## 6 Simulation of Microdroplet Injection Process  
### Analysis of Microdroplet Formation  
A simulation-based qualitative analysis of the jetting processes was performed using furan resin as the jetted fluid. Driving velocity, amplitude, nozzle diameter, and feed attributes were varied to investigate the effects of these parameters on microdroplet formation. In this experiment, the pulse width \(t_{w}\) was set to 10 \(\mu\)s, and \(V_{max}\) was set to 2.500 m/s. Several steps of the microdroplet formation process are demonstrated in Figure 5.  
Figure 5 depicts the steps of microdroplet development: column elongation, neck contraction, column rupture to create the droplet, and steady movement. As shown, a satellite droplet split from the original droplet. This was caused by the working chamber's excess fluid leak, which was triggered by the composite diaphragm's movement. It is challenging to guarantee that the deformation of the composite diaphragm is exactly in line with the physicochemical parameters of the fluid during an actual printing process since numerous elements influence the motions of the fluid in the printhead. The satellite droplet, however, may be able to catch up with the primary droplet throughout its flight to the substrate because it has a lower volume and is consequently susceptible to less drag. Therefore, by either decreasing the satellite droplet's volume or increasing its flight speed, the satellite and main droplets can be combined. The correlation between the satellite and primary droplet flying speeds is seen in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & **Density** & **Viscosity** & **Characteristic Length** & **Reynolds Number** & **Turbulence Intensity** & **Surface Tension** \\  & **(kg/m\({}^{3}\))** & **(cp)** & **(mm)** & **(\%)** & **(dyn/m)** \\ \hline Furan resin & 2100.00 & 9.00 & 0.03 & 15,750.00 & 0.04780 & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters when \(V_{max}\) = 2.200 m/s.  
Figure 4: Simplified simulation model.  
Figure 5: Microdroplet states during the jetting process.  
Figure 6: Physical behaviors of droplet jetting, (**a**) pressure contour plot; (**b**) velocity contour plot.  
Figure 6 shows that the inertia of the satellite droplet is significantly larger than that of the main droplet. As shown in Figure 6b, the satellite droplet merges with the main droplet because the flying speed of the former is significantly greater than that of the latter during the \(t=18\)-\(22\)\(\upmu\)s period. However, the satellite droplet slows down to the same speed as the main droplet from \(t=22\)\(\upmu\)s to \(t=24\)\(\upmu\)s and is slower than the main droplet from \(t=24\)\(\upmu\)s to \(t=30\)\(\upmu\)s. Therefore, the satellite droplet cannot merge with the main droplet.  
As shown in Figure 7, after detaching from the nozzle and flying towards the substrate, the main droplet came in contact with the substrate (sand particles) at \(t=82\)\(\upmu\)s and was deposited onto the substrate (infiltrate the substrate). The droplet surface was perpendicular to the substrate initially and then swelled as it infiltrated the substrate (\(t=84\)-\(86\)\(\upmu\)s). The droplet shrank towards the end of the infiltration process (\(t=86\)-\(88\)\(\upmu\)s) and fully infiltrated the substrate at \(t=90\)\(\upmu\)s.