Pattern Identification on the Nonlinear Radial Motion of an Oscillating Spherical Bubble Using Neural Networks

Abstract

The main goal of this article is to study the oscillatory motion of a spherical gas bubble immersed in a Newtonian liquid subjected to a harmonic pressure excitation. We use the classical Rayleigh-Plesset equation to study the radial motion of the bubble undergoing a forcing acoustic pressure field. The second order nonlinear ordinary differential equation that governs the bubble motion is solved through a robust fifth order Runge-Kutta scheme with adaptive time-step. Several interesting patterns are identified. First we develop an asymptotic solution for low amplitudes of excitation pressure to validate our numerical code. Then we develop a bifurcation diagram in order to show how the parameters of the flow modify the vibrational patterns of the bubble. We also train a neural network to identify the vibrational pattern through its FFT data. The combination of neural networks with a bifurcation diagram could be useful for the identification of the flow physical parameters in practical applications. For each pattern we also provide an analysis of the motion of the bubble on the phase-space and interpret physically the system behavior with its FFT. In addition, we analyze nonlinear patterns using standard tools of dynamical systems such as Poincaré sections and calculate the Lyapunov exponents of the system. Based on that, we have identified topological transitions in phase plane using for instance the analysis of Poincaré sections and the solution in the frequency spectrum. We have seen that the mechanisms that dominate the dynamics of the oscillating bubble is the competition of the acoustic field excitation with surface tension forces and momentum diffusion by the action of the surrounding fluid viscosity.