Nonlinear modal evolution of microbubbles confined by rigid capillary constraint

The nonlinear modal evolution of confined bubbles was investigated by applying modulated acoustic pressure in capillaries. Within a modulation envelope of the pressure, the bubble undergoes a morphological evolution cycle, including spherical oscillation, unstable deformation, surface mode and recovery to breathing mode. This cyclic process depends on bubble size, acoustic pressure and frequency, as well as capillary shape, while also relating to the translational state of the bubble. However, acoustic condition and constrains should be the dominant factors. The time-resolved radius reveals the presence of subharmonic responses, and fundamental oscillations may be suppressed. The oscillations of a pair of bubbles are always out of phase. Bubbles with sub-resonant size have more complicated modal oscillation, accompanying a fast translational motion. Strong capillary confinement may suppress translational motion. At higher drive frequencies, bubbles may be more likely to evolving into chaotic states, and modal superposition becomes pronounced during unstable deformation. The energy distribution spectrum of the three motion components of bubbles indicates that energy conversion and transfer are occurring simultaneously. When the stable surface mode appears, the other two components may be suppressed. A method for estimating unstable deformation thresholds and modal thresholds is proposed, revealing only a slight difference between the two. Consequently, stable surface modes can only exist within a narrow range of acoustic pressure, offering new insights into elucidating the evolutionary mechanisms of bubble shape patterns.