Oscillation of a bubble in a liquid confined in an elastic solid

Abstract

A simple theoretical model is described for the oscillation of a gas bubble in a liquid in a cavity confined by an elastic solid. The phenomenon occurs in nature and technology but has only recently received attention. The compressibility effects in the continuity equation are shown to be negligible, using dimensional analysis. However, the volume change of the confined liquid has to be considered since the associated pressure variation is large. The variation of the cavity volume is assumed to be proportional to the change of the liquid pressure at the confinement wall. The Rayleigh-Plesset-like equation describing the dynamics of a confined bubble is derived, considering the viscous and surface tension effects. Using the linear stability analysis, we show that the bubble undergoes stable damping oscillation when it is subject to small disturbances. The natural frequency of oscillation is obtained analytically. The theory agrees well with recent experiments. The analyses show that the natural frequency of oscillation for a bubble in an elastic confinement is larger, in order of magnitude, than that in an unbounded liquid. The amplitude and period of oscillation of a transient bubble decrease significantly owing to the presence of a confinement, reaching a steady state in a much longer period and at a larger equilibrium radius. When subject to an acoustic wave, a bubble in a confinement oscillates at smaller amplitude. The effects of the confinement increase with the bulk modulus of the confinement and decrease rapidly with the cavity size but are still significant for a large cavity whose size is an order of magnitude larger than the bubble.