Periodic nonlinear oscillations of a bubble between rigid parallel plates

Abstract

A linear analysis of forced bubble oscillations between rigid parallel plates that was presented at the previous ASA meeting is extended to include periodic nonlinear oscillations of the bubble. An equation of Rayleigh–Plesset form is obtained that satisfies the boundary conditions on the plates. Compressibility of the liquid is taken into account with radiation damping, and with time delays associated with acoustic reflections from the plates. The equation is expanded to cubic order in the perturbation of the bubble radius, and the radius is expanded as a Fourier series in harmonics of the acoustic drive frequency. The result is a system of coupled nonlinear algebraic equations for the Fourier coefficients, the solutions of which describe the steady‐state response of the bubble. Solutions were obtained numerically by iteration. An advantage of this frequency‐domain approach is that the infinite sequence of pressure waves reflected from the plates can be summed analytically. Time histories and frequency responses are presented as functions of plate separation, the amplitude of the drive, and its frequency. This general approach can also be used to investigate moderate subharmonic generation. [Work supported by NIH Grant EB004047 and ARL IR&D funds.]