# Versatile analysis of closed-end capillary invasion of viscous fluids

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## Abstract

When a fluid invades a closed-end capillary, the fluid flows owing to surface tension overcoming the viscous resistance, gravity, and the gas pressure from the entrapped gas, and the fluid spontaneously reaches a certain height. An experiment was conducted to observe the rise of the viscous fluid in a closed-end capillary, and the results are compared with a one-dimensional theoretical model and a numerical simulation solved using a level set method. When a static contact angle was used in the theoretical model, the result was different from the experimental result, because the radius of curvature increased as the fluid rose. Therefore, the dynamic contact angle during the increase was measured, and the geometrical equation was developed. This empirical equation was applied to the modified theoretical model, and the result agreed well with the experiment and simulation. Additionally, the effect of the shear stress at the inner wall was investigated, and it was directly affected by the velocity profile of the fluid, especially at the early stage when *t* < 0.01 s. Therefore, the effects of dynamic contact angle change and the entrance length is important to analyze accurately the dynamics of viscous fluid invasion in a closed-end capillary.

## Keywords

Closed-end capillary invasion Dynamic contact angle Level set method Surface tension## 1 Introduction

Capillary invasion is one of the most well-known phenomena in nature and has been investigated in areas such as water distribution in soil [1], oil recovery [2], and paper fluidic devices [3, 4]. In the early 20th century, Lucas [5] and Washburn [6] derived the equation \(\left( {\ell \left( t \right) = \left( {{{r_{c} \sigma \cos \theta } \mathord{\left/ {\vphantom {{r_{c} \sigma \cos \theta } {2\mu }}} \right. \kern-0pt} {2\mu }}} \right)^{1/2} t^{1/2} } \right)\) for open capillary invasion, where *ℓ*(*t*) is the invasion distance, *r*_{c} is the inner radius of the capillary, *μ* is the viscosity of the fluid, *σ* is the surface tension of fluid, and *θ* is the contact angle at the fluid–gas interface. Since then, many modifications of this Lucas–Washburn equation have been introduced by numerical, theoretical, and experimental processes. Quéré et al. [7] experimentally showed the existence of the inertial regime at the early stage and found a viscous effect on the occurring oscillation. Fries and Dreyer [8] investigated the effect of inertial force and viscous force during capillary invasion and divided the temporal stage depending on the force balance. Das [9] also determined the oscillation regime and Washburn regime using two dimensionless parameters: the Ohnesorge number and Bond number. Bird et al. [10] and Zhmud et al. [11] conducted fluid invasion experiments with different solutions by varying viscosity and including surfactant. In addition, many researchers have studied the influence of nonuniform capillary cross-sectional shape [12, 13, 14, 15] and capillary invasion in porous media [16, 17, 18, 19].

A few researches were done on closed-end capillaries as well. Fazio et al. [20] compared the analytical and numerical results of closed-end capillary invasion. Also, our previous contribution presented the harmonic oscillation model and one-dimensional modified equation according to experimental data for closed-end capillary invasion [21]. However, these closed-end capillary studies did not provide numerical validation of the results considering the change in fluid properties.

Yet, the closed-end capillary is not investigated thoroughly to extend possible applications, such as controlling the microscale/nanoscale volume of the fluid, estimating the fluid properties, liquid printing on the surface having nanoscale cracks [22], and developing new cell pressure probes [23] which measures the pressure of soft membrane materials, such as a cell or vesicle, by measuring the invasion length. In this research, an experiment was conducted, and the results were compared with those of a modified theoretical model and numerical simulation using level set method to confirm the effects of dynamic contact angle and fluid viscosity on closed-end capillary invasion using two different solutions, 30% and 60% glycerol solutions. And finally, we found the entrance length of each case. In this manuscript, we describe the materials and methods in Sect. 2. In Sect. 3, the theoretical equation and numerical scheme are shown. In Sect. 4, we show the theoretical and numerical results without considering the dynamic contact angle reduction showing a large deviation and then we modify the theory by applying the experimental results of dynamic contact angle measurement. The entrance length of each case is also found. In Sect. 5, the conclusion is provided.

## 2 Materials and methods

*ℓ*

_{c}and

*r*

_{c}represent the length and inner radius of the capillary, respectively. The fluid starts to invade through the closed-end capillary when the inlet of the capillary meets the fluid, where the rise is accompanied by gas compression. A numerical simulation was performed by a two-phase flow axisymmetric module with the level set method of COMSOL Multiphysics. The geometry and boundary conditions are shown in Fig. 1b. The experiments were carried out using a high-speed camera (FastCam mini100, Photron) equipped with a zoom lens (VU-50486, Navitar) to observe the fluid invasion in a closed-end capillary. An LED lamp was used as the light source, and an XYZ linear stage was used to hold and fine tune the position of the capillary, as shown in Fig. 1c. Glass capillaries manufactured by VWR International were used, and they had an inner radius and height of 172 μm and 127 mm, respectively. Before the experiment was conducted, the capillaries were cleaned by an oxygen plasma cleaner (PDC-32G, Harrick Plasma) for 60 s to obtain hydrophilic conditions without impurities and contaminants from the surface after rinsing with ethanol (Sigma-Aldrich) and deionized water. The temperature and humidity were kept at 24 °C ± 1 °C and 50% ± 5% during the experiments. The images of fluid invasion were captured at 2000–4000 fps and with 1280 × 1024 pixel resolution. A glycerol–water mixture was used as the invasion fluid. It is well known that such a solution can control the viscosity with a slight change in hydrophilicity [24]. The viscosity and surface tension of the mixtures were measured by a viscometer (R180, ProRheo) and SmartDrop (SDL200TEZD, Femtofab), respectively. Fluids of 30% and 60% glycerol mixtures having different viscosities but similar density and surface tension were used to observe the viscosity effect mainly on fluid invasion in a closed-end capillary: viscosity

*μ*= 2.5 and 10.8 mPa s, density

*ρ*= 1090 and 1160 kg/m

^{3}, and surface tension

*σ*= 70 and 67 mN/m, respectively. This gives a different Ohnesorge number (\(Oh = \mu /\sqrt {\rho \sigma r_{c} }\)), which is a dimensionless number indicating the ratio of the viscous forces to the inertial and surface tension forces; its value for the 30% glycerol solution is approximately four times larger than that for the 60% glycerol solution;

*Oh*= 0.022 for 30% and

*Oh*= 0.093 for 60% glycerol.

## 3 Theory and numerical simulation

### 3.1 Momentum balance of the closed-end capillary invasion

*ℓ*(

*t*) using a balance of the inertial, viscous, surface tension, hydrostatic, and entrapped gas pressure due to the closed-end capillary as follows:

*ℓ*is the distance of the rising fluid,

*h*is the depth at which the capillary is immersed in the fluid, as shown in Fig. 1a,

*g*is gravitational acceleration, and

*θ*is the contact angle. In a closed-end capillary, the pressure of the trapped gas

*P*

_{g}(

*t*) can be converted as \(P_{atm} (\ell_{c} /(\ell_{c} - \ell (t)))\) for an ideal gas expression. Equation (1) can be transformed using dimensionless parameters

*L*=

*ℓ*/

*r*

_{c},

*H*=

*h*/

*r*

_{c},

*T*=

*σt*/

*μ*

*r*

_{c}, \(\chi_{c}\) =

*ℓ*

_{c}/

*r*

_{c}as follows:

*Bo*= \(\left( {\rho gr_{c}^{2} } \right)/\sigma\) is the Bond number, and \(\beta = \left( {P_{atm} r_{c} } \right)/\sigma\) is the ratio of the atmospheric to capillary pressure. This differential equation was solved with various fluid parameters using MATLAB.

### 3.2 Numerical simulation

*h*

_{m}. The contact line and gas–fluid interface can be tracked using the level set method defined as follows:

*γ*is the amount of reinitialization, and

*ε*is the interface thickness. Through this equation, one can represent the interface between the gas and fluid with \(\phi = 0.5\), in gas \(\phi = 0\), and in fluid \(\phi = 1\). The optimal values of the reinitialization parameter and interface thickness should be considered for avoiding numerical instability. It was estimated that the reinitialization parameter was the maximum expected velocity magnitude, and the interface thickness was 0.75

*h*

_{m}.

## 4 Results

### 4.1 Theoretical investigation of closed-end capillary invasion: constant contact angle

*t*< 0.003 s, which occurred when the inertia force was dominant. However, the 60% glycerol did not show oscillation, because it had a higher viscosity than the 30% glycerol occurring the high shear stress; nevertheless, it did not follow the experimental data at the early stage as the dynamic contact angle increased during invasion. The details modifying the contact angle are described in the next section.

### 4.2 Theoretical investigation of closed-end capillary invasion: dynamic contact angle

*θ*during invasion, the geometrical analysis used a fitting circle with radius (

*R*′) and measuring the sagitta (

*h*

_{s}) from the circular arc, as shown in Fig. 3a. During the measurement, optical distortion was observed when the refraction of the light passed into different media: gas, glass, and fluid. The distortion increased in the radial direction, i.e., the image of

*R*′ taken by camera was larger than the actual radius of curvature, but there was no distortion in the invasion direction [26]. Therefore, the sagitta (

*h*

_{s}) from the circular arc did not change, and the dynamic contact angle could be calculated from the sagitta and the measured capillary radius (

*r*

_{c}) using the following equation:

### 4.3 Shear stress on wall in closed-end capillary invasion

*L*

_{e}~ 0.05

*dRe*~ 707

*μ*m and 36

*μ*m if the fluid flows maintain their maximum speed. These calculated entrance length values should not be same with the actual entrance length, because capillary invasion at the early stage has a large shear stress but invasion stops quickly by gas pressure. In addition, the dynamic contact angle and the meniscus shape affect the velocity profile. In the theoretical model in Eq. (1), it was assumed that the velocity profile is parabolic for the whole area, assuming a Hagen–Poiseuille flow. However, the actual velocity profile changed during invasion, and the entrance length and meniscus shape were found by numerical simulation. Figure 5a shows the shear stress at the inner wall by time: 2, 5, 15, and 30 ms. The shear stress is larger in the early stage, because the velocity profile is plug shaped, as in Fig. 5b, and the meniscus experiences high shear stress owing to the shape of meniscus. From this simulation, we obtain the points where the rate of change of shear stress becomes ~ 0 as shown in the inset of Fig. 5a, and we found that the entrance length was ~ 69

*μ*m for the 60% glycerol solution. Before 0.008 s (gray region), there are points that cannot be considered as the entrance length, even if the change of the shear stress becomes ~ 0, because this zero value is induced by a small invasion length and meniscus effect. In the same way, we found that the entrance length was ~ 284

*μ*m for 30% glycerol solution.

Figure 5c shows the rise length by time at the early stage. The theoretical and numerical results had oscillation which was not observed in experiment. High shear stress was generated by unsteady, plug shape, velocity profile at the entrance length region in the experiment, but simulation or theory was not able to reflect this phenomenon accurately. Thus, an extra amount of damping in experiment occurred resulting in the monotonic increase of the length. Although there is a small difference between them, the modified theory and simulation still well predict the experimental results, since this discrepancy exists only below 0.01 s.

## 5 Conclusions

Fluid invasion in closed-end capillary was analyzed through an experiment, theoretical modeling, and numerical simulation. In this study, two different concentrations of glycerol solutions (30% and 60%) were used as the invasion fluid for investigating the effect of viscosity. Analytically, a one-dimensional equation with a static contact angle was solved, and a two-phase flow module with a level set method was used for the numerical simulation. Using the dynamic contact angles varying from 56° to 12° for 30% and from 66° to 17° for 60%, the empirical equation of the dynamic contact angle was developed and applied to modify the theory and simulation. It was noted that the results of the modified theoretical model and numerical simulation with the dynamic contact angle matched well with the experimental data, although there exists small deviation at a very early stage. Furthermore, the effect of the shear stress that was induced by a velocity profile near the inlet and that of meniscus was investigated through numerical simulation. Consequently, the entrance length for the 30% and 60% glycerol solutions was found to be ~ 284 μm and ~ 69 μm. These results could be used as a foundation work for the microscale/nanoscale fluid volume control, such as the liquid printing on the surface having nanoscale cracks and pressure measurement of a cell or vesicle by measuring the invasion length.

## Notes

### Acknowledgements

This research was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2014M3C1B1033982, 2017R1A2B2006964), and by the National Research Council of Science & Technology (NST) grant by the Korean government (MSIT) [No. CMP-16-04-KITECH].

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