## 1 Introduction  
Droplets on surfaces are a phenomenon observed in everyday life, as well as in many environmental or industrial applications: coating processes [1; 2; 3], combustion processes [4; 5; 6], printing [7; 8], self-cleaning surfaces [9; 10; 11], self-catchment surfaces [12; 13; 14], protein adsorption chips, etc. [15; 16; 17]. The study of the motion states of droplets on an inclined, hydrophilic rough surface in gravity is a fundamental problem in the mechanics of wetting and spreading [18; 19; 20; 21; 22; 23; 24; 25], which facilitates a better understanding of how to manipulate a droplet on a rough surface. Obviously and simply, a small droplet on an inclined hydrophilic rough surface has two main motion states: pinned and inchworm sliding. However, predicting the motion state of a droplet on an inclined, hydrophilic rough surface is more complicated and difficult, because it concerns the surface inclination, the droplet shape, the droplet's contact angle hysteresis, and the dynamic behavior of the droplet's three-phase contact line. There have been many studies to predict the pinned or sliding state of a Newtonian fluid (water, glycerol etc.) droplet on an inclined, smooth surface. The earliest theoretical work was attributed to [PERSON] et al. [26]. In their theory, when a droplet stays on an inclined, smooth surface, the component of the gravitational force along the inclined surface \(F_{\mathrm{g}}\) can be expressed by  
\[F_{\mathrm{g}}=mg\sin\phi \tag{1}\]  
where \(mg\) is the gravitational force. For a Newtonian fluid droplet, the pinned force \(F_{\rm P}\) is equal to the capillary force, and can be expressed by  
\[F_{\rm P}=\gamma_{\rm lv}w\times[\cos\theta_{\rm r}-\cos\theta_{\rm a}] \tag{2}\]  
where \(\gamma_{\rm lv}\) the liquid-vapor surface tension, \(w\) the width of the drop perpendicular to the motion, \(\theta_{\rm a}\) is the advancing contact angle, and \(\theta_{\rm r}\) is the receding contact angle, respectively. If \(F_{\rm g}\geq F_{\rm P}\) the droplet slips, whereas if \(F_{\rm g}<F_{\rm P}\) the droplet is pinned. Subsequently, [PERSON] et al. [27] studied the motion states of a droplet on an inclined, rough surface. They used almost the same method as [PERSON]'s method to predict the droplet's motion state, only replacing \(\theta_{\rm a}\) and \(\theta_{\rm r}\) on a smooth surface with that on a rough surface. In the [PERSON] and [PERSON] methods, \(w\) is always replaced with the contact circle diameter of a droplet on a horizontal plane. Their predictions did not consider the shape change of a droplet's shape when the droplet stays on an inclined surface; therefore, their prediction results contain many errors. After these works, [PERSON] et al. [28; 29; 30] used the minimum energy dissipation principle to analyze the evolution of the droplet's shape when a droplet begins to stay on an inclined surface. They derived the droplet's motion state by solving a series of equations of contact line evolution. However, for a rough surface the equations of the droplet's three-phase contact line evolution are very complicated and not easily solved. Then, [PERSON] et al. [31; 32; 33] developed a volume of fluid (VOF) method and corresponding JADIM software solver, which could numerically simulate the changes of the droplet shapes and the dynamic front/rear contact angles (\(\theta_{1-s}/\theta_{2-s}\)), when a droplet stays on an inclined, smooth surface. By comparing \(\theta_{1-s}/\theta_{2-s}\) with \(\theta_{\rm a}/\theta_{\rm r}\), they judged the motion state of a droplet on an inclined surface in gravity. However, for a rough surface, the VOF method needs to finely mesh the bottom of a droplet, due to micro or nano structures on the hydrophilic rough surface. Maybe it is complicated and trivial. [PERSON] et al. [34] did many experiments of droplets on the inclined, smooth surfaces and showed the relationship between the droplets' motion states and the changes of contact angles. However, they did not give the theoretical models.