# Transient Phenomena of Dynamic Contact Angle in Micro Capillary Flows +
## 4 Results and Discussion
In a similar method, the present study's time stages were separated by momentum previously calculated, and the formulas are given as follows:  
\[ \begin{array}{l}\text{Purely inertial time stage}\\ \frac{inertial-viscous}{capillary}=0.97\\ \\ \frac{viscous-inertial}{capillary}=0.97\\ \end{array}  \tag{7}\]  
Figure 12 shows the times for the dominant momentum according to the various viscosities in comparison with [PERSON] approach [23]. As viscosity lowers, the interval between two different approaches is greater, but overall initial time becomes shorter at a  
Figure 11: Transient behavior of four forces acting on the capillary flow in the case of \(\mu=0.21\) Pa-s.  
Figure 10: Transient behavior of viscous force for various viscosities, which is the third term of the left\(-\)hand side in Equation (6). higher viscosity. This result brings to mind that the capillary force varying with dynamic contact angle plays an important role in determining the maximum velocity. Due to assuming a constant contact angle, the constant capillary force caused high initial velocity in a short time, and it slowed down fairly fast. However, rapidly changing velocity within a short time does not occur in actual flow; thus, a wider time interval for the present approach is closer to the actual flow. As proof of this, the time at maximum velocity exists in the viscous-inertial region as positive acceleration, but it is not included in the purely viscous region as negative acceleration. In order to compare the magnitude of interfacial force, inertial force, and viscous force, [PERSON] [9] used the dimensionless numbers with regard to the capillary flows, such as the capillary number (Ca = \(\mu V/\sigma\)) and Weber number (We = \(2\rho VL/\sigma\)). In the same way, Figure 13 shows the magnitude of three forces for various viscosities as a function of the dimensionless number. It is found that the data lines are overlapped in a certain interval, which is separated by three asymptotic lines obtained from time stages [8; 9]. In the plot, the capillary number and Weber number rapidly increase and reach maximum until the viscous-inertial regime (\(t_{0}<t<t_{viscous}\)), but the numbers continually decrease after that until the end of the process (\(t>t_{viscous}\)). This is the same manner with meniscus velocity due to the dimensionless number as a function of velocity. In terms of dominant force, at low capillary and Weber numbers, the interfacial force is absolutely dominant, which agrees with the result of [PERSON]'s review table [9]. In addition, there exists only the inertial force as the dominant force at purely inertial regime (\(t_{0}<t<t_{inertial}\)), and the viscous forces are significantly increased at viscous-inertial regime (\(t_{inertial}<t<t_{viscous}\)), where inertial and viscous forces exist as dominant forces but proceed in opposite directions. Thus, lower viscosity increases inertial force, and higher viscosity increases interfacial and viscous forces. The capillary force determines the balance of other forces, and it has various parameters, such as density, surface tension, and contact angle. If variations of density and surface tension are neglected, the capillary force varies with the contact angle from the equations. The contact angle depends on the surface conditions inside the tube, which causes a considerable problem in liquid penetration tests. [PERSON]'s experiment [15] has two different conditions, in which the surfaces of the inner tube are dry or wet. In their study, the result showed that the experimental data for the wet condition agrees quite well with the theory. However, in Figure 14, the experimental data are found to be similar to the dry condition rather than the wet condition. This result comes from the consideration of the dynamic contact angle. Generally, if the contact angle is ignored or considered as  
Figure 12: Separated time stages according to the various viscosities. a static contact angle, its value is calculated using a 0\({}^{\circ}\) or fixed equilibrium value.