3.2.2.1 MilliQ water
In Figure 3.7 is represented the detachment of a milliQ water drop. We observe the formation of a neck between the reservoir and the drop. The neck gets thinner and longer until the detachment of the drop at Figure3.7(f). The neck is detached from the reservoir and forms a satellite drop in Figure3.7(g). The thinning dynamics is shown in Figure3.8.
We observe the two expected regimes: first an exponential thinning (blue dashes), the experimental curve is well fitted by the expression Hmin/D = B − et−tc−t0tR with B, tR
and t0 as fitting parameters. We obtain B = 0.98, tR = 19.46 ms and t0 = 6.05 ms.
t0 corresponds to the breakup time if the exponential thinning was valid until the drop detachment, see Figure3.8. In the last milliseconds, the thinning speeds up with a power law in 2/3 (red dots) corresponding to the inertio-capillary regime. The experimental data are well fitted by the expression: Hmin/D = 0.7
tc−t tγ
23
with tγ as unique fitting.
We find tγ=12.86 ms. In conclusion, the thinning dynamics of a water drop detaching from a reservoir is correctly predicted by the exponential thinning announced by the linear analysis and, in the last milliseconds, by the inertio-capillary regime predicted by the non-linear analysis.
3.2.2.2 High-molecular-weight polymer solution
In Figure 3.9 is represented the detachment of a drop of the PEO solution. Until 9 ms before the breakup, the destabilization is precisely similar to the one of a water
3.2. Detachment of a pendant drop - Experimental results 53
-60 -28 -8 -3 -1 0 0.5 2 ms
1 mm
b c d e f g h
a
Figure 3.7: Detachment of a milliQ water drop. Q = 2 mL/min, D0 = 2 mm and Re = 20. The origin of time is taken at the moment of the drop detachment.
-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.0
0.2 0.4 0.6 0.8 1.0
H min/D
t-tc (s)
t0
Figure 3.8: Thinning dynamics of a milliQ water drop. Q = 2 mL/min, D0 = 2 mm and Re = 20. The blue dashes correspond to the exponential fit of the experimental data of the form Hmin/D = B − et−tc−t0tR with fitting parameters B = 0.98, tR = 19.46 ms and t0 = 6.05 ms. The red dots correspond to the power law fit of the experimental data of the form Hmin/D = 0.7
tc−t tγ
23
with fitting parameter tγ = 12.86 ms.
drop, Figure 3.7. However, on Figure 3.9(e), at -8.5 ms, a very thin filament is formed connecting the upper reservoir to the main drop. A satellite drop is suspended on this filament that gets longer and thinner. Eventually, the filament breaks up liberating the satellite drop. This destabilization mechanism is similar to the "bead-on-string" structure described by [Amarouchene 2001, Wagner 2005, Christanti 2001].
The thinning dynamics plotted in the Figure3.10 exhibits the same first two regimes as those observed for water drops. First an exponentially thinning well fitted by an
ex-54 Chapter 3. Destabilization of liquid ligaments into drops
-74.5 -24.5 -12 -9 -8.5 -8 -7 -4.5 -0.5 0 ms
a b c d e f g h i j1 mm
Figure 3.9: Detachment of a drop of PEO solution 106 g/mol 0.05%w. Q = 2 mL/min, D0 = 2 mm and Re = 16. The origin of time is taken at the moment of the drop detachment.
pression of the form Hmin/D = B− et−tc−t0tR with fitting parameters B = 0.95, tR = 19.83 ms and t0 = −1.53 ms. t0 corresponds to the breakup time is the exponential thinning was valid until the drop detachment. Here t0 < 0 as the breakup occurs later than pre-dicted by the linear analysis. The exponential thinning is followed by the inertio-capillary regime with a power law in 2/3 well fitted by the expression Hmin/D = 0.7
tc−t tγ
23 with fitting parameter tγ = 14.97 ms. The breakup time is however delayed with the appear-ance of a new regime. In order to investigate this new regime, which lasts about 8 ms, we have performed experiments with a higher acquisition rate (up to 20,000 frame/s).
These data are represented in the inset of Figure 3.10 in a semi-log graph. This rep-resentation highlights the existence of the new regime. We observe a very good repro-ducibility of our data: the data recorded at higher frame rate are also very well fitted by the fits previously determined. The new regime observed corresponds to the thin-ning of the filament formed between the main drop and the reservoir and it follows an exponential thinning indicated by the green dashes, as experimentally measured by [Amarouchene 2001, Tirtaatmadja 2006]. The thickness of the filament is closed to the spatial resolution of the set-up, this is why the data are less well-defined in this regime but we can still note an exponential evolution well fitted by the expression Hmin/D = Ce−(t−tc)3τ with fitting parameters C=0.005 and τ =1.28 ms. Following Equation 3.6, τ should be equal to the relaxation time of the polymer chains calculated from the Zimm model. Our experimental value is really closed to the one obtained by [Christanti 2002] for PEO so-lution of same molecular weight and concentration. They experimentally found τ = 2.4 ms and they compared this value to the longest relaxation times of the polymer chain according to the Zimm model: τZimm = 0.92 ms. Our experimental value is in good agreement with these two values. A consequence of the formation of this filament is to retard the breakup. In the case of a Newtonian fluid, the drop would have been detached at the intersection of the red dotted line with the abscissa axis (Figure3.10) but for the PEO drop the formation of the filament prolongs the process and therefore retards the breakup.
3.2. Detachment of a pendant drop - Experimental results 55
Figure 3.10: Thinning dynamics of a drop of 106 g/mol PEO solution at a concentration of 0.05 % w/w, Q = 2 mL/min, D0 = 2 mm and Re = 16. The blue dashes correspond to the exponential fit of the experimental data of the form Hmin/D = B− et−tc−t0tR with fitting parameters B = 0.95, tR = 19.83 ms and t0 =−1.53 ms. The red dots correspond fitting parameter tγ = 14.97 ms. (Inset) Thinning dynamics recorded at 20,000 frame/s in a log/lin graph to highlight the exponential thinning of the long-living filament formed.
The short green dashes correspond to the exponential fit of the experimental data of the form: Hmin/D = Ce−(t−tc)3τ with fitting parameters C=0.005 and τ =1.28 ms.
3.2.2.3 Industrial emulsion
We show in Figure 3.11 the detachment of a pendant drop of the industrial emul-sion. The destabilization process is in all aspects similar to the one of a water drop (Figure 3.7). In the same way, the thinning dynamics of the industrial emulsion drop (presented in Figure 3.12) is identical to the water drop dynamics (see inset Figure3.12 for the superimposition of the two thinning dynamics). First, the fluid neck exponen-tially thins: the experimental data are well fitted by the expression Hmin/D = B−et−tc−t0tR with fitting parameters B = 0.99, tR = 19.46 ms and t0 = 6.00 ms. Second, the thinning speeds up with the inertio-capillary regime and the data are well fitted by the expression Hmin/D = 0.7
tc−t tγ
23
with fitting parameter tγ = 13.32 ms.
56 Chapter 3. Destabilization of liquid ligaments into drops
-79 -39 -24 -18 -4 -1.5 -0.5 0 0.5 1.5 ms
1 mm
a b c d e f g h i j
Figure 3.11: Detachment of an industrial emulsion drop. Q = 2 mL/min, D0 = 2 mm and Re = 20. The origin of time is taken at the moment of the drop detachment.
-0.08 -0.06 -0.04 -0.02 0.00
0.0 0.2 0.4 0.6 0.8 1.0
H min/D
t-tc (s)
-0.08 -0.06 -0.04 -0.02 0.00 0.0
0.5 1.0
Hmin/D
t-tc (s) milliQ water industrial emulsion
Figure 3.12: Thinning dynamics of an industrial emulsion drop. Q = 2 mL/min, D0 = 2 mm and Re = 20. The blue dashes correspond to the exponential fit of the experimental data of the form Hmin/D = B − et−tc−t0tR with fitting parameters B = 0.99, tR = 19.46 ms and t0 = 6.00 ms. The red dots correspond to the power law fit of the experimental data of the form Hmin/D = 0.7
tc−t tγ
23
with fitting parameter tγ = 13.32 ms. (Inset) Superimposition of the dynamics of a milliQ water drop (orange squares) and of an industrial emulsion drop (white circles).
3.2.3 Conclusion
We have seen that a water and an industrial emulsion pendant drops present the same breakup mechanism and follow the same thinning dynamics. The first stages of the
3.3. Destabilization of stretched ligaments - Experimental results 57