Bubble front

The expected surfactant distribution is also portrayed qualitatively in Figure 2. At low Ca, recirculation eddies in the liquid phase lead to two stagnation rings around the bubble, as shown by the two pairs of heavy black dots on the interface (18>19). Near the bubble front, surfactant molecules are swept along the interface and away from the stagnation perimeter. They are not instantaneously replenished from the bulk solution. Accordingly, a surface stress, rg, develops along the interface... 

According to the Young-Laplace expression, the pressure difference across the bubble-front interface is 2aQ/(RT-h ). 

In this case, however, the bubble aft translates over the constant thickness film deposited by the bubble front so that h is already fixed. 

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

Figure 3. Schematic of matching to the spherical cap at the bubble front. The radius of the flow-altered sphere is For a static bubble, the bubble radius is R. ...

Figure 4. The surfactant distribution at the bubble front expressed as a deviation from equilibrium.

Fig. 13.33 Fractional coverage predicted by simulations (solid circles) in comparison with the experiments of Taylor (62) (open diamond) and Hyzyak and Koelling (67). [Reprinted by permission from V. Gauri and K. W. Koelling, Gas-assisted Displacement of Viscoelastic Fluids Flow Dynamics at the Bubble Front, J. Non-Newt. Fluid Meek, 83, 183-203 (1999).]...

V. Gauri and K. W. Koelling, Gas-assisted Displacement of Viscoelastic Fluids Flow Dynamics at the Bubble Front, J. Non-Newt. Fluid Mech., 83, 183-203 (1999). 

The theory of effective viscosity has been developed by Betherton [172], Hirasaki and Lawson [173], Falls et al. [171] and Kovscek and Radke [153]. It was shown that the effective viscosity is a sum of three terms the first accounts for the contribution of the slugs of liquid between bubbles, the second is the resistance against surface deformation in the advancement of bubbles through the capillaries (pores) and the third is the gradient of surface tension caused by the withdrawal of the surfactant (from the bubble front to the bubble back). The experimental data of Falls et al. [171], Hirasaki and Lawson [173], Ettinger and Radke [166] for bead packs and Berea core agree with the calculations from Hirasaki and Lawson s models [173],... 

Arriola (8) and Ni (5) have observed a second mechanism for snap off in strongly constricted square capillaries. At low liquid flow rates, a bubble is trapped in the converging section of the constriction and liquid flows past the bubble. As liquid flow rate increases, waves developed in the film profile and at some critical liquid flow rate these oscillations become unstable and bubbles snap off. In these experiments, the bubble front is located upstream of the constriction neck. Therefore, no driving force for the drainage mechanism exists. Bubbles formed by this mechanism are produced at a high rate and have a radius on the order of the constriction neck. No attempt has previously been made to model snap-off rate by this mechanism in noncircular constrictions. 

Snap off by the instability mechanism may occur in the following way. A bubble in an angular constriction such as a square channel will flatten against the walls as shown in Figure 2. The radius of the circular arcs in the corners for a static bubble is about one half of the tube half width. At low liquid flow rates, a bubble trapped behind a constriction has this nonaxisymmetric shape. As liquid flow rate increases, the bubble moves farther into the constriction and the fraction of cross sectional area open to liquid flow at the front of the bubble increases until the thread becomes axisymmetric at some point near the bubble front. 

Thus, the retardation of the movement of a reflected particle by the liquid countercurrent is very sensitive to the degree of retardation of the bubble surface, i.e. to the DAL structure in the vicinity of the bubble front pole. The movement is fast at a weak surface retardation and is slow at a strong surface retardation. Thus, at strong surface retardation, the inertial path of the reflected particle can exceed that for the case of weakly retarded surface. The greater the tangential velocity, the shorter is the sliding time. 

The simulations by Giavedoni and Saita showed that the bubble front tended to adopt the shape of an arc of a circle for decreasing values of Ca. Undulations, which increased with Re, appeared at the back of the bubble. The size of the undulations depended on both Ca and Re, and for Ca > 0.5, no undulations were observed. For all the values of Re that were tested, the rear of the meniscus was convex at first, became flat as Ca increased, and then adopted a concave shape, while as Re increased, the change from flat to concave shape appeared at lower values of Ca. The interface at the back of the bubble close to the meniscus tip is an almost perfect hemisphere for Ca < 10. 

The presence of bubbles affects the flow field within the liquid slugs and results in fluid recirculation [33]. At low Ca, there is a stagnation ring at each bubble cap (Figure 8.2). For 0.6 < Ca < 0.7 there is still recirculation in the liquid accompanied by two stagnation points on the bubble front and inside the liquid. At Ca > 0.7, complete bypass of the liquid occurs with a single stagnation point at the bubble front (34,35). Thulasidas et al. [34] found theoretically that complete bypass occurs at Ca> 0.5 in upward flow and at Ca > 0.6 in downward flow. In square channels, liquid bypass was found to occur at Ca > 0.54 for horizontal flow [36], whereas in upward and downward flows complete bypass occurred at Ca> 0.5 and Ca>0.57, respectively [34]. 

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