Translational instability of a bubble undergoing shape oscillations

Abstract

This paper studies the translational instability of an oscillating bubble. It is shown that when a spherical bubble undergoing volume oscillation becomes unstable, giving rise to shape oscillations of two neighboring modes, the translational mode is intimately coupled with the two shape modes and this results in translational instability of the bubble. The main contribution is twofold. First, the integral relations for motions of bubbles in an infinite perfect liquid are not relied on, hence result is applicable to liquids with weak viscous effect. Second, the method of deriving the amplitude equations, which is similar to that of normal form calculations for ordinary differential equations, has not been applied to partial differential equations before.