Two Bubbles

In the case of two bubbles the situation becomes more complex since the bubbles will interact with each other through acoustic coupling. Typical experimental results of two bubbles subjected to Pm = —2 MPa, but with different interdistances d, are shown in Fig. 7.6. In the left case (d = 400 pm), both bubbles remain essentially spherical during their whole lifetime (so-called weak 

Indeed, the P(t) curve resulting from this extended Rayleigh-Plesset equation (7.2) leads to a delayed bubble collapse for the two-bubble case, see Fig. 7.5, in good quantitative agreement with the measurements, despite the fact that the bubbles feel an anisotropic pressure field around their surface, especially in the last stage of their collapse. When the bubbles come closer to each other (middle 

---

The preceding conclusion is easily verified experimentally by arranging two bubbles with a common air connection, as illustrated in Fig. II-2. The arrangement is unstable, and the smaller of the two bubbles will shrink while the other enlarges. Note, however, that the smaller bubble does not shrink indefinitely once its radius equals that of the tube, its radius of curvature will increase as it continues to shrink until the final stage, where mechanical equilibrium is satisfied, and the two radii of curvature are equal as shown by the dotted lines. 

Interfacial Forces. Neighboring bubbles in a foam interact through a variety of forces which depend on the composition and thickness of Hquid between them, and on the physical chemistry of their Hquid—vapor interfaces. For a foam to be relatively stable, the net interaction must be sufficiently repulsive at short distances to maintain a significant layer of Hquid in between neighboring bubbles. Otherwise two bubbles could approach so closely as to expel all the Hquid and fuse into one larger bubble. Repulsive interactions typically become important only for bubble separations smaller than a few hundredths of a micrometer, a length small in comparison with typical bubble sizes. Thus attention can be restricted to the vapor—Hquid—vapor film stmcture formed between neighboring bubbles, and this stmcture can be considered essentially flat. 

If, by one of the above procedures, a few or even many bubbles have been introduced into a liquid, there is still no foam. In a foam, films between the bubbles are thin otherwise, the system is a gas emulsion. How, then, can a true foam be achieved If it is assumed that, because of some kind of stirring, two bubbles move to meet each other and the liquid layer between them gets thinner and thinner and if this process continues for a sufficient time, the two bubbles will touch and, eventually, coalesce. Many such encounters would destroy the foam. It is clear, therefore, that bubbles should be free to approach each other closely, but should be unable to cross the last short fraction of the initial distance. 

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). 

Fig. 5 shows the simulated air-bubble formation and rising behavior in water. For the first three bubbles, the formation process is characterized by three distinct stages of expansion, detachment, and deformation. In comparison with the bubble formation in the air-hydrocarbon fluid (Paratherm) system, the coalescence of the first two bubbles occurs much earlier in the air-water system. Note that the physical properties of the Paratherm are p — 870kg/m3, Pi — 0.032 Pa - s, and a — 0.029 N/m at 25 °C and 0.1 MPa. This is due to the fact that, compared to that in the air-Paratherm system, the first bubble in the air-water system is much larger in size and hence higher in rise velocity leading... 

Record the shape of this simulated molecule and measure the bond angle between the centres of the two bubbles where the nuclei of the atoms would be located. 

Further, if an insonation frequency of 50 kHz is employed neither of these two bubbles undergo collapse (Figs. 2.18 and 2.19). Only a bubble close to the resonance size (0.6 X 10 cm) will undergo collapse at the higher frequency (Fig. 2.20). 

Figure 2.7 shows a system that initially shows two bubbles of different curvature. After the tap is opened, the smaller bubble is found to shrink, while the larger bubble (with lower AP) increases in size until equilibrium is reached (when the curvature of the two bubbles become equal in magnitude). This type of equilibrium is the... 

FIGURE 2.6 Coalescence of two bubbles with different radii. 

FIGURE 2.7a,b Equilibrium state of two bubbles of different radii (see text). 

The foam as TLF has a very intriguing structure. If (1) two bubbles of the same radius come into contact with each other, this leads to (2) the formation of contact area and subsequently to (3) formation of one large bubble. 

Use the Laplace equation to calculate the spherical radius of the soap film which is formed by the contact of two bubbles with radii of 1 and 3 cm. Assume that the soap bubbles have a surface tension of 30 mj m . Draw a sketch of the contacting bubbles to help you. 

In order to understand the basis for the prevention of bubble coalescence and hence the formation of foams, let us examine the mechanical process involved in the initial stage of bubble coalescence. The relatively low Laplace pressure inside bubbles of reasonable size, say over 1 mm for air bubbles in water, means that the force required to drain the water between the approaching bubbles is sufficient to deform the bubbles as illustrated in Figure 8.2. The process which now occurs in the thin draining film is interesting and has been carefully studied. In water, it appears that the film ruptures, joining the two bubbles, when the film is still relatively thick, at about lOOnm thickness. However, van der Waals forces, which are attractive in this system (i.e. of air/water/air), are effectively insignificant at these film thicknesses. 

Pass a slow stream of thoroughly dried hydrogen chloride through the tube at a rate of one or two bubbles in three seconds during four hours. Gradually raise the temperature in the first zone to 1000 °C, and in the second zone to 800 °C. Do this during two hours, retain the maximum temperature in the furnace for half an hour, and... 

The dry air is best introduced through a tube leading to the bottom of the flask it is well not to disconnect the condenser, but to note the point at which no more drops condense. The current of dry air should be quite slow—not more than two bubbles per second in the sulfuric acid wash bottle. 

The specific ability of certain finely divided, insoluble solids to stabilize foam has long been known [Berkman and Egloff, op. cit., p. 133 and Bikerman, op. cit., Chap. 11]. Bartsch [Kolloidchem. Beih, 20, 1 (1925)] found that the presence of fine galena greatly extended the life of air foam in aqueous isoamyl alcohol, and the finer the solids, the greater the stability. Particles on the order of 50 pm length extended the life from 17 seconds to several hours. This behavior is consistent with theory, which indicates that a solid particle of medium contact angle with the liquid will prevent the coalescence of two bubbles with which it is in simultaneous contact. Quantitative observations of this phenomenon are scanty. 

In order to prevent the decomposition of the higher chlorides of silicon, it is very important that only a small part of the reaction tube should be heated at a time. When the chlorine is first passed through the reaction tube, the temperature of the heating coil is about 250°C. when the reaction is well started, it is lowered to about 150°C. The best rate of flow of chlorine is less than two bubbles per second. Under these conditions, in about 12 or 14 days, all the calcium-silicon will be used up, and about 700 ml. of liquid silicon chlorides will be collected. 

The transition between these two bubble categories was unclear and strongly dependent on the presence of surfactants. An expression for calculation of the mass transfer coefficients in this transition region was not given. 

Coalescence occurs when two bubbles approach each other, collide, and become one bigger bubble. Two important factors are ... 

In the case of a liquid film separating two bubbles in a foam, and where only the electrical and van der Waals forces are considered,... 

In pure liquids, gas bubbles will rise up and separate, more or less according to Stokes law. When two or more bubbles come together coalescence occurs very rapidly, without detectable flattening of the interface between them, i.e., there is no thin-film persistence. It is the adsorption of surfactant, at the gas-liquid interface, that promotes thin-film stability between the bubbles and lends a certain persistence to the foam structure. Here, when two bubbles of gas approach, the liquid film thins down to a persistent lamella instead of rupturing at the point of closest approach. In carefully controlled environments, it has been possible to make surfactant-stabilized, static, bubbles, and films with lifetimes on the order of months to years [45],... 

Figure 15 shows an example of typical images of the illuminated tracer particles for PIV/LIF and the SIT images of two bubbles in three representative positions. It is clearly seen that the tracer particles in the wake and a partial outline of the bubble boundaries are visible on the PIV image. The number of tracer particles is sufficient enough to calculate velocity vectors through the normal cross-correlation-based PIV algorithm. 

Figure 15 An example of PIV/LIF/SIT taken images (a) schematic of experimental conditions and ranges (b) typical PIV image showing illuminated tracer particles around the two bubbles (c) typical shadow images of the two bubbles at three instants (Tokuhiro et al, 1999).

Figure 16 An example of PIV/LIF/SIT taken turbulent velocity fields in the wake region of two bubbles together with the bubble shadows at four instants (Tokuhiro...

Two bubbles, one of radius 1 cm and the other of radius 2 cm, stick together. In which direction does the interface between the two bubbles curve ... 

The active surface of the platinized foil electrode should be covered completely by the solution when in use. It should be supplied with pure hydrogen gas at the rate of one to two bubbles per second from a jet about 1 mm in diameter. To avoid changes in concentration, the hydrogen can be passed through a saturator that contains water or cell solution before it enters the cell. 

---