# A Machine Learning Approach to Predict Fluid Viscosity Based on Droplet Dynamics Features
## 3 Results and Discussion
### Droplet Dynamics Analysis  
Figure 3 shows drop experiments with five different viscosities of glycerol-water (G-W) solutions of 70%, 75%, 85%, 90%, and 99%, exploring the drop process induced by gravity. For the lower viscosity \(Oh=0.15\), \(Oh=0.25\), and \(Oh=0.48\) fluids, as shown in Figure 3a-c, as the flow pump slowly pushes in the fluid, the droplet begins to accelerate away from the needle when the gravity of the droplet exceeds the surface tension. At this point there is the appearance \(h_{min}<R\) of the liquid neck, and the length of the droplet increases, the thickness of the droplet \(h_{min}\) decreases rapidly, and then, as shown in the small dashed line in Figure 3a at the moment of 0 ms, the liquid neck appears asymmetric and inhomogeneous at the upper and lower ends, and an END-PINCHING fracture occurs [18]. As the lower end breaks, shrinkage occurs first; when the upper end breaks, the resulting independent liquid filaments have undergone shrinkage at the lower end, manifesting as a rounded sphere, and the upper side manifests as a sharp cone when it just breaks, as shown in the small green dashed line in Figure 3a at \(-0.25\) ms. For the higher viscosity \(Oh=0.79\) versus \(Oh=3.12\) G-W solutions, the droplets fall more slowly, the viscosity impedes fluid flow, fluid flow inside the liquid neck is more difficult, the droplets fall further, and the liquid filaments are stretched longer. Figure 4 demonstrates that as the time before pinch-off decreases, the liquid neck thickness \(h_{min}\) decreases. In Figure 4a, the neck thickness decreases as the time from pinch-off decreases, and the neck thickness decreases faster and faster as the time decreases. It can be seen that it takes more time for the droplet with higher viscosity to clamp off because the viscosity of the fluid hinders the flow of the fluid, the flow inside the neck is inhibited, and it is more difficult for the fluid to flow outward from the neck, which slows down the contraction of the neck. Figure 4b shows \(h_{min}/R=(0.0709/Oh)\tau+a\) against the experimental data curve. The constant a \((h_{min}/R=(0.0709/Oh)\tau+a)\) was increased to correct the overlap with the experimental data. In the study of [PERSON] et al. [16], the correction constants for the viscous regime corresponding to \(Oh=0.23\) are \(-0.005\), a is 0.03 for \(Oh=0.55\), and a is 0.33 for \(Oh=1.81\). In our experiments, the similar viscosities and the correction constants correspond to, respectively, \(-0.02\) for \(Oh=0.25\), a is 0.03 for \(Oh=0.79\), and a of 0.25 for \(Oh=3.12\). The slight difference with the Castrejon-Pita results is a negligible error in the mapping process. In this regime, viscosity is the main action in the contraction regime. Figure 2: Definition of \(h_{min}\) and R (\(Bo=0.10\) and \(Oh=0.15\)). (**a**) \(h_{min}\) and R definition; (**b**) contraction process of the local liquid neck of the drop; (**c**) evolution of the outer contour of the drop process. When the contraction process shifts from the inertial to the viscous regime, there exists a transient that makes the inertial, viscous action balanced [35], and usually the transition occurs at \(h_{min}/R\sim\partial h^{2}\)[20,35]. The dashed line in Figure 4b represents \(h_{min}/R\sim\partial h^{2}\) the position of this critical value in the longitudinal coordinate, and the colour of the dashed line for the same viscosity is the same as that of the liquid neck contraction line. For example, when \(\partial h=0.79\), the horizontal light blue dashed line represents the \(h_{min}/R\sim\partial h^{2}\) size and the light blue dashed line intersects with the light blue neck shrinkage line, which indicates that the effect of stickiness increases and affects the shrinkage process after the neck has theoretically shrunk to this size threshold. The theoretical transition thresholds of \(\partial h=0.15\), \(\partial h=0.25\), \(\partial h=0.48\), and \(\partial h=0.7\) are marked in the figure \(h_{min}/R\sim\partial h^{2}\).