Satellite drops

Fig. 1.19 Experiments showing the shape of the filament between the two main drops (satellite droplet) (a) before the firtst pinch-off and (b) after the last pinch-off, for droplet to surrounding fluid viscosity ratio of 0.4. The initial filament breaks up into rntwe smaller satellite droplets due to the capillary action [94] (Courtesy of Cambridge University Press)...

For the case of a thread at rest, the initial growth of a disturbance can be relatively well characterized by linear stability theory. In the initial stages, the deformation of the thread follows the growth of the fastest growing disturbance (Tomotika, 1935). Eventually the interfacial tension driven flow becomes nonlinear, leading to the formation of the smaller satellite drops (Tjahjadi et al., 1992). 

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). 

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... 

Upon breakup, the filament breaks into a set of primary or mother drops whose sizes are, to a first approximation, proportional to R. The size of drops produced when the filament breaks can then be obtained from the distribution of R. Each mother drop produced upon breakup carries a distribution of satellites of diminishing size for example, each mother drop of radius r has associated with it one large satellite of radius r, two smaller satellites of radius i 2 four satellites of radius r(3), and so on. For breakup at rest, the distribution of smaller drops is a unique function of the viscosity ratio. 

Fig. 21. Formation of satellite drops during the breakup of a filament at rest. A comparison between computations and experimental results is shown (Tjahjadi, Stone, and Ottino, 1992). Numbers refer to dimensionless times with ( = 0 corresponding to Fig (a).

Fig. 23. (a) Distribution of drop sizes for mother droplets and satellite droplets (solid lines) produced during the breakup of a filament (average size = 2 x 10 5 m) in a chaotic flow. The total distribution is also shown (dashed line). A log-normal distribution of stretching with a mean stretch of 10 4 was used, (b) The cumulative distribution of mother droplets and satellite droplets (solid line) approaches a log-normal distribution (dashed line). 

The number of satellite drops produced upon breakup by capillary instabilities decreases as p increases (minimum of 3 to maximum of 16). 

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. 

Since the satellite drops constitute only a small proportion of the total liquid flowrate, a cup operating under these conditions effectively produces a monodisperse spray. Under these conditions the drop size from sharp-edged discs has been given by Walton and... 

Figure 3.3. TMAFM images of nanovolcanoes formed at the surface of an EDT-TTF-(CONHMe)2 microcrystal after eruption of the CH3CN solvent to the atmosphere. The microcrystal was formed after the deposition of a drop of a saturated solution of EDT-TTF-(CONHMe)2 in CH3CN on a glass coverslip and exposed to ambient conditions, (a) 10 pm x 10 pm and (b) 2 pm x 2 pm. The illumination is set in such a way that the feature appears like a volcano seen by a satellite.

Fig. 14-85. As shown the actual breakup is quite close to prediction, although smaller satellite drops are also formed. The prime advan-tage of this type of breakup is the greater uniformity of drop size.

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. 

All this does not mean that the combination of human and natural influences on ice cap formation is easy to understand after all, satellite observation of polar ice started only in 1979, and the ice cap area around Antarctica is at record highs. Yet, for the first time, scientists have confirmed that the ice on Earth is melting at both ends. How is that possible How can the southern ice cap grow and the ice be melting at both ends It seems that the area of the southern ice cap is growing because of increased humidity in the air, and its thickness is dropping because of the warming of the waters below. 

---