# 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  
Figure 4 shows time series of images for DW, demonstrating the movements of the meniscus during 0.25 s, with different capillary tube radii. In every result obtained at the tube diameter--at 0.4 mm (Figure 4a), 1.2 mm (Figure 4b), and 1.7 mm (Figure 4c)--the location of the meniscus exhibited a common trend with time. In the very early stage of the capillary rise near 0.05 s, as shown in Figure 4c, the rise profile in the vertically oriented tube increased nonlinearly with time. Then, the rise degraded over time. The rise height corresponding to the meniscus location at 0.25 s decreases from 29 to 16 mm, with increase in the tube radius from 0.4 to 0.85 mm. As shown in Figures 3 and 4a, the maximum rise height at \(r\) = 0.4 and 0.6 mm was 29 and 23 mm at 0.25 s, respectively. However, the maximum value (17 mm) was observed at 0.08 s and not at 0.25 s, as shown in Figure 4c. This phenomenon, called meniscus oscillation, has been previously reported by researchers [13; 26; 27]. The main reason for the oscillation is that the kinetic force may lead to the oscillation. More precisely, meniscus oscillation occurs when the boundary layer of viscous grows at a slow pace with low viscous forces, and keeping liquid inertia to balance the capillary and gravitational force only. In other words, the viscous force restrains the meniscus oscillation energy out while fluids rise up in the tube having radius [23]. In addition, this oscillation is affected by liquid temperature and capillary tube diameters [28]. According to the study by [PERSON] et al. [29], such oscillations occur when the aspect ratio (tube radius/equilibrium height), \(r/h_{\text{e}}\), is approximately greater than 0.05. In this study, aspect ratios for tube radius/equilibrium height, 0.4/36.1, 0.6/24.1, and 0.85 mm/16.1 mm, were 0.011, 0.025, and 0.053, respectively. Another criterion proposed by [PERSON] et al. [30] describes the occurrence of oscillations and involves using a dimensionless number, \(\omega\), as follows:  
\[\omega=\frac{\rho^{2}g\tau^{4}}{64\mu^{2}h_{\text{e}}}, \tag{2}\]  
where \(\rho\) is the density of the liquid (kg/m\({}^{3}\)), \(g\) is the standard gravity (m/s\({}^{2}\)), \(r\) is the capillary radius (mm), \(\mu\) is the viscosity of the liquid (m Pa\(\cdot\)s), and \(h_{\text{e}}\) is the equilibrium height (mm). The critical value of \(\omega\) was 0.25, according to [PERSON]'s results (oscillation at \(\omega\) > 0.25). The test case for \(r\) = 0.85, in which oscillation occurs in this study, has \(\omega\) = 5.7. Figure 5 shows the rise height (\(h\)) of DW in time (\(t\)), which is plotted as log 10 to magnify the differences in the h at the test case of \(r\) = 0.4 mm. The capillary rise has several stages [31]: the first stage is purely dominated by inertial forces, the second is where the influence of viscosity increases (visco-inertial flow), and the third is when the effect of inertia becomes nil and the flow becomes purely viscous. According to proposed analytical solutions in the previous studies [16; 31; 32; 33], the capillary rise for the stages I-III can be expressed by Equations (3)-(5), respectively:  
\[h=t\sqrt{\frac{2\sigma\cos\theta}{\rho r}}, \tag{3}\]  
\[h=\sqrt{\frac{\sigma r\cos\theta}{2\mu}\left[t-\frac{\rho\tau^{2}}{8\mu} \left(1-e^{-\frac{8\mu}{\rho\tau^{2}}t}\right)\right]}, \tag{4}\]\[h=\sqrt{\frac{\sigma r\cos\theta}{2\mu}}t. \tag{5}\]  
Figure 5a shows both \(h\) and \(t\) expressed in log scales for identifying the early stage of the rise by inertial forces, as found in previous studies [16; 19; 34]. In Figure 5a, the red  
Figure 4: Time series of direct photographs for DW showing the height with different tube radii. (a) 0.4 mm, (b) 0.6 mm, and (c) 0.85 mm. Figure 5: Variation of height (\(h\)) versus time (\(t\)) for the DW rise in capillary tube at test case of \(2r=0.8\). (a) \(h\) (log 10 mm) versus \(t\) (log 10 s) and (b) \(h\) (mm) versus \(t\) (log 10 s).