Secondary Bjerknes forces between two bubbles and the phenomenon of acoustic streamers

Abstract

The translational velocities of two spherical gas bubbles oscillating in water, which is irradiated by a high-intensity acoustic wave field, are calculated. The two bubbles are assumed to be located far enough apart so that shape oscillations can be neglected. Viscous effects are included owing to the small size of the bubbles. An asymptotic solution is obtained that accounts for the viscous drag on each bubble, for large ${\it Re}$ based on the radial part of the motion, in a form similar to the leading-order prediction by Levich (1962), $C_{D} = 48/{\it Re}_{T}$; ${\it Re}_{T} \to \infty$ based on the translational velocity. In this context the translational velocity of each bubble, which is a direct measure of the secondary Bjerknes force between the two bubbles, is evaluated asymptotically and calculated numerically for sound intensities as large as the Blake threshold. Two cases are examined. First, two bubbles of unequal size with radii on the order of $100\,\umu$m are subjected to a sound wave with amplitude $P_{A} < 1.0$ bar and forcing frequency $\omega_{f} = 0.51\omega_{10}$, so that the second harmonic falls within the range defined by the eigenfrequencies of the two bubbles, $\omega_{10} < 2\omega_{f} < \omega_{20}$. It is shown that their translational velocity changes sign, becoming repulsive as $P_{A}$ increases from 0.05 to 0.1 bar due to the growing second harmonic, $2\omega_{f}$, of the forcing frequency. However, as the amplitude of sound further increases, $P_{A} \approx 0.5$ bar, the two bubbles attract each other due to the growth of even higher harmonics that fall outside the range defined by the eigenfrequencies of the two bubbles. Second, the case of much smaller bubbles is examined, radii on the order of $10\,\umu$m, driven well below resonance, $\omega_{f}/2\pi = 20$ kHz, at very large sound intensities, $P_{A} \approx 1$ bar. Numerical simulations show that the forces between the two bubbles tend to be attractive, except for a narrow region of bubble size corresponding to a nonlinear resonance related to the Blake threshold. As the distance between them decreases, the region of repulsion is shifted, indicating sign inversion of their mutual force. Extensive numerical simulations indicate the formation of bubble pairs with constant average inter-bubble distance, consisting of bubbles with equilibrium radii determined by the primary and secondary resonance frequencies for small and moderate sound amplitudes or by the Blake threshold for large sound amplitudes. It is conjectured that in experiments where ‘acoustic streamers’ are observed, which are filamentary structures consisting of bubbles that are aligned and move rapidly in a cavitating fluid at nearly constant distances from each other, bubbles with size determined by the Blake threshold are predominant because those with size determined by linear resonance are larger and therefore become unstable due to shape oscillations.