Bubble rear

Matching procedures for the bubble rear follow by analogy. 

Figure 5. The surfactant distribution at the bubble rear expressed as a deviation from equilibrium.

Experimental observations (S3) indicate that a stagnant cap is formed over the rear of the droplet as surface-active agents are added, and that this cap tends to enlarge with increasing concentrations until the entire droplet is enveloped. Thus, circulation may occur only in the front portion of the bubble. In contrast to this mechanism, Thorsen and Terjesen (T3) and Gamer (Gil) concluded that most of the mass transfer takes place at the rear of the bubble. 

The absorption is assumed to occur into elements of liquid moving around the bubble from front to rear in accordance with the penetration theory (H13). These elements maintain their identity for a distance into the fluid greater than the effective penetration of dissolving gas during the time required for this journey. The differential equation and initial and boundary conditions for the rate of absorption are then... 

Slug/semi-annular flow. Here both slug and semi-annular flows were present. The vapor velocity increased with the heat flux and the rear of elongated bubbles began to break up (Fig. 2.30d). Coalescence was no longer clean and created a churn-like zone where the liquid slug had been. 

Fig. 4.3 SEM micrograph of the rear side of an n-(lOO) Si wafer polished on one side. The presence of inverted truncated square pyramidal stmctures fuUy covering the surface can be observed. This pyramidal texturing was attributed to the combination of anisotropic etching of the sdicon and to hydrogen bubbles evolved during the etching reaction. (Reprinted from [23] Copyright 2009, with permission from Elsevier)...

The shape of the front and rear menisci change as a result of the resistance to bubble flow. Calculation of this deviation in bubble shape establishes the dynamic pressure drop across the bubble. 

In region III near the tube center, viscous stresses scale by the tube radius and for small capillary numbers do not significantly distort the bubble shape from a spherical segment. Thus, even though surfactant collects near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension, aQ. Region... 

Figures 4 and depict the calculated surfactant distribution, expressed as 0, for the bubble front and rear, respectively.

Again, as expected, in Figure 5 there is an excess of surfactant near the rear stagnation ring due to surface convection towards that point. Forward from that location, however, there is also a depletion relative to equilibrium adsorption. This is caused by the traveling wave in the rear bubble profile as demonstrated in Figure 2 and in Figure 7 to follow. 

Figure 7. The bubble shape at the rear for the elasticity number equal to 0, 0.1, 0.5, and 1.0.

The rate of mass-transfer, unlike the terminal velocity, may reach its lower limit only when the whole surface of the drop or bubble is covered by the adsorbed film. In the absence of surface-active material, the freshly exposed interface at the front of the moving drop (due to circulation here) could well be responsible for as much mass transfer as occurs in the turbulent wake of the drop. The results of Baird and Davidson 67a) on mass transfer from spherical-cap bubbles are not inconsistent with this idea, and further experiments on smaller drops are in progress in the author s laboratory. In general, if these ideas are correct, while the rear half of the drop is noncirculating (and the terminal velocity has reached the limit of that for a solid sphere), the mass transfer at the front half of the drop may still be much higher, due to the circulation, than for a stagnant drop. Only when sufficient surface-active material is present to cover the whole of the surface and eliminate all circulation will the rate of mass-transfer approach its lower limit. 

In the narrow tubes used by Beek and van Heuven, the bubbles assumed the shape of Dumitrescu (or Taylor) bubbles. Using the hydrodynamics of bubble rise and the penetration theory of absorption, an expression was developed for the total absorption rate from one bubble. The liquid surface velocity was assumed to be that of free fall, and the bubble surface area was approximated by a spherical section and a hyperbola of revolution. Values calculated from this model were 30% above the measured absorption rates. Further experiments indicated that velocities are reduced at the rear of the bubble, and are certainly much less than free fall velocities. A reduction in surface tension was also indicated by extreme curvature at the rear of the bubble. 

By assuming that the surface tension on the surface of a fluid sphere varied from the surfactant-free value, at the nose to zero at the rear, Savic also deduced a relationship between velocity and Eotvos number, shown in Fig. 3.7, which agrees qualitatively with the experimental results of Bond and Newton. Modifications of this approach for cases where the maximum change in local interfacial tension is less than have been devised for bubbles (D5, G7) and... 

Treatment of liquid drops is considerably more complex than bubbles, since the internal motion must be considered and internal boundary layers are difficult to handle. Early attempts to deal with boundary layers on liquid drops were made by Conkie and Savic (C8), McDonald (M9), and Chao (C4, W7). More useful results have been obtained by Harper and Moore (HIO) and Parlange (PI). The unperturbed internal flow field is given by Hill s spherical vortex (HI3) which, coupled with irrotational flow of the external fluid, leads to a first estimate of drag for a spherical droplet for Re 1 and Rep 1. The internal flow field is then modified to account for convection of vorticity by the internal fluid to the front of the drop from the rear. The drag coefficient. 

For systems in which skirt formation can occur and is slightly less than required for skirt formation, large bubbles or drops tend to be indented at the rear. Skirt formation occurs when viscous forces acting at the rim or corner of the dimpled bubble or drop are strong enough to overcome interfacial tension forces and pull the rim out into a thin sheet (B3, H5, Wl, W5). The onset of skirts is dependent both on the ratio We/Re = fiUj/a, sometimes called a capillary or skirt number, and on Re. Figure 8.4 shows data for the transition from unskirted to skirted bubbles or drops. For bubbles, skirts exist for Re > 9 and... 

Experimental measurements of skirt thickness (B3, B5, G8, W2) show reasonable agreement with Eq. (8-18). In practice, skirts become thinner with increasing distance from the rear of the bubble or drop (B3, H5). Skirts behind bubbles are of order 50 fim thick, while the thickness of liquid skirts behind drops is of order 1 mm. 

Gas bubbles in liquid metals and in fluidized beds have been the subject of special studies because of their practical importance and because of the experimental difficulties associated with studying bubble properties in opaque media. Much of the work has been carried out in so-called two-dimensional columns, where a sheet of liquid or fluidized particles, typically 1 cm thick, is confined between two parallel transparent walls. Bubbles span the gap between the front and rear faces and can be observed with backlighting. 

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