Bifurcation and dynamics of periodic solutions to the Rayleigh–Plesset equation: Theory and numerical simulation

Abstract

In this paper, we study the oscillation of a gas-filled spherical bubble immersed in an infinite domain of incompressible liquid under the influence of a time-periodic acoustic field. The oscillation is described by the Rayleigh–Plesset equation, which is derived from the Navier–Stokes equation under the assumptions of spherical symmetry. Taking the initial radius R0, or the initial instantaneous partial pressure Pg0, of the gas-filled bubble as a parameter, we present a saddle–node bifurcation of periodic solutions to the Rayleigh–Plesset equation in the parameter space, provided the mean value of applied pressure is greater than the vapor pressure. Further, we show that the Rayleigh–Plesset equation has two classes of periodic solutions, as R0 (resp. Pg0) tends to +∞, the first class of solutions uniformly diverges to +∞ at the rate of R05 (resp. Pg05/2), while the second class converges to 0 at the rate of R0−5 (resp. Pg0−5/2). As for the case where the density of the liquid ρ is chosen as a parameter, we observe an interesting nonlinear phenomenon that, for some ρ, the Rayleigh–Plesset equation may possess four positive T-periodic solutions: two of them are stable, while the rest of them are unstable. This is quite different from the phenomenon that occurs when R0 (or Pg0) is chosen as a parameter.