Nonlinear normal modes and localization in two bubble oscillators

Abstract

We investigated a bifurcation structure of coupled nonlinear oscillation of two spherical gas bubbles subject to a stationary sound field by means of nonlinear modal analysis. The goal of this paper is to describe an energy localization phenomenon of coupled two-bubble oscillators, resulting from symmetry-breaking bifurcation of the steady-state oscillation. Approximate asymptotic solutions of nonlinear normal modes (NNMs) and steady state oscillation are obtained based on the method of multiple scales. It is found that localized oscillation arises in a neighborhood of the localized normal modes. The analytical solutions of the amplitude and the phase shift of the steady-state oscillation are compared to numerical results and found to be in good agreement within the limit of small-amplitude oscillation. For larger amplitude oscillation, a bifurcation diagram of the localized solution as a function of the driving frequency and the separation distance between the bubbles is provided in the presence of the thermal damping. The numerical results show that the localized oscillation can occur for a fairly typical parameter range used in practical experiments and simulations in the early literatures.

The Numerical Investigation on Bubble Interaction Dynamics in Hydrodynamic Cavitation

“…[22]. Sugita and Sugiura [23] investigated the bifurcation of two spherical bubbles by the method of nonlinear model analysis. Valier-Brasier and Conoir [24] analyzed the resonant acoustic scattering of two spherical bubbles using the addition theorem and spherical harmonics expansions.…”

Asymmetricity and sign reversal of secondary Bjerknes force from strong nonlinear coupling in cavitation bubble pairs

“…Thus, it can be inferred that the nonlinearity inherent in bubble dynamics make them a subject difficult for investigation [28,29]. This can be ascertained from the fact that the study of bubble dynamics has benefited chaos theory and viceversa [29][30][31].…”

“…Besides, the coupling effects between bubbles separated by small distances in strong acoustic fields may be significant enough to alter the behavior of the whole cluster [41]. Unfortunately, the nonlinearity in bubble-dynamics has mostly been approximated by the use of linear theories [28,42]. The inadequacy of such an approach has already been accepted by the scientific community [13,38,43,44].…”

In-phase synchronization between two auto-oscillating bubbles

“…At the heart of these spatiotemporal structures is the mutual interaction between neighboring bubbles. While the interaction of many bubbles has received a great deal of interest [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], the study of the specific interaction between two oscillating bubbles remains an experimental challenge [5,9,13,[16][17][18]. Besides experimental complexity, a more fundamental problem is that the acoustic interaction of two bubbles is clouded by the external sound field driving the bubbles into oscillations.…”