On energy transfer in resonant bubble oscillations

Abstract

In this paper, energy transfer mechanism due to resonant interactions of bubble shape and volume deformation modes is studied. In the process, a number of recent investigations that have focused on specific sets of initial conditions are generalized. Using phase-space analysis and a reduction of the mode dynamics equations to an integrable Hamiltonian, two types of resonant interactions are considered. The first is the so-called one–two resonance, when the natural frequency of the breathing mode is twice (or approximately twice) that of a shape mode. In this case, it is found that there is nearly always a continuous, periodic exchange of energy between the shape and volume modes. However, there exists a particular class of initial conditions, that was the subject of most earlier studies, for which the energy transfer is one way, from the shape mode to the volume mode. The second type of resonant interaction occurs when the natural frequency of the breathing mode is approximately equal to that of the shape mode. Here, it is found that periodic energy exchange can take place if equilibrium shape of the bubble is nonspherical due, for example, to a surface pressure distribution or a steady extensional flow. The analysis of the one–two resonance case is used to explain the stability of an oscillating spherical bubble to small perturbations of shape. It is found that instability of small volume oscillations can lead to large shape oscillation.