Breakup necking

Illustration Satellite formation in capillary breakup. The distribution of drops produced upon disintegration of a thread at rest is a unique function of the viscosity ratio. Tjahjadi et al. (1992) showed through inspection of experiments and numerical simulations that up to 19 satellite drops between the two larger mother drops could be formed. The number of satellite drops decreased as the viscosity ratio was increased. In low-viscosity systems p < 0(0.1)] the breakup mechanism is self-repeating Every pinch-off results in the formation of a rounded surface and a conical one the conical surface then becomes bulbous and a neck forms near the end, which again pinches off and the process repeats (Fig. 21). There is excellent agreement between numerical simulations and the experimental results (Fig. 21). 

Illustration Comparison between necking and capillary breakup. 

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... 

Breakup by necking occurs in sustained flows when Ca is close to Cacrit. The number of drops produced upon breakup by necking is generally less than 10. 

There is little evidence showing the mode of breakup in turbulent flow fields. Hinze (HI7) speaks of a bulgy mode of breakup. Published photographs (C7, T12) show highly deformed bubbles and necking drops, protuberances and cell-like surface structures (see Fig. 12.8). Experimental evidence regarding single bubbles and drops in well-characterized turbulent fields would be most welcome. 

Besides deformation, fracture is the other response of materials to a stress. Fracture is the stress-induced breakup of a material. Two types of fracture are commonly defined. A brittle fracture is breakup which occurs abruptly without localized reduction in area. A ductile fracture is the failure of the material which is preceded by appreciable plastic deformation and localized reduction in area (necked region). The brittle fracture and ductile fracture are schematically illustrated in Fig. 1.10. 

Figure 7.21 shows the breakup of a 0.35-mm-diameter castor oil thread in quiescent silicone oil. Both phases are Newtonian. We notice between two main droplets a second generation of droplets formed from the instability of the extended neck, and there is also a hint of a third generation of droplets as well. 

FIGURE 5.2 Jet breakup (a) Necking in a liquid stream withi > 1.5R. (b) Disturbances of the circumference of a liquid jet of diameter D. Breakup occurs [4] when the amplitude of the disturbance is equal to D/2, which occurs first at a wavelength of 4.51 x D. (c) 1-2 pm filament forms between two drops as liquid jet nears the breakup point. Drawn from a photo by Castleman [6]. 

This pressure drop has a strong influence on the d5mamics of breakup, as once the main channel is obstructed by the growing droplet, the upstream interface of the droplet is pushed downstream by the continuous fluid. Once the droplet interface is pushed against the downstream wall of the channel, the neck connecting the orthogonal stream of dispersed fluid with the droplet narrows and eventually breaks, and the shaped droplet is released. 

Since liquid surface tension is strongly dependent on temperature, it can be controlled by controlling the liquid temperature. This technique was utilized [101,102] to control the breakup of a water jet. Furlani et al. [103] have conducted a linear analysis of a jet subject to a spatially periodic variation of surface tension imposed along its length. It is shown that as the jet approaches breakup it swells at the points of maximum stuface tension, and necks at the points of minimum surface tension. A periodic variation of temperature can induce a time-harmonic modulation of the surface tension cr of the jet, which has an equatimi of state of the form cr(7) = (To - P(T - To) where jS is a property constant. Instability of an evaporating jet is considered by Saroka et al. [104], who showed that the evaporation increases the growth rate of instability. 

Fig. 1.20 Time evolution of surfaces of four jets with (a) Bi = 0, (b) Bi = 1.37, (c) Bi = 1.38, and (d) Bi = oo. For all jets a surface displacement was applied accompanied by a thermal resistance which increased sinusoidally from 0 at the neck to 1 at the swell of the initial surface disturbance. The initial temperature was T = 1, Rep = 20, Ma = 200, Ca = 0.2, k = 0.7, and Cq = 0 05 for all jets. The numbers on the figure represent the time. Critical breakup of the jets occurs in the interval 1.37 < Bi < 1.38 [98, Fig. 2] (Courtesy of Cambridge University Press)...

By contrast, it is clear that dUatant liquids should demonstrate an increased stability in the necking sections of capillary jets and a deceleration of the later stages of the capillary breakup. A relatively rapid growth of the initial axisymmetric perturbations leads to an increase of the effective viscosity in the necking sections of the jet and its transformation into a net of practically spherical droplets connected by tiny threads. The results of the numerical calculations for dilatant liquids by Yarin [29] are depicted in Fig. 1.23. 

Grace [286] found that the breakup time decreases upon exceeding the critical Weber number. Similar experiments by Elemans [307] did not show a decrease in breakup time upon exceeding the critical Weber number. The breakup time for viscosity ratios between 0.1 and 1 was found to be around 50 to 100, i.e., tb = 50-100. Experimental work in an opposed-jet device [66] found that with flow occurring a droplet will stretch, while during the no-flow condition, breakup occurs via necking. This is illustrated in Fig. 7.155. 

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