# Simulations of Microbubble Shell Dynamics

Microbubble surface modes appear when contrast agents are insonified with specific ultrasound pulses. In this case, the radius becomes a space-dependent function which can be expanded on the basis of spherical harmonics describing the spatial vibrational modes of the bubble. The radial symmetry breaking, which appears through a Modulational Instability (MI), is typical of many extended nonlinear systems subjected to an external driving.

Here we present a mechanical analogue of bubble pulsations, consisting in a macroscopic ring of coupled oscillators driven by parametric forcing. Depending on the amplitude and the frequency of the driving, the mechanical ring presents a parametric instability leading to surface modes and localized modes oscillating at subharmonics of the parametric excitation. Bubbles are used as contrast agent in medical ultrasound imaging and to carry some drug to special locations. The perspective of this analysis is to define practical optimized ultrasound pulses exciting the bubble and leading to drug delivery applications.

Modeling acoustically driven microbubbles by macroscopic discrete-mechanical analoguesV. Sánchez-Morcillo, N. Jiménez, S. D. Santos, J. Chaline, A. Bouakaz, N. Gonzalez,

*Modelling in Science Education and Learning*,

**6**(7), pp 75–87, (2013)

Macroscopic acousto-mechanical analogy of a microbubble

J. Chaline, N. Jiménez, A. Mehrem, A. Bouakaz, S. D. Santos, V. Sánchez-Morcillo,

*The Journal of the Acoustical Society of America*,

**138**(6), pp 3600-3606, (2015)

## Numerical solutions:

Localized mode (ILM):

This video shows the nonlinear dynamics of a coupled pendulum round chain. A localized mode (a breather) is observed in the numerical solution for a parametrical exitation of 1.95 $f_0$ of the small amplitude pendulum oscilation (f0).

Kink:

This video shows the nonlinear dynamics of a coupled pendulum round chain. A "kink" is observed in the numerical solution for a parametrical exitation of 2.7 of the small amplitude pendulum oscilation (f0).

Surface mode (m=2)

This video shows the nonlinear dynamics of a coupled pendulum round chain. The second surface mode Ym=2 is observed in the numerical solution for a parametrical exitation of 2.03 of the small amplitude pendulum oscilation (f0).

Surface mode (m=3)

This video shows the nonlinear dynamics of a coupled pendulum round chain. The third surface mode Ym=3 is observed in the numerical solution for a parametrical exitation of 2.07 of the small amplitude pendulum oscilation (f0).

Surface mode (m=4)

This video shows the nonlinear dynamics of a coupled pendulum round chain. A surface mode Ym=4 is observed in the numerical solution for a parametrical exitation of 2.09 of the small amplitude pendulum oscilation (f0).