a bubble under axisymmetric deformation

In this work the nonlinear interactions between the axisymmetric shape distortions, the axial translational motion, and the volume oscillations of a gas bubble in an inviscid, incompressible liquid are considered. Representing the surface deformation by a complete set of Legendre polynomials and assuming that both this deformation and the translational motion are small, a Lagrangian energy formulation is used to derive a system of equations valid to third order in these interaction terms. The effects of surface tension and pressure are also accounted for to this order. No restriction is placed on the size of the volume oscillations. Examination of the translational motion equation indicates that while at second order the interaction of two neighboring shape modes can cause the bubble to move, at third order any combination of three odd shape modes or one odd and two even shape modes can result in the bubble moving. Third order interactions between any three even shape modes or one even and two odd modes will contribute to the volume oscillations of the bubble. Restricting attention to the first fifteen shape modes, a truncated subset of equations is obtained. For this system, the bubble is given an initial static deformation in shape and the process by which the other modes become excited is considered in detail for some chosen examples. The role of second and third order interactions is identified and general relations which determine which terms can interact to excite a given shape mode are given.