## Materials and Methods
### Droplet Dynamics Analysis in Multi-Airflow Synergy
#### 2.2.2 Model of Droplet Motion in Space  
(a) Construction of motion model  
According to the particle dynamics theory, the motion equation of a single droplet can be simplified as follows:  
\[m_{p}\frac{d\overset{\rightharpoonup}{v}_{p}}{dt}=\frac{\pi}{6}d_{p}^{3}\rho_{p }g+\frac{\pi}{8}C_{d}d_{d}\rho_{f}\Big{|}\overset{\rightharpoonup}{v}_{f}- \overset{\rightharpoonup}{v}_{p}\Big{|}\Big{(}\overset{\rightharpoonup}{v}_{f}- \overset{\rightharpoonup}{v}_{p}\Big{)} \tag{1}\]  
where \(m_{p}\) is individual droplet mass (kg), \(\overset{\rightharpoonup}{v}_{p}\) is individual droplet velocity (m/s), \(d_{p}\) is the diameter of an individual droplet (m), \(\rho_{p}\) is the density of droplets (kg/m\({}^{3}\)), \(C_{d}\) is coefficient of air resistance, \(\rho_{f}\) is air density (kg/m\({}^{3}\)), \(\overset{\rightharpoonup}{v}_{f}\) is air velocity (m/s), and \(g\) is the acceleration of gravity (m/s\({}^{2}\)).  
According to the vector decomposition theorem, the motion equations in the three directions of \(X\), \(Y\), and \(Z\) are as follows:  
\[\left\{\begin{array}{l}m_{p}\frac{d\overset{\rightharpoonup}{v}_{px}}{dt}= \frac{\pi}{8}C_{d}d_{p}\rho_{f}\Big{|}\overset{\rightharpoonup}{v}_{fx}- \overset{\rightharpoonup}{v}_{px}\Big{|}\Big{(}\overset{\rightharpoonup}{v}_{ fx}-\overset{\rightharpoonup}{v}_{px}\Big{)}\\ m_{p}\frac{d\overset{\rightharpoonup}{v}_{px}}{dt}=\frac{\pi}{8}C_{d}d_{p} \rho_{f}\Big{|}\overset{\rightharpoonup}{v}_{fy}-\overset{\rightharpoonup}{v} _{py}\Big{|}\Big{(}\overset{\rightharpoonup}{v}_{fy}-\overset{\rightharpoonup}{v} _{py}\Big{)}\\ m_{p}\frac{d\overset{\rightharpoonup}{v}_{px}}{dt}=m_{p}g-\frac{\pi}{8}C_{d}d_{p} \rho_{f}\overset{\rightharpoonup}{v}_{pz}\end{array}\right. \tag{2}\]  
where \(\overset{\rightharpoonup}{v}_{px}\), \(\overset{\rightharpoonup}{v}_{py}\), and \(\overset{\rightharpoonup}{v}_{pz}\) are the velocity components of the droplet along each direction, respectively.  
Therefore, by the local approximation method, the displacement of a droplet is as follows:  
\[\left\{\begin{array}{l}x_{p}=x_{p_{0}}+\int\limits_{0}^{\Delta t}\overset{ \rightharpoonup}{v}_{px}dt\\ y_{p}=y_{p_{0}}+\int\limits_{0}^{\Delta t}\overset{\rightharpoonup}{v}_{py} dt\\ z_{p}=z_{p_{0}}+\int\limits_{0}^{\Delta t}\overset{\rightharpoonup}{v}_{pz} dt\end{array}\right. \tag{3}\]  
where \(\Delta t\) is the time step, \(x_{p_{0}}\), \(y_{p_{0}}\) and \(z_{p_{0}}\) are the coordinates of the initial droplet position, respectively.  
(b) Solution of motion model  
Given the initial conditions and time steps of the droplets, the droplet displacement was calculated by Matlab.  
According to Ref. [33], most of the droplets have the same speed when they are ejected through the nozzle. Therefore, the initial droplet velocity is set to 20 m/s [34]. It is supposed that the angle between the initial droplet velocity and the positive direction of the \(Y\)-axis is \(\alpha\). The initial droplet velocities in the \(Y\) and \(Z\) directions, respectively, are as follows:  
\[\begin{array}{l}v_{py_{0}}=v_{p_{0}}\times\cos\alpha\\ v_{pz_{0}}=v_{p_{0}}\times\sin\alpha\end{array} \tag{4}\]  
Because the spray angle of the machine's fan nozzle selected is 80\({}^{\circ}\), \(\alpha\) is from \(-40^{\circ}\) to 40\({}^{\circ}\).  
Droplet diameter is another important factor affecting droplet deposition. It is mainly in the range of \(D_{10}\)-\(D_{90}\)[35], and the droplet size distribution can be approximately considered a normal distribution. Therefore, the initial droplet diameter is as follows:  
\[d_{p}=normrnd(D_{50},\sigma) \tag{5}\]where \(D_{50}\) is the median diameter of the droplet and \(\alpha\) is the standard deviation. By adjusting the spray pressure, the droplet diameter changes. According to Ref. [36], \(D_{50}\) is about 50-110 \(\upmu\)m when the spray pressure is 0.3-0.7 MPa.