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

Abstract

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.

Viscous effects on the oscillations of two equal and deformable bubbles under a step change in pressure

“…Now, the bubble shapes that arise towards the end of the simulations are wavy in both their front and rear sides. Such shapes were called 'globally deformed shapes' by Pelekasis & Tsamopoulos (1993a) and earlier observed by Kornefeld & Suvorov (1944) for isolated bubbles. These are clearly distinct from the 'spherical-cap' shapes predicted above when ε = 1 (figure 12).…”

“…in Pelekasis & Tsamopoulos (1993a), where the liquid surrounding the two bubbles was assumed to be inviscid and a much larger dimensionless hydrostatic pressure was used, resulting in larger eigenvalues of the zeroth mode. In Newtonian liquids as well, we find that as D decreases, both parts of the eigenvalue corresponding to in-phase oscillations decrease, while those corresponding to out-of-phase oscillations increase.…”

“…This occurs as in Pelekasis & Tsamopoulos (1993a), in spite of the fact that the bubble radii are very small, and viscous effects are accounted in full here. In order to directly compare with the results in figure 9 of Pelekasis & Tsamopoulos (1993a), we increased the bubble radius to 1 mm (Oh −1 = 269.63, P = 1375.51), set their initial distance at 2.5 and the amplitude to 0.3.…”

“…We find that the time needed for the two bubbles to approach each other increases, in the case of ε = 0.2, allowing the simulations to proceed to much longer times. In order to examine whether the dependence of the acceleration on the disturbance still follows predictions for inviscid liquids (Batchelor 1967;Pelekasis & Tsamopoulos 1993a), i.e. it increases proportionately to ε 2 , we also carried out simulations for ε = 0.1, 0.3, 0.5, 0.7 and 0.8.…”

“…In order to directly compare with the results in figure 9 of Pelekasis & Tsamopoulos (1993a), we increased the bubble radius to 1 mm (Oh −1 = 269.63, P = 1375.51), set their initial distance at 2.5 and the amplitude to 0.3. In Pelekasis & Tsamopoulos (1993a), the average acceleration over the first period of volume oscillations was found to be 7.198, whereas it decreases here, as it should, to 6.72, because of viscous damping. Retaining all the physical parameters to their previous values, except for the initial interbubble distance, which we increase to 2.8, we find that the two bubbles deform at their rear side and flatten at the front, a picture similar to that of ε = 1.…”