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.

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

“…(iv) Even a large step increase in the ambient pressure is ineffective for bubbles in pure water with R * b < 5 µm and for more viscous than water liquids even for larger bubbles. (v) A step change in pressure always leads to bubble attraction, although the amplitude of the disturbance was very large, in agreement with the linear analysis, possibly because the viscous damping decreased the oscillation amplitudes and the duration of the interaction, both of which are necessary to observe deviations from this analysis (Pelekasis et al 2004). (vi) Our predictions are in agreement with those of Doinikov (1999), although we have allowed the bubbles to be very close to each other and to deform.…”

“…In other words, fluid viscosity enhances the stabilizing effect of the extensional flow at the front side of the bubble. Table 4 also demonstrates that both Re vi and Re tr increase with the bubble radius and the latter is more than an order of magnitude smaller than the former, validating the assumptions of Pelekasis et al (2004).…”

“…In an attempt to explain the formation of the so-called bubble grapes, it has been shown by Pelekasis et al (2004) and Mettin et al (1997) that bubbles at very large distances from each other in an oscillatory pressure field can accelerate towards or away from each other or remain at a constant distance after a very large number of cycles of volume oscillations. One of the basic assumptions made in these studies was that the bubbles could retain their spherical shape and weak viscous effects were introduced, in the form of boundary layers around each bubble in the first of the above studies.…”

“…For the problem of bubble-bubble interaction, efforts to evaluate the secondary Bjerknes force in the context of linear oscillations were reported by Zabolotskaya (1984) and later on by Doinikov & Zavtrak (1995) and Doinikov (1999), who included viscous Viscous effects on two deformable bubbles under a step change in pressure 515 effects on interacting spherical bubbles. More recently, Pelekasis et al (2004) have made a systematic effort to evaluate the two available mechanisms that explain the formation of stable bubble clusters, when bubbles oscillate below resonance under an intense acoustic disturbance, extending the earlier work of Oguz & Prosperetti (1990) and Mettin et al (1997). In this effort, the viscous boundary layers around each bubble were systematically accounted for, but the bubbles were still assumed to remain spherical because of the large interbubble distance compared to their radii.…”

Streaming flow by oscillating bubbles: quantitative diagnostics via particle tracking velocimetry

“…However, bubbles in arrays or clouds interact in a non-trivial way. Particularly complex is the case of high-amplitude and nonlinear oscillations regime (high-power driving), where the bubble might eventually pinch off and emit smaller bubbles (Pelekasis et al 1999;Fernandez Rivas et al 2013b;Stricker et al 2013). The case of two bubbles driven in the low-amplitude oscillations regime has been recently studied numerically by Doinikov & Bouakaz (2016), revealing that larger velocities (and therefore larger wall shear stresses) should be expected as the bubbles are located closer to each other.…”