Regular and chaotic dynamics of a spherical bubble

Abstract

The problem of the translational motion and oscillations of the radius of a spherical gas bubble in a spherical flask filled with a weakly compressible liquid is considered when there is forced radial perturbation of the flask wall. The translational motion is that of the bubble under the action of an acoustic field when there is viscous drag and the effect of added mass. A system of differential equations is proposed which describes the combined oscillations of the radius and the translational motion of the bubble. This system includes an ordinary differential equation of the Herring-Flinn-Gilmore type which describes the evolution of the radius of the bubble. Bifurcation diagrams are constructed for the radius of the bubble, which reveal a repeating structure of the bifurcation set within the limits of the harmonic resonances of the system. The dynamic characteristics of the bubble for different equilibrium radii are investigated numerically in connection with the analysis of the stability of its position in space.