Effects of the second harmonic on the secondary Bjerknes force

The time-averaged interaction force exerted by an acoustic field between two gas bubbles, known as the secondary Bjerknes force, is derived with an accuracy up to a component induced by the second harmonic of bubble oscillations. The surrounding medium is assumed to be an incompressible viscous liquid and the distance between the bubbles much larger than their radii. It is shown that the second-harmonic component of the interaction force in many cases prevents the bubbles from coalescing, causing them either to repel each other or to form a bound pair with some stable separation. This can occur providing the imposed field is strong enough so that the second-harmonic force component is comparable to the ``linear'' (produced by the linear oscillations of the bubbles) interaction force. The obtained results are of interest in understanding collective bubble phenomena in strong acoustic fields, such as cavitation streamer formation.