Modulational Instability of Microbubbles Surface Modes

Starting from the continuous Marmottant model[1], for the vibration of the bubble radius R(t), and a discrete nonlinear model of the discretized radius with periodic boundary conditions and nonlinear coupling between surface modes, MI in bubbles is investigated numerically. After a first step of Symmetry Analysis[2] applied to the nonlinear equations associated to the models, invariant properties allow an identification of dimensionless similarity variables that link both continuous and discrete models. Thanks to this similarity analysis, the bubble can be studied with a macroscopic mechanical 1D ring chain (diameter = 73 cm) using N=48 nonlinearly coupled pendulum of mass m=6g and length =3.2cm with periodic conditions. The second step consists in studying numerically the MI criterion versus the amplitude of external field applied to the bubble. The analysis and the experiments reveals the existence of Intrinsic Localized Modes (ILMs), similar to those found in other more generic systems of nonlinearly vibratory lattices. The observation of ILM in 1D chain of coupled oscillators[3, 4], presenting many similarities with the discrete bubble model, showed to play a role in creating spatio-temporal localized excitations leading to the breaking of the bubble. In our periodic pendulum lattice apparatus describing the macroscopic equivalence of the bubble, the system parametrically driven at 5.4 Hz presents subharmonic behavior and localized modes oscillating at 2.7 Hz. Versus the amplitude and the frequency of the driving, the mechanical ring presents parametric instability surface modes and localized modes oscillating at subharmonics of the parametric excitation.