Dynamics and stability of a hemispherical bubble under vertical oscillation

The study of shape oscillation in gas bubbles immersed within a liquid domain has a long history in hydrodynamics, tracing back to Lord Rayleigh's work on their natural vibration modes. We focus on the dynamics of a hemispherical bubble attached to a rigid wall under external periodic forcing, which is of interest in numerous practical applications. We combine quasi-potential theory and multiscale perturbation methods to investigate the frequency response of the bubble when the forcing amplitude is small. We identify the parameter regimes for the stability of different shape oscillation modes, which can exhibit both harmonic and subharmonic responses depending on the forcing magnitude. We discuss the role of bubble geometry and fluid properties on the emergence of these modes, and their potential to generate directed propulsive motion. Theoretical predictions are compared to experiments and simulations.