Numerical investigation of the strength of collapse of a harmonically excited bubble

Abstract

The nonlinear dynamics of an acoustically excited spherical gas bubble in water is being investigated numerically. The applied model to describe the motion of the bubble radius is the Keller–Miksis equation, a second order ordinary differential equation, which takes into account the compressibility of the liquid. During the radial oscillations of the bubble, it may enlarge and collapse violently causing high temperature and pressure or even launch a strong pressure wave at the collapse site. These extreme conditions are exploited by many applications, for instance, in sonochemistry to generate oxidising free radicals. The recorded properties, such as the very high bubble wall velocity, and maximum bubble radius of the periodic and chaotic solutions are good indicators for the strength of the collapse. The main aim is to determine the domains of the collapse-like behaviour in the excitation pressure amplitude–frequency parameter space. Results show that at lower driving frequencies the collapse is stronger than at higher frequencies, which is in good agreement with many experimental observations (Kanthale et al., 2007, Tatake and Pandit, 2002). To find all the co-existing stable solutions, at each parameter pair the model was solved numerically with a simple initial value problem solver (4th order Runge–Kutta scheme with 5th order embedded error estimation) by applying 5 randomly chosen initial conditions. These co-existing attractors have different behaviour in the sense of the collapse strength.