Chapter 3. Simulation results and discussion
3.5 Suppression of satellite drops
The breaking up of freely flying liquid thread has two modes – multiple breaking up because of wave-like instability and end pinching where the liquid thread pinches off from a bulbous end16, 43. The mechanism of end pinching may be a consequence of the fluid motion induced by capillary pressure gradients near the end of liquid thread43. The two modes of the breaking up of free liquid thread could be observed in the
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present simulations. The examples of end pinching and multiple breaking up are found in Fig. 3.3 and Fig. 3.17 respectively. During multiple breaking up, a wave-like disturbance appears along the freely flying liquid thread. This disturbance grows until the liquid thread breaks up at several places and varied times. The liquid thread in multiple breaking up tends to form numerous satellite drops of varied size.
The breaking up of the freely flying liquid thread is related closely to the length of the liquid thread at pinching off, which is defined as the distance between the leading-edge position and tail tip position of the thread. In their DOD dispensing experiments Dong et al. 16 observed that, for the freely flying water thread of small length at pinching off, the formation of a satellite results from end pinching and for a long thread a wave-like instability occurs and multiple breaking up is dominant.
Figure 3.18 shows the variation with cases of liquid thread length at pinching off in
the current simulations. For a ratio of thread length to nozzle radius (17.15 µm) greater than about 9.67, multiple breaking up occurs through a wave-like instability.
When this ratio is smaller than 8.8, the breaking up becomes an end-pinching mode, as shown in Fig. 3.18(a). In addition, the longer the length of liquid thread, the more satellite drops are formed. Experiment 5, for example, shows five satellite drops, experiment 6 three satellite drops and experiment 7 two satellite drops. The thread length at pinching off is positively correlated with the value of Wef. With the same
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value of Wef , an increased Df might yield an increased length of thread at pinching off. The causes of these phenomena might be that, when Wef increases,
the forward momentum density increases, thus accounting for an increasing difference in axial velocity and then a more elongated thread. When Wef is constant, the larger
Df implies a larger forward momentum density and then greater elongation of the thread. A decreased ∆τp2 could slightly shorten the liquid thread length at pinching
off by accelerating the rate of necking, shown in Fig. 3.18(b).
To investigate further the effect of actuation conditions on the thread length at pinching off, we performed additional experiments, as illustrated in Table 7. Figure 3.19 shows the variation in liquid thread length at pinching off for these experiments.
The values of l /b Rnoz in these cases are all above 9.67; the multiple breaking-up
mode is then dominant. These results indicate that the varying parameters in the pause and backward stages cause a slight variation of thread length at pinching off. From the discussion above, we conclude that the thread length at pinching off from the nozzle outlet is governed mainly by the conditions of the forward stroke – Df and Wef.
In most applications, the satellite drops would degrade the printing quality or increase the difficulty of a precise microfludic control; for this reason a suppression of satellite drops has considerable practical significance. As noted above, the breaking-up mode of the freely flying liquid thread and its thread length at pinching
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off have been shown to be mutually positively correlated. In the end-pinching mode, the formation of a satellite drop seems to be predictable and its size is comparable.
Several authors 16, 44-46 have suggested criteria to observe satellite formation by end pinching; for instance, the numerical research done by Notz and Basaran 45 shows that the liquid thread with sufficiently large initial aspect ratio defined as the ratio of a half thread length to thread radius pinched off daughter drops from almost spherical ends by end pinching when Oh <O(0.1). In this study, the shape of the liquid thread was
assumed to be a cylinder with hemispherical caps at its two ends. In contrast, satellite drops caused by multiple breaking up tend to occur arbitrarily and have varied size.
To investigate the suppression of satellite drops, we focus our attention on the end-pinching breaking up of the freely flying liquid thread into the primary drop and the free secondary liquid thread. This secondary thread contracts into a single satellite drop. In the following analysis based on work of Dong et al. 16, we denote the pinching-off time as t , the breaking-up time b1 t , the thread length at pinching off b2
l , the position b z of the tail tip of the thread, the position t zp, of the leading edge of the thread, the average speed
1
schematically in Fig. 3.20. The freely flying liquid thread would contract into a single drop without satellite formation provided that the thread length at pinching off from
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the nozzle is less than a critical value l . If we denote the radius of this final single b* drop as r , d
Scaling the period with the capillary time and the velocity with the capillary speed, we obtain
to the capillary duration and a the ratio of retreating velocity to the capillary speed. In the present study, the time scale is tca ≈8.341 µs and velocity scale vca ≈2.056 m
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Table 8 presents this prediction in our simulations. Except for experiment 15,
noz
b R
l / is larger than the critical value l /b* Rnoz; then one satellite drop can be observed. Notice that although the value l /b Rnoz in experiment 15 is less than the
critical value l /b* Rnoz, the prediction in this case shows one satellite formed;
however, the life time of this satellite is quite short compared to experiment 8, as shown in Fig. 3.13. One explanation for this is that in order to highlight the significant variables of the critical value l /b* Rnoz, we approximate the radius of the final single drop r as nozzle radius and neglect the term d νp /νr in Eq. (18), which would
slightly enlarge the upper limits l /b* Rnoz. It can be seen in Table 8 that the ratio of the radius rp of the primary drop followed by one satellite to nozzle radius (17.15
µm) in experiment 15 is approximately 0.746, which can be expected to be slightly smaller than r /d Rnoz. From Eq. (19), the critical length of the thread at pinching off without formation of satellite drops depends mainly on c , 2 c and 1 a . The value
c represents the time at which the primary drop is formed through either end 2
pinching or multiple breaking up. As recommended by Dong et al. 16, c and 1 c are 2
closely related to the liquid properties, nozzle radius and the waveform of the transducer pulse. The variation of c in our current experiments is given in Fig. 2 3.21 and Table 8. c in experiment 1 is approximately equal to that in experiment 5 2 and c in experiment 3 to that in experiment 6. The value 2 c in experiment 4 is 2
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almost the same as that in experiment 7. These results indicate that c depends 2 strongly on Df . The relation between c and the waveform of the transducer pulse 1
is shown in preceding sections. Compared to the experimental results of Dong et al. 16, the ratio of the average speed of the retreating thread tail to the capillary speed, as shown in Table 8, ranges from 2.2 to 3.2 showing reasonable values. It is noted that in a few simulation cases, upon the pinching off of liquid thread from nozzle outlet, a tiny isolated liquid fragment called flotsam is observed. This flotsam is characterized by the vanishing velocity and size comparable to a single numerical cell. The appearance of the flotsam would interfere with the measurement of the retreating speed and the length of liquid thread at pinching off, and should be excluded from the data analysis. As already noted, experiments 12, 13, 16 and 17 appear to show no satellite drop because the tail of the liquid thread contracts into the thread head.
Experiments 12 and 13 show an interval t of pinching off smaller than in b1
experiment 1. To estimate the critical value of experiments 12 and 13, we took c = 2 4.559 and a = 2.635 as obtained in experiment 1 with the same value ofDf and
because the value of a in experiments 14 and 15 is larger than that in experiment 8, as indicated in Table 8. We thus obtain the critical thread length at pinching off
noz
b R
l /* = 4.055, larger than l /b Rnoz = 3.994 in experiment 12, in agreement with a prediction obtained from Eq. (19). An estimate of the value l /b* Rnoz in experiments
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16 and 17 is similarly obtained on taking c and 2 a identical to values in
experiment 9. Table 9 contains the estimate of a critical thread length l /b* Rnoz in experiments 12, 13, 16 and 17. It is seen that a decrease in pause stage ∆τp2 could
both shorten the liquid thread length lb at pinching off and enlarge the critical value
b*
l , thus damping the satellite formation.
Based on the analysis above and from Eq. (19), we prefer the larger c , the 2 smaller c and the larger a to induce the larger 1 l /b* Rnoz. The larger the value
noz
b R
l /* , the wider the range of liquid thread length at pinching off without satellite formation. As shown in Figs. 3.21 and 3.7, both c and 1 c appear to increase when 2
D increases with constant f We . An increase in f We leads to a decrease in f c 1
when D is fixed, as shown in Fig. 3.7. However, the larger f We would cause the f longer length l /b Rnoz of liquid thread at pinching off, which may contribute to the formation of satellite drops as depicted in Fig. 3.18. Both l /b Rnoz and c tend to 1 decrease as the pause stage ∆τp2 decreases. According to research of Dong et al.16,
the interval between c and 1 c increases when liquid viscosity increases, liquid 2
surface tension decreases or nozzle radius decreases. The parameter a is shown not to be significantly related to the liquid surface tension, nozzle radius and the waveform of the transducer pulse. However, as the liquid viscosity increases, the value of a would slightly increase. In conclusion, for a DOD drop generator with a given liquid,
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the waveform of transducer pulse could carefully be designed to obtain a longer tb2,
a shorter tb1 and then larger upper limits lb*.