## 3 Results and Discussion
### Forces Acting on the Droplet  
Firstly, the droplet motion on the windshield should be deduced from the force analysis in the airflow. In the presence of the airflow, the droplets originally attached to the wall begin to deform, causing the advancing contact angle \(\theta_{adv}\), along the direction of motion to increase above the static contact angle \(\theta_{e}\). The receding contact angle \(\theta_{rec}\) on the side away from the direction of motion decreases. The contact angle hysteresis force \(F_{adh}\) resulting from this phenomenon can be expressed as a function of the droplet wetting length \(L_{d}\), the surface tension of the droplet \(\sigma\), and the contact angle hysteresis [41]. \[F_{adh}=KL_{d}\sigma(cos\theta_{rec}-cos\theta_{adv}) \tag{20}\]  
where \(K\) is denoted as an adjustable parameter describing the variation of the droplet contact line. According to the study of [PERSON] et al. [15], the value of \(K\) is set as 1. When a droplet moves rapidly on the windshield, there is internal friction in the droplet, which forms a resistance to motion. The internal flow field is rotating [42, 43]. The rotating flow field and the mean flow field inside the droplet can overlap each other, which will form a boundary layer of thickness \(\delta\). When the droplet diameter is larger than the capillary length \(\sqrt{\frac{\sigma}{P\delta}}\), the viscous dissipation force \(F_{\tau}\) generated by the shear stress associated with the main part of the droplet will dominate the viscous dissipation. The viscous dissipation force \(F_{\tau}\) of the droplet is determined by the velocity gradient inside the droplet. According to the modification of the droplet velocity model proposed by [PERSON] and [PERSON] [44], the viscous dissipation force \(F_{\tau}\) of the droplet will be improved as:  
\[F_{\tau}=\mu A\frac{\partial u}{\partial y}=\mu A\frac{U_{d}\sqrt{Re}}{\sqrt{ r\Delta y}}=\mu U_{d}V^{\prime 1}\sqrt{Re}c_{4}(\theta_{e}) \tag{21}\]  
where the Reynolds number \(Re\) is given by \(Re=\frac{\rho_{l}U_{d}\tau}{\mu}\), where \(\rho_{l}\) is the density of water, \(U_{d}\) is the velocity of droplet, and \(r\) is half of the wetting length \(L_{d}\), \(r=L_{d}/2\). The geometric parameters describing the shape and motion of the droplet, such as the droplet volume \(V^{\prime}\), the contact area \(A\) between the droplet and the inclined surface, and the droplet motion velocity \(u\), are brought into the expression for the viscous dissipative  
Figure 9: Comparison of the experiment and numerical stimulation results on the movement of a droplet under airflow: (**a**) Evolution of the droplet’s contours; (**b**) Evolution of the advancing and receding contact lines. force \(F_{\tau}\) of the droplet. It is necessary to introduce the expression \(c_{4}\) (\(\theta_{e}\)) for the equilibrium contact angle \(\theta_{e}\). \[c_{4}(\theta_{e})=\frac{A}{V^{\frac{1}{3}}\sqrt{\tau\Delta y}}=\pi\bigg{(}\frac {3}{\pi}\bigg{)}^{\frac{1}{3}}\bigg{(}\frac{sin^{3}\theta_{e}}{2-3 cos\theta_{e }+cos^{3}\theta_{e}}\bigg{)}^{\frac{1}{2}}\bigg{(}\frac{2+cos\theta_{e}}{1- cos\theta_{e}}\bigg{)}^{\frac{1}{6}} \tag{22}\]  
The wind direction is parallel to the windshield, so there is a certain wind traction force \(F_{wind}\) when the droplet moves on the windshield, which is defined by the equation:  
\[F_{wind}=\frac{1}{2}\rho_{g}C_{D}SU^{2} \tag{23}\]  
where \(\rho_{g}\) is the air density, \(C_{D}\) is the drag coefficient, \(S\) is the projected area of the droplet, and U is the velocity of the airflow. The value of the drag coefficient \(C_{D}\) is related to the droplet shape and Reynolds number in the airflow. In order to improve the efficiency of calculation, the drag coefficient \(C_{D}\) of the rigid sphere model is set as 0.4. As shown in Figure 10, the droplet attached to the windshield will be subjected to four forces when moving, namely, its own gravity \(F_{g}\), droplet contact angle hysteresis \(F_{adh}\), viscous dissipation force \(F_{\tau}\) and wind traction force \(F_{wind}\).