# Transient Phenomena of Dynamic Contact Angle in Micro Capillary Flows +
## 4 Results and Discussion  
In Figure 4, the movement of the flow meniscus in the unsteady capillary flow is visualized for the whole measurement length in the case of a 99% glycerin-water mixture. In the visualization images, the meniscus has a maximum width of 174 pixels and a maximum thickness of 37 pixels. When the meniscus lies at the entrance region, the contact angle is nearly 90\({}^{\circ}\), but it gradually decreases as the flow goes on. In many studies, approximate equations were developed for dynamic contact angles, such as empirical equations and the function of exponential. However, any of these equations do not have a perfect form for the decaying rate as time goes on. To minimize error, in the present study, the dynamic contact angle was determined by a polynomial equation with the least-square fitting method. Based on the visualized images in Figure 4, the variations of contact angle for all fluid viscosities are plotted in Figure 5. Here, the symbols denote the experimental data. The solid lines are least-squares fitted data to a polynomial function, respectively. The present data indicate that it approaches 90\({}^{\circ}\) at \(t=0\), and then it decreases according to the time until it reaches an equilibrium value. The higher viscosity of the liquid takes more time to reach the equilibrium contact angle. \begin{table}
\begin{tabular}{c c c} \hline \hline \begin{tabular}{c} **Volume Fraction** \\ **of Glycerin (\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Static Equilibrium Contact Angle (\({}^{\circ}\))** \\ \end{tabular} &
\begin{tabular}{c} **Viscosity** \\ **(Pa-s)** \\ \end{tabular} \\ \hline
87.5 & 7.95\({}^{\circ}\) & 0.21 \\ \hline
93.75 & 7.98\({}^{\circ}\) & 0.54 \\ \hline
96.87 & 8.45\({}^{\circ}\) & 0.82 \\ \hline
99.0 & 8.95\({}^{\circ}\) & 1.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Static equilibrium contact angles and viscosities for various glycerin–water mixtures. Figure 3: Illustration of contact angle with respect to the thickness \(T\). Figure 2: Schematic of the experimental system. Figure 6 shows the wetting distances for various viscosities as a function of the filling time of the capillary meniscus. The measured data are averaged by five experiments in each case. The measured initial length is 700 \(\upmu\)m from the inlet of the tube, and this initial length is also taken into account in the theoretical calculation. The overall patterns of wetting distance show that the meniscus moves fast just after fluid is dispensed at the edge, and then its traveling speed becomes slow. As the viscosity decreases, it takes more time for the meniscus to travel a given wetting distance. Theoretically, due to the viscous force acting as a resistance of the capillary flow, the fluid with a higher viscosity takes a shorter time to travel the same distance. From the movement of the meniscus position in Figure 6, the meniscus velocity can be obtained by differentiating the meniscus position with time as shown in Figure 7. The velocity is very high at the entrance region, and then it decreases gradually. As the viscosity becomes lower, this results in higher velocity and a higher decreasing rate. Despite such a drastic change in the initial velocity, the theoretical data by Equation (6) agree well with experimental data. This agreement results from the consideration of all force terms acting on the capillary flow. However, it is still unknown which force is dominant in the flow. Thus, it is necessary to investigate the variation of each force in Equation (6) separately and search for the dominant force in capillary flow. Figure 8 shows the inertial force at the inlet of the tube for various viscosities according to the time on a log scale. The inertial force starts from zero at the entrance of the tube and increases until it becomes maximum. Then, it decreases to zero within a short time. Figure 4: Movement of flow meniscus and dynamic variation of contact angle in the unsteady capillary flow in the case of a 99% glycerin–water mixture.