## 3 Results and Discussion
### Spread Radius Model  
For the spread radius, we first built a model between the trapezoidal wave and droplet volume via numerical simulation. Subsequently, the model was validated by experimental observation through the camera. In the CFD simulation, the collision speed of the droplet \(U\) and the droplet radius \(R_{s}\) were adjusted by changing the driving signal of the piezoelectric actuator. In this study, the two-phase flow model of inkjet from the arrayed nozzle was developed based on the theoretical model of two-phase flow. When setting up the model, mass and momentum conservation are described by the Navier-Stokes equations. Because the fluid velocity is smaller than the speed of sound, both the ink and air are considered as incompressible. The driving waveform of the piezoelectric nozzle is trapezoidal, as shown in Figure 3 below. This control signal was used to eject the ink. In the same period, as the duty cycle of the rising edge becomes smaller, the amount of the ejected ink increases. The simulation software used in this article was COMSOL5.4. The simulation results are explained below. Fluid was considered as a rigid body in the simulation. To form the droplet, the fluid must overcome surface tension. With a high-flow speed, a droplet with the required volume will be able to form at the nozzle earlier. In addition, a larger momentum makes it easier to overcome the surface tension and therefore form a larger droplet. Conversely, with lower flow speeds, only a part of the fluid that is already outside the nozzle will be able to accumulate into a droplet. Therefore, the fluid will keep accumulating while overcoming the surface tension, therefore showing two different types of flow behaviors. In particular, only the fluid that has already reached the exit of the nozzle will be able to overcome the surface tension and form a droplet. The rest of the fluid will keep oscillating under the pressure wave and the surface tension until the kinetic energy is converted to the internal energy completely. As a consequence, the Figure 4 is obtained as the simulation result. \begin{table}
\begin{tabular}{c c} \hline \hline
**Drop Speed** & \(\mathbf{U}\) \\ \hline Collision direction & \(x\) \\ Drop diameter & \(D_{0}\) \\ Droplet spreading diameter & \(d_{max}\) \\ Precurring temperature & \(T_{s}\) \\ Ink viscosity & \(\mu_{d}\) \\ Nozzle traveling speed & \(v\) \\ Surface tension coefficient & \(\sigma_{ink}\) \\ Surface tension & \(F_{st}\) \\ Ink density & \(\rho_{ink}\) \\ Characteristic time of viscous dissipation & \(t_{c}\) \\ Dissipation function & \(\Phi\) \\ Spreading height & H \\ Contact angle & \(\theta_{ink}\) \\ Drop volume & \(V\) \\ Slope angle of area & \(\theta_{k}\) \\ The kinetic energy of the droplet before it hits the substrate & \(E_{K0}\) \\ The surface energy of the droplet before it hits the substrate & \(E_{S0}\) \\ The kinetic energy after maximum spreading of droplets & \(E_{K1}\) \\ The surface energy of the droplet as it spreads to the maximum diameter. & \(E_{S1}\) \\ The change in the surface energy before and after the droplet hits the substrate. & \(E_{p}\) \\ The viscous dissipation of droplet & \(W\) \\ Weber number & We \\ Reynolds number & \(Re\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parametric models and symbols. In this study, by varying the rising-edge duty cycle of the trapezoidal wave, the droplet volume and initial velocity in the simulation were adjusted to be within 8-24 pL and within 4.75-8.21 m/s, respectively. With decreasing rising-edge duty cycle, more fluid volume is accumulated at the nozzle during the same amount of time, which yields a larger kinetic energy and a greater gravity effect. Therefore, it becomes easier for the fluid to overcome the surface tension and form the droplet. Thus, an increasing rising-edge duty cycle results in a slower velocity. The simulation results are validated by experimental measurements in this study.