# Effects of Tube Radius and Surface Tension on Capillary Rise Dynamics of Water/Butanol Mixtures
## 3 Results and Discussion
### Effect of Tube Inner Radius on Capillary Dynamics
dotted line represents a linear variation of \(h\) versus \(t\) in the inertial stage of the capillary rise dynamics. However, it was difficult to divide the stages after the first stage using inertial forces. In this study, we modified Figure 5a as Figure 5b, where \(h\) is plotted as a function of \(t\) and expressed in millimeters to clearly describe the various stages of a capillary rise phenomenon. The capillary rise can be divided into three characteristic stages: Stage I is the purely inertial region, Stage II is inertial + viscous region, and Stage III is purely viscous region). Similar studies in terms of segregation of the capillary rise process have been reported [18; 35]. Stage II shows a linear increase in \(h\) with \(t\). Meniscus height increases as per the parabolic law were observed at both Stages I and III. The \(h_{1}\) represents the transition between Stages I and II, and \(h_{2}\) represents the transition between Stages II and III. The \(h_{\rm e}\) was calculated as the position at 0.99\(h_{\rm e}\) at the start of initiating the capillary rise. End positions of individual stages (Stages I, II, and III), such as heights (\(h_{1}\), \(h_{2}\), and \(h_{\rm e}\)), can also be expressed for their corresponding times \(t_{1}\), \(t_{2}\), and \(t_{\rm e}\), respectively. As shown in Figure 6, comparing the experimental results of this study with Equations (3)-(5) shows that the capillary rise height of water is relatively well described with the models in the early stage. Therefore, the transition time between Stages I and II was calculated by \(t_{1}=\rho r^{2}/4\mu\)[31; 34] based on the [PERSON]'s equation. The value of \(t_{1}\) was 0.045 in rather good agreement with this study, as shown in Figure 5b. For the transition time between Stages II and III, \(t_{2}\) was determined at the point where the linear region ends at the start point of \(t_{1}\). The lengths (including percentages of corresponding Stages I, II, and III) of the inertial, inertial + viscous, and viscous stages, and their end positions are given in Table 1 with different tube radii. The results in Table 1 were obtained from the data (\(h\) versus \(t\)) in Figure 7a and using the method shown in Figure 5b. As shown in Figure 5b and Table 1 at the test case of \(r=0.4\) mm, the sequence for the length of each stage was as follows: Stage II (inertial + viscous) \(>\) Stage I (inertial) \(>\) Stage III (viscos region). In the capillary rise processes for \(r=0.4\) mm, the lengths of Stages I, II, and III were 11.2, 16.4, and 8.3 mm, corresponding to 31.2%, 45.7%, and 23.1%, respectively. Stage II was completed at time point (\(t_{2}\)) of approximately 0.23 s. This period is a relatively short time for the entire rise \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(r\) (mm) & Stage I (mm) & Stage II (mm) & Stage III (mm) & \(h_{1}\) (mm) & \(h_{2}\) (mm) & \(h_{\rm e}\) (mm) \\ \hline
0.4 & 11.2 (31.2\%) & 16.4 (45.7\%) & 8.3 (23.1\%) & 11.2 & 27.6 & 35.9 \\
0.6 & 8.7 (36.1\%) & 10.1 (41.9\%) & 5.3 (22\%) & 8.7 & 18.8 & 24.1 \\
0.8 & 7.2 (44.7\%) & 7.4 (46\%) & 1.5 (9.3\%) & 7.2 & 14.6 & 16.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lengths of characteristic stages including percentages of corresponding stages and their end positions for DW with different capillary tube radii. Figure 6: Comparison of analytic solutions with experimental results of this study (water in a capillary tube with \(r=0.4\) mm). The variable value of cos\(\theta\), as extracted from Equation (6), was used. process, therefore, it is an early stage. These results indicate that the capillary rise at the early stage is dominantly influenced by the pure inertia and hybrid (inertia + viscous) regions rather than by the pure viscous region. The length of Stages I + II and III was 27.6 and 8.3 mm, corresponding to 76.9%, and 23.1%, respectively. This influence was more noticeable at higher tube radius.