# Study of Deformation and Breakup of Submillimeter Droplets' Spray in a Supersonic Nozzle Flow
## 3 Results
### \"Pulling\" Breakup  
Another breakup morphology, many cases of which were observed in the flow, resembled so-called \"pulling\", distinguished as a separate regime in [17] and noted in [19] as an early stage of breakup in underexpanded jet. This regime deserves a more detailed discussion, as little attention was paid to it earlier. In case of pulling, a stamen emerges in the wake of a droplet, then grows into a long fiber and disrupts into a number of much smaller droplets with seemingly equal intervals between them, resembling round jet breakup in Rayleigh regime or, more likely in first wind-induced regime [31], which occurs at jet Weber numbers of \(0.4<We_{j}<13\). \(We_{j}\) is expressed similar to Equation (1), but with a jet diameter instead of the droplet size. In some cases at an early stage, a liquid film begins to form, collapsing into a stem shortly after. The subsequent scenario does not differ from the pulling breakup scenario without initial film formation. Early liquid film formation may indicate the beginning of a competing bag breakup regime that switches to a pulling breakup. An illustration of pulling breakup stages, combined from several events, is shown in Figure 10.  
The experimental technique allowed capturing only two subsequent images, which is not suitable for tracking the evolution of the droplet's breakup. As a result, only snapshots of the pulling process could be detected in the captured breakup events. This leaves the nature of the observed pulling breakup somewhat unclear, though the most probable explanation, proposed in the paper [17], is a viscous drag that transports the liquid to the wake of the droplet due to the high shear stress. Indeed, for the shear stress in the laminar boundary layer:  
\[\tau\sim\mu\frac{v}{d}\sqrt{Re}=\sqrt{\frac{\mu\sigma\Delta v}{d^{2}}We} \tag{11}\]  
Figure 9: Example of the developed bag rupture and underdeveloped bag collapse. Thus, when scaling a droplet size by a factor of \(k\), while preserving a Weber number, a shear stress value changes as \(k^{5/4}\) (\(\Delta v\) scales as \(k^{-1/2}\) to preserve _We_). Scaling a droplet size \(d\) from 1 mm down to 50 \(\upmu\)m yields an increase in \(\tau\) by a factor of \(\approx 22\). Note that in the experiments by [PERSON] et al. [20], due to the conditions (highly rarefied flow, relative Mach number \(\approx 3\), large droplets), the increase in \(\tau\), estimated in the same manner, was about 4.2, which also supposedly led to a manifestation of the pulling regime. A statistic of about 300 fully developed pulling events were extracted from images and analyzed. Droplets in pulling breakup state were mainly in the range of _We_ from 4 to 14. The equivalent size of the droplet was calculated as:  
\[d=\sqrt{\frac{6\big{(}V_{body}+V_{stem}\big{)}}{\pi}} \tag{12}\]  
where \(V_{body}\) and \(V_{stem}\) are the volumes of the droplets body and the stem, respectively. A stem shape was approximated as a cylinder for this purpose, and the droplet's body shape was approximated as a spheroid. A cumulative percentage of detected pulling breakup events against _We_ is shown in Figure 11a. Few events were detected at the _We_ above 14, but their presence is most likely can be attributed to the error in determining a _We_ number for the droplet. Still, an overlap between bag breakup range and observed pulling breakup range exists at the Weber numbers of 11-14. This overlap is reasonable in the light of some previous works (see, for example, a reference to the paper by [PERSON] [14] or [PERSON] et al. [3] in the introductory section), claiming that transition between regimes is a continuous process, rather than a process with a single-value threshold. Figure 11: Cumulative percentage of pulling breakup events against Weber number (**a**) and the statistics of stem length against Reynolds number for pulling regime (**b**). Figure 10: Illustration of pulling breakup stages, combined from several events.