Deformation of the drop

Mechanically imposed oscillations at frequencies of 5-50 cycles sec. cause increases of up to 4 times in the rates of extraction of acetic acid from drops of CCI4 into water (79). The increase is due to the periodic deformation of the drops causing fluid circulation inside and outside, particularly at certain resonant frequencies. 

Translational flow. At low Reynolds and Weber numbers, the axisymmetric problem on the slow translational motion of a drop with steady-state velocity U in a stagnant fluid was studied in [476] under the assumption that We = 0(Re2). Deformations of the drop surface were obtained from the condition that the jump of the normal stress across the drop surface is equal to the pressure increment associated with interfacial tension. It was shown that a drop has the shape of an oblate (in the flow direction) ellipsoid with the ratio of the major to the minor semiaxis equal to... 

Presence of surfactant at the interface will also directly affect deformation. The surfactant allows formation of a /-gradient. This would affect the deformation mode of a drop, which has indeed been observed. Moreover, enlarging the interfacial area causes y to increase, as mentioned. This implies that the interfacial free energy increase includes two terms y dA + A dy. The first term is due to the deformation of the drop being counteracted by its Laplace pressure the second is due to surface enlargement being counteracted by the surface dilational modulus ESD. Making use of Eq. (10.20) for we obtain... 

This pressure reaches the greatest value at 9 = 0 and 9 = n, that is, at the drop poles lying on a straight line parallel to Eq. Therefore the deformation of the drop becomes is maximized in the vicinity of these points. As a result, the drop assumes the shape of an ellipsoid with the greater semi-axis directed along Eq. 

The equilibrium condition for the drop is the equality of the electric pressure and surface tension. If the force of electric pressure surpasses the force of surface tension, the drop will continue to deform until the decreasing main radiuses of curvature at the drop poles causes an increase of the surface tension that is sufficient to balance the internal pressure. A significant deformation of the drop, however, may destabilize the drop and cause it to break Therefore the critical value of external electric field strength Ea- can be estimated from the equality... 

The simple estimation discussed above gives us the critical parameter value Xcr 4-y/nfi. A more accurate calculation that takes into account the deformation of the drop [58, 79] gives Xcr = 1-625. For water in oil, S = 3 10 H/m and for water droplet of 1 cm radius, the critical strength of the electric field E. = 2.67 KV/cm. Therefore an electric field with strength Eg < 3 KV/cm will not cause any noticeable deformation of a drop with radius li 1 cm placed far away from the other drops. 

If the dynamic pressure difference (11.122) exceeds the surface tension of the drop, if can affect a significant deformation of the drop and the drop can break. Therefore the drop breakage condition will be... 

At > 0, the condition (13.101) can be violated. However, at such distances, the force of molecular attraction becomes essential, and it is this force that is responsible for the final stage of drop coalescence. Therefore the possible deformation of the drop can be noticeable only at final stage of drops mutual approach, and it can only slow coalescence by a little, but it will have no significant effect on the collision frequency of drops. 

Significant deformation of the drop surface precedes the process of drop breakage. This deformation is caused by the action on the surface of stresses from external and internal flow, for which significant gradients of velocity and dynamic... 

Local deformations of the drop result in irregular dents and ledges. The protruding parts eventually separate and form smaller droplets (Fig. 17.11, c). 

Low values of Ca mean that the stresses due to interfacial tension are prevalent if compared to viscous stresses, and drops in low Ca regimes tend to minimize their surface area by producing spherical shapes. On the contrary, in regimes described by high values of Ca, viscous effects dominate, and we can observe large deformations of the drops and asymmetric shapes. 

The first stage of breakup is deformation of the drop into a shape that resembles an oblate spheroid (see Fig. 6.2). Given the similarity to Newtonian drops, it is reasonable to assume that the same physical mechanisms apply, namely, unequal static pressure distribution over the drop surface. 

Taylor describes experiments with rf jr] having values of0.0003, 0.5, 0.9, and 20. He observed that, in the parallel-band apparatus that produced a shearing flow, when (t] /rf) is 0.0003 (i.e., a very low viscosity drop is placed in a high-viscosity matrix) the drop is elongated indefinitely in flow. However, (L—B)/ L+B) increases less rapidly than Tjay/K, unlike the expectation from Eq. (6.28b). In the four-roller elongational flow apparatus, the deformation of the drop (I—B)/(I + B) was a unique function of tjaE/K and obeyed Eq. (6.28b) at low stretch rates. At higher stretch rates (I—B)/(I+ B) increases less rapidly. The results were quantitatively similar to the shear flow behavior. 

The film stability was measured either directly in the capillaries, by observing the time needed for the breakdown of (oil or water) film, or in separate measurements carried out on glass plates of the same material. The latter method which is more accurate, consists of determining the time that elapses from placing water (or solution) drops of 100 mm volume on the surface until a sudden deformation of the drop shape takes place. This time is indicative of the rupture of the oil film due to gravity. 

The affine deformation of the drops causes the drops to extend into long thin threads, which is referred to as fibrillation. This process continues until the local radii become so small that the Weber (Capillary) number starts to approach the critical Weber number. At this point the threads become unstable and disintegrate as a result of interfacial tension-driven processes the interfaces are now active. The most important mechanisms are the growth of Rayleigh disturbances in the midpart of the thread, end-pinching, retraction, and necking in the case of relatively short dumbbell-shaped threads. 

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