Analysis of the perforation

As shown in Figure 5.4(a), holes are eventually found to occur and grow in liquid sheets made of model emulsion. In the following, we discuss the different methods to characterize the perforation of the liquid sheet.

5.1.3.1 Perforated area of the liquid sheet

The first method consists in quantifying the time evolution of the cumulated area of holes and that of intact (non perforated) sheet. In Figure5.4(a) is shown a typical image of the destabilization of an emulsion-based liquid sheet 3.40 ms after the tear impact. The liquid sheet is partially perforated. Using ImageJ, we can define a continuous area of the sheet, which corresponds to the blue selection in Figure 5.4(b). The perforated area of the sheet is calculated as the sum of the area of the holes (red selections in Figure5.4(c)).

From Figure 5.4(c), one can determine the number of holes (manually counted) present in the sheet at a given time,N .

a). b). c).

Figure 5.4: a). Image of the destabilization of a model emulsion-based liquid sheet, C=0.3 % v/v, doil=39.1 µm, 3.40 ms after the tear impact. b). Same image with the blue selection representing the continuous area of the sheet. c). Same image with in red the perforated area.

In Figure5.5(a) is represented the time evolution of the continuous and the perforated areas. For a typical experiment, t = 0 corresponds to the time of impact of the tear on

5.1. Experimental set-up, samples and image analysis 109 the target. The continuous area increases with time up to a maximum obtained here at t=3.4 ms and then decreases due to the surface tension-driven retraction of the sheet.

The perforated area is equal to 0 at short time as no hole has yet perforated the sheet.

From t=1 ms, the perforated area increases with time up to a maximum (t=4.8 ms) and then slightly decreases. From 4.8 ms to the complete destabilization of the liquid sheet (t=5.2 ms), the perforated area decreases because the holes at the sheet periphery disappear. One can define the intact area of the liquid sheet as the difference between the continuous area and the perforated area. The intact area is also shown in Figure 5.5(a), it follows the same trend as the continuous area however the maximum expansion area is shifted towards shorter times (t=3.0 ms).

In Figure 5.5(b) is shown the evolution of the number of holes present in the sheet, N , as a function of time. N increases with time up to a maximum of 56 holes at 4.2 ms.

Then N decreases down to 20 holes at t =5.2 ms. From 4.4 to 4.8 ms, the perforated area increases whereas the holes number decreases. This means that holes are coalescing, leading to a reduction of the number of holes and concomitantly to an increase of the total holes area.

Figure 5.5: a) Time evolution of the continuous, perforated and intact areas of a model emulsion-based liquid sheet (C=0.3 % v/v, doil=39.1 µm). b). Time evolution of the number of holes present in the sheet, N .

5.1.3.2 Cumulated number of perforation events

In the previous section we have defined the parameter N as the number of holes present in the sheet at a given time. N is the result of two contributions. A positive one due to the nucleation of new holes and a negative one due to the merging of growing holes as they come into contact and the disappearance of the border holes. An alternative way to characterize the perforation mechanism is to count the number of nucleation events that perforate the sheet during the whole experiment (from the impact of the tear to

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Chapter 5. Destabilization of emulsion-based liquid sheet: single-tear and standard spray experiments the complete disintegration of the sheet into drops). We define λ as the instantaneous nucleation rate, i.e. the number of holes that have nucleated between t and t + dt. N is defined as the cumulated number of perforation events (i.e. the integrated value of λ over time). On contrary to N , λ and N quantify only one process, that is the nucleation of new holes in the sheet, and do not take into account the coalescence of holes that impacts onN .

In practice, the number of nucleation events is counted manually. It is really easy to determine, from an image to the next one, the number of new holes created. In Figure 5.6(a) is shown the time evolution of the instantaneous perforation rate, λ. We observe a delay time before the nucleation of holes (around 0.6 ms), then the instantaneous perforation rate oscillates without a clear trend between 3 to 15 holes per time interval, dt, taken here equal to 0.2 ms. In the same graph (right Y axis) is plotted the evolution of the intact area of the sheet. We observe that there is no correlation between the instantaneous perforation rate and the intact area of the sheet. Hence, the instantaneous perforation rate seems to be independent of the intact surface available. On average from the beginning of the perforation to the whole destabilization, 7.1± 2.9 holes nucleate in the sheet every 0.2 ms. We can define a perforation density per unit of intact area,D, by dividing the instantaneous perforation rate, λ, by the intact area of the sheet. The time evolution of the perforation density is plotted in Figure 5.6(b). The perforation density fluctuates around a mean value (D=0.032 ± 0.016 mm⁻¹) during the whole perforation process without a clear trend. We can estimate the perforation density to be constant over time, which means that the perforation density is independent of the thickness of the sheet. Indeed, as we have seen in Chapter 4 the thickness of the sheet varies with time. At t=1 ms, the sheet thickness is in average equal to (105.3± 55.3 µm) (depending on the radial position) and at t=4.5 ms equal to (29.9± 5.2 µm). The perforation density is of the same order of magnitude at t=1 ms and 4.5 ms whereas the sheet thickness values are very different, which implies that the perforation density is independent of the thickness of the sheet, a not trivial result.

We show in Figure 5.7(a) the time evolution of the cumulated number of perforation events, N . As already observed, we note that there is a delay time, td, before the nucleation of the first hole is detected. For t > td, the cumulated number of perforation events increases almost linearly with time up to a maximum value. Note that the linear increase is the direct consequence of a time independent instantaneous perforation rate.

We define Ntot as the total number of perforation events that occur in the liquid sheet during the whole experiment.

In the following, we decide to characterize the perforation mechanism of a liquid sheet by the values of Ntot, the total number of perforation events, and td the delay time. We prefer this analysis to the one of the perforated area and the number of holes present in the sheet, N , (section 5.1.3.1) as it only considers the nucleation of new holes. As N results from different contributions (nucleation, coalescence and disappearance of holes) it is more difficult to interpret.

5.1. Experimental set-up, samples and image analysis 111

Figure 5.6: a). Left Y axis: Time evolution of the instantaneous perforation rate , λ;

Right Y axis: Time evolution of the intact area of the sheet. b). Time evolution of the density of perforation, D.

Figure 5.7: a). Time evolution of the cumulated number of perforation events, N , that occur in the liquid sheet during the whole experiment with the definition of N_tot the total number of perforation events and td the delay time. The emulsion is a model emulsion at C=0.3 % v/v with doil=39.1 µm. An arrow indicates the time of the maximal sheet expansion for an equivalent pure water liquid sheet. b). Destabilization of a model emulsion-based liquid sheet. The origin of time is taken at the impact of the liquid tear on the target.

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Chapter 5. Destabilization of emulsion-based liquid sheet: single-tear and standard spray experiments