Capillary breakup

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... 

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.

Fig. 43. Capillary breakup of closely spaced molten nylon-6 threads in molten polystyrene. Photograhs at different times are shown (frames a through f correspond to 0, 210, 270, 360, 390, and 510 s). The initial thread diameter is 70 fim (Elemans el at., 1997).

Capillary Breakup Extensional Rheometer (CaBER), 21 740 Capillary columns, 6 377, 408 band broadening, 6 412 instrumentation, 6 424 speciality appbcations, 6 427t Capillary condensation, 1 585, 591 ... 

Recently, Razumovskid441 studied the shape of drops, and satellite droplets formed by forced capillary breakup of a liquid jet. On the basis of an instability analysis, Teng et al.[442] derived a simple equation for the prediction of droplet size from the breakup of cylindrical liquid jets at low-velocities. The equation correlates droplet size to a modified Ohnesorge number, and is applicable to both liquid-in-liquid, and liquid-in-gas jets of Newtonian or non-Newtonian fluids. Yamane et al.[439] measured Sauter mean diameter, and air-entrainment characteristics of non-evaporating unsteady dense sprays by means of an image analysis technique which uses an instantaneous shadow picture of the spray and amount of injected fuel. Influences of injection pressure and ambient gas density on the Sauter mean diameter and air entrainment were investigated parametrically. An empirical equation for the Sauter mean diameter was proposed based on a dimensionless analysis of the experimental results. It was indicated that the Sauter mean diameter decreases with an increase in injection pressure and a decrease in ambient gas density. It was also shown that the air-entrainment characteristics can be predicted from the quasi-steady jet theory. 

Filament stretching and capillary breakup rheometers are two experimental instruments used to impose uniaxial extension to fluids [34-39], In both of these devices a fluid is placed between two surfaces or platens, and the spacing between the platens holding the sample is increased, as shown in Figure 4.16. 

A commercial instrument for extensional viscosity measurements is currently offered by the Thermo Electron Corporation [40], The device uses capillary breakup techniques and is called the Haake CaBER . Vilastic Scientific, Inc. also offers an orifice attachment to their oscillatory rheometer for extensional viscosity determinations [41,42], The principle of operation of the rheometer is oscillatory tube flow [43,44], Dynamic mechanical properties can be determined... 

Capillary breakup method The capillary breakup method is based on Tomotika s theory [Tomotika, 1935, 1936]. The author was the first to investigate the development of Rayleigh s instabilities in cylinders of one fluid imbedded in another (see Pigs. 4.10 and 4.11) [Rayleigh, 1879]. The amplitude variations of the sinusoidal distortions, a, can be described by ... 

Table 4.3. Interfacial tension coefficient as determined using the capillary breakup method p ,uciani et al., 1997]...

Abstract This chapter deals with capillary instability of straight free liquid jets moving in air. It begins with linear stability theory for small perturbations of Newtonian liquid jets and discusses the unstable modes, characteristic growth rates, temporal and spatial instabilities and their underlying physical mechanisms. The linear theory also provides an estimate of the main droplet size emerging from capillary breakup. Formation of satellite modes is treated in the framework of either asymptotic methods or direct numerical simulations. Then, such additional effects like thermocapiUarity, or swirl are taken into account. In addition, quasi-one-dimensional approach for description of capillary breakup is introduced and illustrated in detail for Newtonian and rheologically complex liquid jets (pseudoplastic, dilatant, and viscoelastic polymeric liquids). 

Grant and Middleman [22] reported that (1.36) correlated experimental data for capillary breakup of low viscosity liquid jets when a value of C = 13 was selected. It is instructive to use this value of C to evaluate the initial perturbation amplitude Co- Taking for the estimate the unperturbed cross-sectional radius a = 1 mm, one can find Co = 10 exp( 13) = 2.26 x 10 m = 2.26 nm. How plausible such estimates are for liquid jets whose profile is visibly perturbed at the nozzle exit, remains an open question. 

The previously noted analyses did not consider the effect of the velocity profile on capillary breakup. Reitz-Bracco [25] relaxed this assumption, and considered a liquid jet in a gas with a velocity profile in the radial direction. They obtained the following general characteristic equation ... 

Quasi-One-Dimensional Approximation to the Jet Equations in the Case of Capillary Breakup... 

Equations 1.49-1.51 represent themselves a slightly modified version of the equations used in [30-32]. The modification introduced in [29] and references therein involves the exact (not the asymptotic) expressions for the capillary force at the jet surface and capillary pressure in the jet cross-section, which allows description of the capillary breakup until formation of drops. A detailed derivation of such equations based on the above-mentioned physical assumptions and the integral mass and momentum balances can be found in the monograph [29]... 

Summarizing, in the linear stability theory of capillary breakup of thin free liquid jets, the quasi-one-dimensional approach allows for a simple and straightforward derivation of the results almost exactly coinciding with those obtained in the framework of a rather tedious analysis of the three-dimensional equations of fluid mechanics. This serves as an important argument for further applications of the quasi-one-dimensional equations to more complex problems, which do not allow or almost do not allow exact solutions, in particular, to the nonlinear stages of the capillary breakup of straight thin liquid jets in air (considered below in this chapter). 

Nonlinear Analysis of Capillary Breakup of Liquid Jets... 

On the other hand, capillary breakup of sufficiently viscous liquid jets is a longwave phenomenon, and its description in the framework of the quasi-one-dimensional equations of the dynamics of liquid jets is sufficiently accurate. The effect of the viscosity on the capillary breakup of highly-viscous liquid jets was studied numerically by Yarin [29]. The initial perturbation of the jet surface was imposed as a harmonic... 

At the late stage of capillary breakup near the jet cross-section where the breakup will eventually occur, liquid flow completely forgets the initial conditions. It is dominated by the local flow conditions and becomes self-similar. The numerical description of the latest stages of capillary breakup is unreliable near the cross-sections where the cross-sectional radius tends to zero. A theoretical description of such self-similar final jet pinching is given in [79-84], assuming either inertia or viscosity dominated flows in the tiny threads and, in particular, using quasi-one-dimensional equations. 

Fig. 1.6 Capillary breakup of a glycerin jet (Z = 1.755 Yarin [29]). (a) Jet profile corresponding to (me half of the perturbation wavelength. The cross-sectional radius R is rendered dimensi(mless by a = 6 X 10 m, the axial coordinate x - by =...

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