Bjerknes force

Secondary Bjerknes forces arise when two oscillating bubbles are present in a pressure field. Attractive forces between bubbles with inphase pulsation cause coalescence. Bubbles oscillating out of phase are repelled. The bubble oscillation is in phase when both bubbles are smaller or larger than the resonance size and attractive forces dominate. If one bubble is smaller and one larger than the resonance size, they oscillate in and out of phase and repel one another. 

Primary and secondary Bjerknes forces lead to structures known as cavitation streamers. A large bubble above the resonance size oscillating with surface insta- 

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The coalescence of bubbles is driven by the two mechanisms. One is the attractive radiation force between bubbles called secondary Bjerknes force. The other is the other radiation force called the primary Bjerknes force which drives active bubbles to the pressure antinode of a standing wave field, ft should be noted, however, too strong acoustic wave repels bubbles from the pressure antinode as described in the next section [29, 30]. 

Radiation Forces on a Bubble (Primary and Secondary Bjerknes Forces)... 

Both the primary and secondary Bjerknes forces are originated from the pressure gradient across a bubble [35]. 

When the driving ultrasound is a standing wave, p(x t) is expressed as follows for the primary Bjerknes force. 

While the secondary Bjerknes force is always attractive if the ambient radius is the same between bubbles, it can be repulsive if the ambient radius is different [38]. The magnitude as well as the sign of the secondary Bjerknes force is a strong function of the ambient bubble radii of two bubbles, the acoustic pressure amplitude, and the acoustic frequency. It is calculated by (1.5). 

Matula TJ, Cordry SM, Roy RA, Crum LA (1997) Bjerknes force and bubble levitation under single-bubble sonoluminescence conditions. J Acoust Soc Am 102 1522-1527... 

Mettin R, Akhatov I, Parlitz U, Ohl CD, Lauterbom W (1997) Bjerknes forces between small cavitation bubbles in a strong acoustic field. Phys Rev E 56 2924—2931... 

As particles gather and the interparticle distances become small, the influence of scattered sound from neighbouring particles increases, which in turn gives rise to an interparticle acoustic force that is commonly named secondary forces, or Bjerknes forces named after Bjerknes. He first described the theory behind this phenomena, Equation 44.3, where a is the radius of the particle, d is the distance between the particles, and 0 is the angle between the centre line of the particles and the direction of propagation of the incident acoustic wave. 

FIGURE 44.1 Fluorescently labeled polystyrene particles gathered in the pressure nodal plane (in plane with the image focus) of an acoustic standing wave by the primary acoustic radiation force in a 70 xm deep microchannel. The microbeads are also clustered in a dense hexagonal pattern by Bjerknes forces. 

Like the Bjerknes forces, solid particles whose acoustic properties differ from those of the liquid are subject to primary and secondary forces. Primary forces drive particles into pressure nodes, where secondary forces between particles are responsible for further aggregation. 

The number of bubbles is reduced by the approach and coalescence of bubble under the action of secondary Bjerknes forces. New bubbles are created by the violent collapse of cavitating bubbles. Their number b,civ and the number j of daughter bubbles created on collapse have to be calculated along with the total number of bubbles. The coefficients fecou and fecoii have to be adjusted to experimental findings and are generally a function of the applied local sound pressure. 

Keywords Bjerknes force Bubble breakup Bubble interaction Bubble oscillation Chaotic oscillation Damping rate Droplet oscillation Nonlinear oscillation Oscillation frequency RPNNP equation Shape modes Spherical harmonics Volume oscillation... 

Secondary Radiation Forces In addition to the axial and lateral radiation forces attributable to the primary acoustic field, secondary acoustic forces are produced between particles themselves. These particle-particle interactions, known as Bjerknes forces, aid the formaticai of aggregates within a standing wave, but are negligible until the particles are in close proximity. 

Bubbles not only interact with their environment but also with the sound field and the neighboring bubbles. In this section, the problem of the sound field propagation in a bubbly liquid is not addressed because the conditions considered in experimental and theoretical studies are rather far from those encoimtered in sonochemistry. We will focus on the forces between bubbles and the acoustic field (primary Bjerknes force) and between bubbles (mutual or secondary Bjerknes force).H9 Though the primary Bjerknes force is present in progressive... 

Recent theoretical analyses on bubble interaction are worth mentioning. Oguz reported the extreme versatility in bubble behavior in a 20-kHz acoustic field under "stable cavitation" upon slight changes in the bubble s radii, distance, and acoustic pressure.124 Shape oscillations develop as a result of the secondary Bjerknes force and may lead to the break-up of the bubble or a strong jet inside it. Beside the work by d Agostino and Brennen,i25 Chahine et ah studied the... 

Figure 28 - Bjerknes force between bubbles oscillating in phase (left) and out-of-phase (right) at maximum expansion of the left-hand bubble (top) and 50 ps after collapse, when jets are the largest (bottom) maximal bubble diameter 2.5 mm...

One of the important application area of ultrasound (US) is catalytic reactions with the participation of low molecular mass compounds and heterogeneous catalysts. The effect of ultrasound on catalytic reactions in the presence of platinum and rhodium catalysts of various dispersities was investigated in Ref [1]. It was demonstrated that ultrasound can provide for the occmrence of chemical processes that cannot be performed even in the presence of catalysts. It is assumed that the main mechanism of its action on catalytic processes consists in the dispersion of catalyst particles however, as was shown in Ref [1], the adhesion of particles can occur during the action of the so-called Bjerknes forces, that is, forces that promote the attraction of particles (primarily small particles) to a deformed bubble followed by their sticking together. As a consequence, the diffusion of reagents to the surface of a particle becomes more pronoimced and the rate of the process inereases. 

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