Analysis of time delay effects on a linear bubble chain system.

A chain of vertically rising discrete air bubbles represents a transition phenomenon from individual to continuum behavior in a bubbly liquid. Previous studies have reported that there is a preference for acoustic energy to propagate along the bubble chain and that this behavior could be explained by a coupled-oscillator model. However, it has recently been demonstrated that quantitative results from the coupled-oscillator model do not match experimental data. In this paper, it is shown how adding time delays to the coupled-oscillator model can produce results that are in better agreement with experimental data. In addition, the effects of time delays on the natural frequencies and damping of individual eigenmodes of the vertical bubble chain are also investigated. It was found that adding time delays can dramatically change the damping of the different modes of the system while having less dramatic impact on the natural frequencies of the individual eigenmodes. Counterintuitively, it is found that the effects of time delays appear to be more important when the bubbles are closer together than when they are farther apart.