Stability bubble shape

As discussed in Section 1.2.2 the bubble shapes in fairly dry foams and froths (4 gas > 0.83, approximately) are not spheres or distorted spheres, but polyhedrons. In practice there will be distributions of both gas-cell sizes and shapes. In addition to the gas bubbles, froth contains the floated particles, pulp liquor, and a fraction of (hydrophilic) particles that did not float due to bubble attachment, but which were mechanically entrained in the froth. The pulp liquor and these latter particles all have to be allowed to drain back out of the froth. The rate of this drainage will be greatest at the froth-pulp interface (i.e., the bottom of the froth layer) and slowest near the top of the froth layer. Froth drainage equations are discussed elsewhere [53]. The froth needs to be a stable enough foam that some time can be allowed for these drainage processes, and also so that the upper layer(s) of the froth can be swept out of the flotation cell. On the other hand, the froth should not be too stable as a foam so that it will break easily after collection. In addition to the role of the frother, froth stability is also promoted by increasing liquid viscosity. 

Now, I (cos 6) = cos 6. Thus, if a (t) / 0, the center of the bubble translates a distance ii (t) from the origin, but the bubble shape is unchanged. To discuss the stability of the spherical shape, we must determine whether the coefficients Uk(t) for k > 2 increase or decrease with time. 

Thus, for each deformation mode k, there are a discreet set of frequencies for which the bubble shape is unstable. This is consistent with the stability diagram in Fig. 4-17 for the limit eb -> 0. 

Thus, if (b - af)2) > 0, solution (4—339) is unstable because the slowly varying amplitude hmctions grow exponentially in time. On the other hand, for (b - af 2) < 0, the solution is stable with a two-time scale oscillation in the bubble shape. The stability boundary thus occurs at... 

For non-spherical bubbles the effect of polymeric additives becomes essential at lower eoneentrations, as eompared to the spherieal bubbles. For example, retardation of the bubble eollapse near a solid wall was observed in sueh eoncentration interval where dynamies of spherical bubbles has remained unehanged. In Figures 7.2.8 and 7.2.9 the data are reprodueed, where the Rayleigh time, to, was ehosen for a scaling time, and for curves 1- 4 R max/L = 0.5, 0.56,1.39,1.25, respectively. The bubble collapse is accompanied by generation of a microjet towards the wall and addition of polymer led to stabilization of the bubble shape and retardation of the jet formation. This effect is connected with the increase in the elongational viscosity of a polymeric solution in flow around collapsing bubble. 

Relatively large bubbles with low excess pressure are likely to behave as if they were closed, whereas the reverse should hold true for very small bubbles. Thus, upon increasing the bubble size, a transition takes place from instability to practically complete stability. Invoking shape deformations will not necessitate any major modifications of the above discussion as the spherical shape is always associated with minimal free energy for a given bubble volume. 

Foaming by discharging the melt into the atmosphere, on the other hand, produced closed cells with a mean cell diameter of ca. 500 LLm. The foam cells were polyhedral in shape and quite uniform in size, which is comparable to a commercial foam with a mean cell size of ca. 1.0 mm. But, the DDC foams contained slightly collapsed cellular structures. Seemingly, this foaming process created a more rapid and uniform cooling of the melt, causing a rapid build-up of viscosity, which substantially stabilized bubbles. 

In Fig. 1.1, the parameter space for transient and stable cavitation bubbles is shown in R0 (ambient bubble radius) - pa (acoustic amplitude) plane [15]. The ambient bubble radius is defined as the bubble radius when an acoustic wave (ultrasound) is absent. The acoustic amplitude is defined as the pressure amplitude of an acoustic wave (ultrasound). Here, transient and stable cavitation bubbles are defined by their shape stability. This is the result of numerical simulations of bubble pulsations. Above the thickest line, bubbles are those of transient cavitation. Below the thickest line, bubbles are those of stable cavitation. Near the left upper side, there is a region for bubbles of high-energy stable cavitation designated by Stable (strong nf0) . In the brackets, the type of acoustic cavitation noise is indicated. The acoustic cavitation noise is defined as acoustic emissions from... 

Fig. 1.1 The regions for transient cavitation bubbles and stable cavitation bubbles when they are defined by the shape stability of bubbles in the parameter space of ambient bubble radius (R0) and the acoustic amplitude (p ). The ultrasonic frequency is 515 kHz. The thickest line is the border between the region for stable cavitation bubbles and that for transient ones. The type of bubble pulsation has been indicated by the frequency spectrum of acoustic cavitation noise such as nf0 (periodic pulsation with the acoustic period), nfo/2 (doubled acoustic period), nf0/4 (quadrupled acoustic period), and chaotic (non-periodic pulsation). Any transient cavitation bubbles result in the broad-band noise due to the temporal fluctuation in the number of bubbles. Reprinted from Ultrasonics Sonochemistry, vol. 17, K.Yasui, T.Tuziuti, J. Lee, T.Kozuka, A.Towata, and Y. Iida, Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles, pp. 460-472, Copyright (2010), with permission from Elsevier...

Ice cream emulsion has a very characteristic degree of stability. The air bubbles should remain dispersed, but as soon it melts in the mouth, the emulsion should break. This leads to the sensation of taste, which is very essential to enjoy its specialness. The sensation of taste on the surface of the tongue is known to be related to molecular shape and physicochemical properties. As soon as these molecules are separated from the emulsion, the taste sensation is recorded in the brain. Therefore, the various components must stay in the same phase after the breakup of the emulsion. Emulsifiers that are generally used have low HLB values (for W/O), and have been found to have considerable effect on the structure of the ice cream. 

Changing the shape of the dispersed species while flowing also has an impact. Since emulsion droplets and foam bubbles are not rigid spheres, they may deform in shear flow. In the cases of electrostatically interacting species, or those with surfactant or polymeric stabilizing agents at the interface, the species will not be noninteracting, as is assumed in the theory. Thus, Stokes law will not strictly apply and may underestimate or even overestimate the real terminal velocity. 

From the liquid foam stability theory it follows that a liquid foam is most stable when the gas bubbles are strictly spherical in shape since, according to the Laplace and Plateau laws, the interface area and the capillary pressure have minimum values in this case. As it has been shown in Sect. 4.1 for the monodispersed... 

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