Vibration modes of a bubble close to a boundary

Using the recently proposed bubble equation, we study the vibration characteristics of a bubble close to a solid boundary. The results indicate that a rigid boundary has an important effect on large-amplitude bubble vibration. Whether the bubble vibration is stable or unstable strongly depends on the distance between the initial bubble center and the solid boundary. Furthermore, it also depends on both the amplitude and the frequency of the external perturbation. It is found that the smaller the distance between the initial bubble center and the solid boundary, the larger the initial bubble radius and the larger both the amplitude and frequency of the external perturbation, the more easily the bubble vibration becomes unstable. It is shown that this unstable bubble vibration is possibly related to the production of a reentrant microjet for the bubble, which suggests a potential method for estimating bubble size and the distance between the bubble center and the solid boundary by exerting an external perturbation with controllable amplitude and frequency on the liquid. Furthermore, the dependence of the natural frequency of the bubble on the external pressure for small-amplitude vibration can reveal the bubble radius and the distance between the bubble center and the solid boundary. In addition, the vibration characteristics of a bubble close to a solid boundary under a periodic external perturbation are revealed. Several bubble vibration modes are identified; the strongest vibration modes are those with the natural frequency and the external vibration frequency.