Breakup critical capillary number

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Fig. 7.24 Breakup of a droplet of 1 mm diameter in simple shear flow of Newtonian fluids with viscosity ratio of 0.14, just above the critical capillary number. [Reprinted by permission from H. E.H. Meijer and J. M. H. Janssen, Mixing of Immiscible Fluids, in Mixing and Compounding of Polymers, I. Manas-Zloczower and Z. Tadmor, Eds., Hanser, Munich (1994).]...

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 9.7 Photographs of droplet shapes in planar extensional flow for various viscosity ratios M of the dispersed to the continuous phase. The droplets are viewed in the plane normal to the velocity gradient direction. The critical capillary numbers Cac and droplet deformation parameters Dc at breakup are also given. The droplet fluids are silicon oils with viscosities ranging from 5 to 60,000 centistokes, while the continuous fluids are oxidized castor oils both phases are Newtonian. (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

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 ... 

Note that in shear for A, = 1, the critical capillary number = 1, whereas for A, > 1, increases with X and becomes infinite for X > 3.8. This means that the breakup of the dispersed phase in pure shear flow becomes impossible for X > 3.8. This limitation does not exist in extensional flows. 

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. 

To make things more interesting, the experimental observations of De Bruijn [1989] seem to have contradicted the latter conclusion. The author found that the critical capillary number for viscoelastic droplets is always higher (sometimes much higher) than for Newtonian ones, whatever the -value De Bruijn concluded that drop elasticity always hinders drop breakup. 

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. 

Critical capillary numbers for elongational flows are lower than for shear flows. In other words, the elongational flow field is much more effective for droplet breakup in a dispersive mixing regime (Grace 1982). 

The steady-state deformation of isolated droplets decreases with increasing dispersed phase elasticity for the same imposed capillary number. A linear relationship between critical capillary number for droplet breakup (Kn-i,) and dispersed-phase Weissenberg number (Wi[Pg.934]

The microrheology discussed in Section 2.1.2.3 describes the breakup of isolated drops in a Newtonian system. The mechanisms leading to deformation and breakup take into account the three principal variables the viscosity ratio (X), critical capillary number (Kcni), and the reduced time (f ), defined in Eq. (2.19). For application of microrheology to polymer blends the theories developed for Newtonian emulsions need to be extended to viscoelastic systems in the fidl range of composition, that is, they must take into account coalescence. Since the microrheology evolution up to about the year 2000 has been summarized by Utradri and Kamal [3] the following text win focus on more recent developments. 

In practice, in a mixture much larger drops can be found than predicted by the critical capillary number because Grace s observations were based on single drops. In actual systems, where many drops exist, coalescence will occur. Because material elements also undergo varying levels of shear forces in time, the mixing process in polymer systems can be considered as a complex interaction between deformation, drop breakup, coalescence, and retraction. 

Taylor [64] found that in simple shear flow, a dispersed drop with viscosity ratio p = 1 breaks up when the Ca > 0.5. Breakup seems to occur when the shear stress and the interfacial stress are of the same order of magnitude. The critical Capillary number depends on the type of flow and on the viscosity ratio. In the mixing process two regimes can typically be distinguished ... 

Grace [6] has constructed a plot of the critical capillary number as a function of the viscosity ratio, p, under two types of flow a simple shear flow and a hyperbolic (elongational) flow field (Figure 1.2). It is shown that droplets are stable when their Ca number is below a critical value the deformation and breakup are easier at P within a 0.25 to 1 range for shear flow, and the elongational flow field is more effective for breakup and dispersion than the shear flow. It can also be seen that at a viscosity ratio p > 4-5, it is not possible to break up the drop in simple shear flow. 

Stone and Leal found that the surfactant tended to concentrate near the poles of the droplet. This tended to reduce the surface tension at the poles, which, in turn, caused more deformation at the poles. They studied the breakup of droplets and determined critical capillary numbers. 

Droplet breakup and coalescence are the primary physical processes in the mixing of liquids with very different viscosities. There is an extensive literature on the breakup of single droplets of Newtonian fluids in a Newtonian matrix, mostly building on a classic study by Grace that was first pubhshed in 1982 but was based on older work. Grace created a map of the critical capillary number Ca = r mY R/o... 

If the hydrodynamic stress is sufficiently high, Ca exceeds a certain value known as the critical capillary number, CacR. Under these conditions no stable shape can persist in the flow, and the droplet deforms continually until it breaks into daughter droplets. These can undergo consequent deformations and breakups until the droplets are so small that the interfadal stress overrules the hydro-dynamic stress. At Ca < Cocr the droplets are deformed into a shape which is stable in the flow and can be also predicted by the theory of Maffetone and Minale [76]. On the other hand, at very high values of Ca (Ca CacR), a quick affine deformation of the droplets occurs and long cylindrical threads are formed which are rather stable in the flow and decay only at very high deformations or even just after the cessation of flow [77,78]. 

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. 

At a viscosity ratio of unity, Grace 1982 [10] and Bentley and Leal 1986 [13] stated the lowest critical capillary number for steady shear flow, indicating good breakup conditions, while strain stress related breakup is roughly independent of the viscosity ratio. Investigations on breakup are falsified by coalescence. Coalescence was detected by investigating the influence of the disperse phase content at constant viscosity ratio [14], Figure 21.9 shows the shear viscosity curves of these emulsions. The emulsifier concentration of each emulsion was also set to 10% of the oil mass fraction to keep the disperse phase-related emulsifier amount constant above cmc. 

Another important parameter of the droplet breakup process is the time necessary for the interfacial-driven instabilities to cause breakup, tb, when the actual capillary number exceeds the critical capillary number. Grace (1982) provided this information in Figure 6.21 for Newtonian fluids. Note that the dimensionless burst time is denoted as which is equal to tb/r, where r is the time scale of the bursting process and it is equal to Rp-Jy- For example, for a polymer blend with p = 0.1,)/ = 10 mN/m, / = 10 qm, Pc = 1.000 Pa s, and Ca/Cac — 10, the dimensionless burst time is 11, and the time scale is equal to 1 s. Thus, the burst time, tb, is equal to 11 s. 

Bentley and Leal have measured droplet shapes and critical conditions for droplet breakup over a wide range of capillary numbers, viscosity ratios, and flow types. The flow type is conveniently controlled in an apparatus called a four-roll mill, in which a velocity field is generated by the rotation of four rollers in a container of liquid (see Fig. 1-15). By varying the rotation rate of one pair of rollers relative to that of a second pair, velocity fields ranging from planar extension to nearly simple shear can be produced near the stagnation point. 

Figure 35.4 shows the variation of ellipticity with respect to the Weber, Reynolds, and capillary numbers at various axial locations. As observed fi-om these figures, the droplets are big close to the injector and the Ohnesorge numbers are small. The Weber number is much larger than the critical Weber number ( 6) and the drops undergo breakup. The deformation predicted by the above correlation... 

In addition to the critical Weber number for a drop-on-demand breakup, the criterion of We > 8 for a Rayleigh breakup has also been reported [16]. This limit can also be motivated by the lower limit of jet formation in the case of dripping out of a vertical capillary with diameter Dnozzis under the action of gravity. The static pressure pstat inside a hanging droplet is... 

Unlike in NEMD models, the microstructures emerging due to competition between the breakup and coalescence processes can be studied by using DPD modeling. For example, in Figure 26.23, the four principal mechanisms, the same as those responsible for droplets breakup [ 118,119], can be observed in DPD simulation of the R-T instability. As shown in [116,119], moderately extended drops for capillary number close to a critical value, which is a function of dynamic viscosity ratio... 

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