Capillary flows drop breakup

Abstract This chapter reviews atomization modeling works that utilize boundary element methods (BEMs) to compute the transient surface evolution in capillary flows. The BEM, or boundary integral method, represents a class of schemes that incorporate a mesh that is only located on the boundaries of the domain and hence are attractive for free surface problems. Because both primary and secondary atomization phenomena are considered in many free surface problems, BEM is suitable to describe their physical processes and fundamental instabilities. Basic formulations of the BEM are outlined and their application to both low- and highspeed plain jets is presented. Other applications include the aerodynamic breakup of a drop, the pinch-off of an electrified jet, and the breakup of a drop colliding into a wall. 

The previous discussion focused on the breakup of liquid thread suspended in a quiescent Newtonian fluid. In real mixing operations quiescent conditions will usually not occur, except perhaps for short periods of time. The more important issue, therefore, is how the breakup occurs when the system is subjected to flow. Good reviews on the breakup of liquid threads are available from Acrivos [304], Rallison [305], and Stone [306]. Probably the most extensive experimental study on drop breakup was performed by Grace [286] data was obtained over an enormous range of viscosity ratios 10- to 10 Grace determined the critical Weber (Capillary) number for breakup both in simple shear and in 2-D elongation the results are represented in Fig. 7.152. 

For shear flow in Newtonian liquids, oil drop breakup may be described by the capillary number Ca (viscosity of continuous phase tjc, shear rate y interfacial tension y, drop diameter x) ... 

The critical capillary number is the capillary number value below which no oil drop breakup occurs [10]. Grace [10] indicated that file viscosity ratio X (see (21.2)— viscosity of disperse //d and continuous phase is a crucial factor influencing the critical capillary number, especially for simple shear flow. Armhruster [11] and later Jansen et al. [12] showed that for high concentrated systems the viscosity of the emulsion rje, has to be used instead of the viscosity of cmitinuous phase. 

In addition, similar experiments were done as performed by Kitamura and Takahashi [11, 32], The drop size firom threads without stretching was determined by high flow rate through the capillary. The thread breakup was superimposed by a cross-wind flow. By the comparison of the drop volume and the volume of the disintegrated thread segment, the breakup wavelength is calculated with (22.10). 

The tensor L defines the character of the flow. The capillary number for the drop deformation and breakup problem is... 

Consider drops of different sizes in a mixture exposed to a 2D extensional flow. The mode of breakup depends on the drop sizes. Large drops (R > Caa,tal/xcy) are stretched into long threads by the flow and undergo capillary breakup, while smaller drops (R Cacri,oV/vy) experience breakup by necking. As a limit case, we consider necking to result in binary breakup, i.e., two daughter droplets and no satellite droplets are produced on breakup. The drop size of the daughter droplets is then... 

Fig. 20. Radius of drops produced on capillary breakup in hyperbolic extensional flow (Rdrops), radius of the thread at which the disturbance that causes breakup begins to grow (Rent), and the time for growth of the disturbance (fgrow) for different values of the dimensionless parameters p and /xc feao/tr. The time for capillary breakup of the extending thread ((break) can be obtained from these graphs (see Illustration for sample calculations) (Janssen and Meijer, 1993).

Fig. 22. Radius of drops produced by capillary breakup (solid lines) and binary breakup (dotted lines) in a hyperbolic extensional flow for different viscosity ratios (p) and scaled shear rate (p,cylo) (Janssen and Meijer, 1993). The initial amplitude of the surface disturbances is ao = 10 9 m. Note that significantly smaller drops are produced by capillary breakup for high viscosity ratios.

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Figure 2-17. A series of photographs of the time-dependent breakup of a drop hy the capillary-driven flow mechanism that is known as end-pinching. The drop is initially stretched in a flow, and then the flow is stopped, allowing the drop shape to evolve toward breakup. The viscosity of the drop relative to the suspending fluid is 1.2.

U. Olgac, A. Doruk Kayaalp and M. Muradoglu, Buoyancy-Driven Motion and Breakup of Viscous Drops in Constricted Capillaries, Int. J. Multiphase Flow, 32(9), 1055-1071 (2006). 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are well understood. The viscosity ratio, X, critical capillary number, and the reduced time, t, are the controlling parameters. Within the entire range of X, it was found that elongational flow is more efficient than shear flow for breaking the drops. 

Some authors report the next guide principles that may be applied for blend morphology after processing, (i) Drops with viscosity ratios higher than 3.5 cannot be dispersed in shear but can be in extension flow instead, (ii) The larger the interfacial tension coefficient, the less the droplets will deform, (iii) The time necessary to break up a droplet (Tj,) and the critical capillary number (Ca ) are two important parameters describing the breakup process, (iv) The effect of coalescence must be considered even for relatively low concentrations of the dispersed phase. 

Now, consider the droplet formation resulting from the breakup of relatively large drops in a gas flow. The drop is subjected to an external aerodynamic force (pressure pc). It should be counterbalanced by the internal pressure of the drop Pi and the capillary pressure p ... 

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