Nonspherical bubble

Abstract Sonoluminescence from alkali-metal salt solutions reveals excited state alkali - metal atom emission which exhibits asymmetrically-broadened lines. The location of the emission site is of interest as well as how nonvolatile ions are reduced and electronically excited. This chapter reviews sonoluminescence studies on alkali-metal atom emission in various environments. We focus on the emission mechanism does the emission occur in the gas phase within bubbles or in heated fluid at the bubble/liquid interface Many studies support the gas phase origin. The transfer of nonvolatile ions into bubbles is suggested to occur by means of liquid droplets, which are injected into bubbles during nonspherical bubble oscillation, bubble coalescence and/or bubble fragmentation. The line width of the alkali-metal atom emission may provide the relative density of gas at bubble collapse under the assumption of the gas phase origin. 

Changing bubble size with height in the bed Negligible bubble-cloud resistance Negligible cloud-emulsion resistance Nonspherical bubbles... 

The above results apply to spherical bubbles, and analysis for nonspherical bubbles is considerably more complex (P3). Marmur and Rubin (M3) have... 

Heat transfer to macroscopic bubbles has received but little attention. Jakob (J2) presents the growth curves for several steam bubbles up to a radius of 3.6 mm. The main experimental obstacle is that of determining the true volume of a nonspherical bubble. 

Formula (4.10.10) can be used for calculating the mean Sherwood number (one replaces Nu by Sh) for nonspherical bubbles moving in a viscous fluid. 

The role of bubble size is much more involved in viscoelastic fluids than that in Newtonian media. It is customary to use the bubble radius (equivalent volume for nonspherical bubbles) as the characteristic linear dimension in the scaling of the governing equations and the pertinent boundary conditions. The characteristic shear rate associated with a rising bubble (low Reynolds number region) is 0 V/R) and, hence, the radius enters in the calculations of the magnitudes of the viscous and elastic forces in addition to the linear dimension in the Reynolds and Weber numbers. It was conceivable that under appropriate conditions, that is, levels of surface and viscoelastic forces, the no-slip-no-shear changeover may occur as a smooth transition as reported by De Kee et al. (1990) or as an abrupt jump as suggested by the qualitative analysis of Leal et al. (1971). 

The pressure applied produces work on the system, and the creation of the bubble leads to the creation of a surface area increase in the fluid. The Laplace equation relates the pressure difference across any curved fluid surface to the curvature, 1/radius and its surface tension y. In those cases where nonspherical curvatures are present, the more universal equation is obtained ... 

Here, e is a small parameter that will form the basis of an asymptotic approximation for the dynamics of the bubble surface. The question here is whether a bubble with a nonspherical initial shape of small amplitude 0(e) will return to a sphere - that is, fn(9,[Pg.271]

One complication is that the boundary conditions (4-264)-(4-266) must be applied at the bubble surface, which is both unknown [that is, specified in terms of functions R(t) and fn(9,tangent unit vectors n and t, that appear in the boundary conditions are also functions of the bubble shape. In this analysis, we use the small-deformation limit s 1 to simplify the problem by using the method of domain perturbations that was introduced earlier in this chapter. First, we note that the unit normal and tangent vectors can be approximated for small e in the forms... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

Many empirical relations for steady-state velocity of deformed drops and bubbles of various shapes, including shapes more complicated than the ellipsoidal shape, are presented in [94]. Laminar flow past nonspherical drops was studied numerically in [98, 517]. 

Mass Transfer Between Nonspherical Particles or Bubbles and Flow 185... 

The dependence (4.12.3) can also be used to estimate the intensity of transient mass transfer for nonspherical particles, drops, and bubbles at Pe 1. In this case, all dimensionless variables r, Sh, Shst, and Pe must be defined on the basis of the same characteristic length a. Under this condition, the expression (4.12.3) provides valid asymptotic results for small as well as large times. Equation (4.12.3) can be rewritten as follows ... 

Formula (5.3.8) and Eq. (5.3.9) can be used for the calculation of the mean Sherwood number for nonspherical particles, drops, and bubbles at high Peclet numbers. 

Cavitation near the liquid-solid interface differs from cavitation in the pure liquid. Near the solid surface, the bubble collapse becomes nonspherical and causes a shock-wave action on the surface. Velocity of microjets reaches hundreds of meters per second. The shock wave created by jets transfer strong local pulse, which is able to damage the surface of solids and crush brittle materials [248]. 

There are three types of preexisting nuclei we need to consider (1) free approximately spherical bubbles, which generally will be coated to a greater or lesser extent with surface active molecules, (2) similar bubbles, attached to a surface, and (3) portions of gas or vapor trapped in crevices, cracks, pits, and other geometrically nonspherical cavities. Such bubbles may be filled with air... 

At low Reynolds number (Re 1), the experimental values of X for both drops and gas bubbles are included in Fig. 6. The correspondence between predictions and experiments is seen to be about as good as can be expected in this kind of work. Indeed, the results do lend some support to the hypothesis of the importance of the viscous parameters in this regime. In view of the idealizations inherent in theoretical treatments coupled with the uncertainties arising from the inadequacy of the power law, nonspherical shapes, and possible viscoelastic and wall effects, the match seen in Fig. 6 is not bad at all. 

Bubble radius can be measured directly by a film or video camera. Pearson and Middleman (1977) found their bubbles were sightly nonspherical and used an average radius. For opaque liquids one can infer the bubble radius by measuring the small change in gas pressure above the liquid sample caused by the collapsing bubble (Johnson and Middleman, 1978). This method is also simpler and faster to use. 

At low gas volume fraction (<0.01), the forces acting on bubbles in a liquid are similar to those acting on a single bubble. Due to their low inertia as compared with the Hquid phase, bubbles are subject to a large number of forces, i.e., forces due to drag, lift, and virtual mass. In the definition of the interfacial forces, it is customary to characterize the bubble size with the sphere equivalent diameter. That is, ah effects due to nonsphericity are lumped in the closure. The closures that are most commonly applied... 

One of the major difficulties wiU be to find the best suitable parameters that characterize the nonsphericity, the deformabUity, the polydispersity, the clustering, the aUgnment, and the surfactant fouling. These parameters should be sufficiently general to apply to a large range of dispersed particles, bubbles, or droplets, yet at the same time be sufficiently descriptive to distinguish between different systems and aUow for accurate correlations to be... 

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