Surface stability of a bubble in a liquid fully confined in an elastic solid

This paper is concerned with the surface stability of a bubble in a liquid, fully covered by an elastic solid, which is common in natural and engineering applications. A theoretical model is described for the phenomenon, assuming that the bubble gas undergoes a polytropic process, the flow is irrotational, and the variation of the liquid volume is proportionally related to the liquid pressure. The boundary valued problem is modeled using the perturbation method via the Legendre polynomials, and the evolution equations for radial and shape oscillations of a confined bubble are formulated. We derive an explicit analytical expression for the natural frequency of shape modes for a confined bubble. The analyses lead to interesting conclusions including: the natural frequency of shape mode increases due to the confinement only if the cavity dimension is within 2-3 bubble radii, for which the shape oscillation of a bubble is always stable and damps with time; as the cavity dimension is 10 times larger than the bubble radius, the natural frequency for radial oscillation is significantly larger, while the natural frequency for shape mode remains the same as that for an unconfined bubble. An unconfined bubble experiences shape mode oscillation under the Mathieu relation. By contrast, shape mode resonance of confined bubbles occurs only when both the radial resonance condition and the Mathieu relation are satisfied, which is associated with a much larger driving frequency of ultrasound.