Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave

Abstract

Micro-cavitation bubbles generated by ultrasound have wide and important applications in medical ultrasonics and sonochemistry. An approximate theory is developed for nonlinear and non-spherical bubbles in a compressible liquid by using the method of matched asymptotic expansions. The perturbation is performed to the second order in terms of a small parameter, the bubble-wall Mach number. The inner flow near the bubble can be approximated as incompressible at the first and second orders, leading to the use of Laplace's equation, whereas the outer flow far away from the bubble can be described by the linear wave equation, also for the first and second orders. Matching between the two expansions provides the model for the non-spherical bubble behaviour in a compressible fluid. A numerical model using the mixed Eulerian–Lagrangian method and a modified boundary integral method is used to obtain the evolving bubble shapes. The primary advantage of this method is its computational efficiency over using the wave equation throughout the fluid domain. The numerical model is validated against the Keller–Herring equation for spherical bubbles in weakly compressible liquids with excellent agreement being obtained for the bubble radius evolution up to the fourth oscillation. Numerical analyses are further performed for non-spherical oscillating acoustic bubbles. Bubble evolution and jet formation are simulated. Outputs also include the bubble volume, bubble displacement, Kelvin impulse and liquid jet tip velocity. Bubble behaviour is studied in terms of the wave frequency and amplitude. Particular attention is paid to the conditions if/when the bubble jet is formed and when the bubble becomes multiply connected, often forming a toroidal bubble. When subjected to a weak acoustic wave, bubble jets may develop at the two poles of the bubble surface after several cycles of oscillations. A resonant phenomenon occurs when the wave frequency is equal to the natural oscillation frequency of the bubble. When subjected to a strong acoustic wave, a vigorous liquid jet develops along the direction of wave propagation in only a few cycles of the acoustic wave.