Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure

It is well known from experiments in acoustic cavitation that two bubbles pulsating in a liquid may attract or repel each other depending on whether they oscillate in or out of phase, respectively. The forces responsible for this phenomenon are called ‘Bjerknes’ forces. When attractive forces are present the two bubbles are seen to accelerate towards each other and coalesce (Kornfeld & Suvorov 1944) and occasionally even breakup in the process. In the present study the response of two initially equal and spherical bubbles is examined under a step change in the hydrostatic pressure at infinity. A hybrid boundary–finite element method is used in order to follow the shape deformation and change in the potential of the two interfaces. Under the conditions mentioned above the two bubbles are found to attract each other always, with a force inversely proportional to the square of the distance between them when this distance is large, a result known to Bjerknes. As time increases the two bubbles continue accelerating towards each other and often resemble either the spherical-cap shapes observed by Davies & Taylor (1950), or the globally deformed shapes observed by Kornfeld & Suvorov (1944). Such shapes occur for sufficiently large or small values of the Bond number respectively (based on the average acceleration). It is also shown here that spherical-cap shapes arise through a Rayleigh–Taylor instability, whereas globally deformed shapes occur as a result of subharmonic resonance between the volume oscillations of the two bubbles and certain non-spherical harmonics (Hall & Seminara 1980). Eventually, in both cases the two bubbles break up due to severe surface deformation.