Instantaneous phase difference of nonlinear coupled bubbles from nonstationary forcing

Abstract

The attraction and repulsion regimes of acoustically forced bubbles follow from their phase coupling. The direct evaluation of phase difference has received less attention than the secondary Bjerknes force, though it has important implications for nonlinear, nonstationary forcing regimes. We numerically investigate the response of two nonlinearly coupled bubbles undergoing radial oscillations and translation from chirp forcing. The corresponding scattered acoustic responses are analyzed using the empirical mode decomposition (EMD) and Hilbert transform in terms of respective instantaneous amplitude, frequency, and phase. This allows for determination of the phase differences from the coupling between the two bubbles. The phase coupling results agree with previous analytical theory in the linear acoustic limit. Attraction and repulsion regimes for nonstationary forcing can be classified with respect to the instantaneous phase differences unlike the time averaged secondary Bjerknes force. In the nonlinear regime, the chirp direction (forward, backward) results in significantly different radial and translational responses. This result can be explained with respect to influence of the softening jump discontinuity (backbone curve) on the phase coupling. Notably a backward chirp results in overall greater amplitude response and thus more translation during attraction. Implications for optimal waveforms are highlighted for potential applications.