Galloping bubbles: spontaneous self-propulsions of vibrating bubbles along solid boundaries

Bubbles moving in a liquid are simple in appearance yet display a wealth of intriguing dynamics relevant to innumerable practical applications. Here, we demonstrate that vertically vibrated bubbles near a wall may undergo a spontaneous symmetry breaking in their harmonic shape oscillations, leading to steady propulsive motion. We characterize the dynamics of these self-propelled, or `galloping', bubbles in terms of the key system parameters, including bubble volume, driving frequency, and acceleration. Our results reveal that the bubble propulsion is intimately related to their resonant shape oscillations, which can be fine-tuned to produce a myriad of dynamics including rectilinear, circular, and zig-zag motions. We use direct numerical simulations to complement our experimental observations and extend the galloping instability to hemispherical bubbles. Our spectral analysis demonstrates that the galloping instability results from the excitation of modes asymmetric about the vibration axis. We then perform a stability analysis using perturbation theory and uncover the symmetry-breaking mechanism in terms of resonant shape modes. By computing the inertial reaction of an inviscid external liquid to bubble deformation we then describe the propulsion mechanism as a case of swimming in potential flow.