Nonspherical sub-millimetre resonantly excited bubble oscillations

Abstract

We consider the resonant forcing of a gas bubble by a sinusoidal pressure wave near the second harmonic of the oblate/prolate (k = 2) shape mode in an axisymmetric geometry. Employing a model which accounts for nonlinear shape mode interactions to third order, viscous effects to the same order (in the absence of vorticity) and weak compressibility, the transition in the predicted shape deformation as the forcing strength is increased is studied in detail, together with its subsequent impact on both the spherical oscillations of the bubble and any self-propulsion that may occur. For a marked range of forcing strengths the bubble is found to undergo stable oscillatory deformation which consists of even shape modes only and is dominated by the oblate/prolate mode, the amplitude of which increases with the forcing strength. However, if the forcing strength is sufficiently large the k = 3 shape mode is also found to become resonantly excited, the growth of which occurs on a slower timescale and whose oscillation pattern consists of two dominant frequencies, the largest of which corresponds to the driving frequency. Interaction between the stable k = 2 mode and the growing k = 3 mode is found to lead to a marked non-oscillatory displacement of the bubble with the rate of displacement increasing as the amplitude of the oscillations in the k = 3 shape mode increases. Increasing the forcing strength further causes the k = 3 mode to grow faster and thus causes the bubble to translate earlier.