## 2 Experimental and Computational Numerical Methods
### Experimental Setup
#### 2.1.2 Image Acquisition and Post-Processing  
In total, 6 different experimental cases were studied and analyzed (N1H0 to N2H2), as shown in Table 1. In each case, a total of 1000 captured frames were used to measure the sizes of the droplets and their distribution. Fiji (Fiji Is Just Image) is an open-source image-processing package based on ImageJ2 [31], bundling a lot of plugins that facilitate scientific image analysis [28]. Fiji was utilized to post-process 1000 frames for each case to measure the sizes of the droplets. The method for enumerating the sizes of the droplets using Fiji required the image file to be converted from RGB to 8-bit grayscale, as illustrated in Figure 2a,b. The automated counting of droplets used threshold algorithms to discriminate the features of interest from the background. The threshold level was dependent on the algorithm selected, and in this image analysis, the Yen threshold algorithm was used [32]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Exp. Run** & \begin{tabular}{c} **Number of** \\ **Nozzles** \\ \end{tabular} & \begin{tabular}{c} **Nozzle Diameter,** \\ **\(\mathbf{\mu}\)m** \\ \end{tabular} & \begin{tabular}{c} **Nozzle Plate** \\ **Material** \\ \end{tabular} & \
\begin{tabular}{c} **Flow Rate, L/hr** \\ **\end{tabular} &
\begin{tabular}{c} **Vertical Height** \\ **(H), mm** \\ \end{tabular} \\ \hline N1H0 & 17 & 170 \(\pm\) 5 & PEEK & 3.6 & 0 \\ N1H1 & 17 & 170 \(\pm\) 5 & PEEK & 3.6 & 200 \\ N1H2 & 17 & 170 \(\pm\) 5 & PEEK & 3.6 & 400 \\ N2H0 & 30 & 200 \(\pm\) 5 & SS1.4310 & 7 & 0 \\ N2H1 & 30 & 200 \(\pm\) 5 & SS1.4310 & 7 & 200 \\ N2H2 & 30 & 200 \(\pm\) 5 & SS1.4310 & 7 & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of experimental runs. To establish the counting threshold, the procedure began by opening the selected image. The following sequence of steps was then meticulously executed: navigating through 'Image > Adjust > Threshold', selecting the appropriate algorithm, and applying it to convert the image into a binary format, as illustrated in Figure 2b. This conversion was achieved by choosing 'Process > Binary > Make binary'. Subsequently, droplet counting was conducted using 'Analyze > Analyze Particles'. During this process, the droplet size parameters were set with an area range of 50/200-infinity pixels and a circularity range of 0.8-1. Each droplet counted was outlined and numbered in a separate window, as shown in Figure 2d. Subsequently, the pixel area of each of these outlined droplets was calculated, and from this area, the diameter was determined in pixel units. Finally, to obtain the actual droplet size of each identified droplet, the pixel area was converted using a pixel size value of 10.75 um/pixel (calculated from camera calibration). Details regarding camera calibration and nozzle size measurements can be found in the Supplementary Material Figures S1-S3. Notably, this same image analysis technique has been employed in biological cell size measurement, as documented in the literature [33]. From each droplet diameter size, we further calculated various average mean diameters (\(d_{10}\), \(d_{20}\), \(d_{30}\), and \(d_{32}\)) of the liquid droplets. The arithmetic mean diameter, denoted as \(d_{10}\), represents the average diameter of all particles within a given sample. \[d_{10}=\frac{\sum n_{i}d_{i}}{N} \tag{4}\]  
In contrast, the surface mean diameter \(d_{20}\), refers to the diameter of a particle whose surface area, when multiplied by the total number of droplets, equals the overall surface area of the sample. \[d_{20}=\left[\frac{\sum n_{i}d_{i}^{2}}{N}\right]^{1/2} \tag{5}\]  
The volume mean diameter, \(d_{30}\), signifies the diameter of a particle whose volume, when multiplied by the total particle count, matches the total volume of the sample. \[d_{30}=\left[\frac{\sum n_{i}d_{i}^{3}}{N}\right]^{1/3} \tag{6}\]  
Figure 2: Case N1H0: (**a**) Raw image of the acquisition of droplets. (**b**) Conversion of image from RGB to 8-bit grayscale.