Versatile analysis of closed-end capillary invasion of viscous fluids

  • Hosub Lim
  • Minki Lee
  • Jinkee LeeEmail author


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.


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 (t) is the invasion distance, rc 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

A schematic diagram of the closed-end capillary invasion is illustrated in Fig. 1a. c and rc 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/m3, 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.
Fig. 1

a Schematic diagram of system, b geometry and boundary conditions for simulation, and c experimental setup for observing the meniscus of capillary rise

3 Theory and numerical simulation

3.1 Momentum balance of the closed-end capillary invasion

A closed-end capillary invasion of a fluid can be described as a momentum balance of the fluid. It is represented by the one-dimensional differential equation with fluid invasion distance (t) using a balance of the inertial, viscous, surface tension, hydrostatic, and entrapped gas pressure due to the closed-end capillary as follows:
$$- \rho \frac{d}{dt}\left( {\ell \frac{d\ell }{dt}} \right) = \frac{8\mu \ell }{{r_{c}^{2} }}\frac{d\ell }{dt} - \frac{2\sigma \cos \theta }{{r_{c} }} - \rho g\left( {h - \ell } \right) - \left( {P_{atm} - P_{g} } \right)$$
Here, 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 Pg(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 = /rc, H = h/rc, T = σt/μrc, \(\chi_{c}\) = c/rc as follows:
$$\begin{aligned} \frac{1}{{Oh^{ 2} }}\frac{d}{dT}\left( {L\frac{dL}{dT}} \right) + 8L\frac{dL}{dT} - 2\cos \theta - Bo\left( {H - L} \right) \hfill \\ -\, \beta \left( { 1- \left( {\frac{{\chi_{c} }}{{\chi_{c} - L}}} \right)} \right) = 0 \hfill \\ \end{aligned}$$
where 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

The COMSOL Multiphysics using the finite element method was used to simulate this system. 14,306 elements of quadrilateral mapped mesh were generated having a maximum element size of 0.006 mm. The discretization was a first-order element for both velocity and pressure fields. The two-phase flow solver with the level set method was used to solve the closed-end capillary invasion. To solve the motion of two immiscible fluids, the Navier–Stokes equation was applied with surface tension and gravity force as follows:
$$\rho \frac{\partial u}{\partial t} + \rho \left( {u \cdot \nabla } \right)u = \nabla \cdot \left[ { - \,pI + \mu \left( {\nabla u + \left( {\nabla u} \right)^{T} } \right)} \right] + F_{st} + \rho g$$
Because the entrapped gas is compressed by the rising fluid, the pressure of the gas phase is assumed to be changed depending on the invasion length, as in the theoretical model. At the inlet, the boundary condition is the hydrostatic pressure \(\left( {P_{inlet} = - \,\rho_{f} h_{0} g} \right)\). A slip boundary condition at the wall of the capillary is used to overcome the singularity at the moving contact line, and the slip length is arranged depending on the mesh size, hm. The contact line and gas–fluid interface can be tracked using the level set method defined as follows:
$$\frac{\partial \phi }{\partial t} + u \cdot \nabla \phi = \gamma {\kern 1pt} \nabla \cdot \left( {\varepsilon \nabla \phi - \phi (1 - \phi )\frac{\nabla \phi }{{\left| {\nabla \phi } \right|}}} \right)$$
where 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 0.75hm.

4 Results

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

The theoretical solutions of Eq. (1) applied to static contact angle \((\theta = 10^\circ )\) are represented by the solid lines in Fig. 2, and the results are compared with the experiments, shown by symbols. The experimental data had a significant difference compared with the theoretical data, except the final height. As shown in Fig. 2, the theoretical velocity at the early stage was much larger than that in the experimental result. In addition, the theoretical height for the 30% glycerol demonstrated an oscillation of approximately 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.
Fig. 2

Experimental and theoretical (Eq. (1)) invasion height of 30% and 60% glycerol without considering the dynamic contact angle

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

The dynamic contact angle was investigated relative to the capillary number (\(Ca = \mu V/\sigma\)) [25]. As the interface moves, the contact angle decreases while fluid velocity reduces. Therefore, the high-velocity region at the early stage, with a large capillary number, gives a large dynamic contact angle. To measure the dynamic contact angle θ during invasion, the geometrical analysis used a fitting circle with radius (R′) and measuring the sagitta (hs) 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 (hs) from the circular arc did not change, and the dynamic contact angle could be calculated from the sagitta and the measured capillary radius (rc) using the following equation:
$$\theta = \cos^{ - 1} \left( {\frac{{ 2h_{s} r_{c} }}{{h_{s}^{ 2} { + }r_{c}^{ 2} }}} \right)$$
Fig. 3

a Geometrical relationship to measure the dynamic contact angle and b experimental contact angles of case of (i) 30% and (ii) 60% glycerol solutions as a function of time, and their empirical expression in Eqs. (6) and (7)

Because the dynamic contact angle is affected by the capillary number, the dynamic contact angle of each fluid was measured for all times as shown in Fig. 3b. As a result, an empirical expression can be derived by fitting the experimental data with time, as shown below.
$$\theta = 0.22 + 1 \times e^{{\frac{ - t}{0.009}}} \left( {30\% {\text{ glycerol solution}}} \right)$$
$$\theta = 0.32 + 1.1 \times e^{{\frac{ - t}{0.028}}} \left( {60\% {\text{ glycerol solution}}} \right)$$
These empirical equations were used to modify the theory and numerical simulation. As a result, the invasion height was obtained, as shown in Fig. 4a for simulation and experiment. As mentioned previously, the radius of the experimental image was enlarged by optical distortion, but the invasion heights agree well. Figure 4b shows a slower invasion than Fig. 2 at the early stage, because the surface tension force causing a high inertia force decreased when the theory considered the dynamic contact angle. Further, the oscillation from a nonmodified theory for the 30% glycerol case disappeared.
Fig. 4

a Time sequential images of simulation and experiment and b experimental, theoretical, and numerical results of capillary invasion of closed-end capillary considering dynamic contact angle

4.3 Shear stress on wall in closed-end capillary invasion

During invasion, the shear stress varies as the velocity profile changes [21]. However, the shear stress is not related to the final invasion height, but it influences the trend of the early stage fluid invasion, especially within the entrance length where the velocity profile is not parabolic. When Reynolds numbers for the 30% and 60% glycerol solutions were calculated during invasion, the maximum Reynolds numbers were approximately 41 and 2, respectively, which represented a laminar flow. From these values, the entrance length could be estimated as Le ~ 0.05dRe ~ 707 μm and 36 μm 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 μm 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 μm for 30% glycerol solution.
Fig. 5

a Shear stress at the wall along with invasion height of 60% glycerol solution, b velocity profile at a height of 0.12 mm from the entrance (red line of the inset) with time, and c rising height of the experimental data, theoretical model, and numerical simulation

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.



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].


  1. 1.
    M. Tuller, D. Or, L.M. Dudley, Adsorption and capillary condensation in porous media: liquid retention and interfacial configurations in angular pores. Water Resour. Res. 35(7), 1949–1964 (1999)CrossRefGoogle Scholar
  2. 2.
    N.R. Morrow, G. Mason, Recovery of oil by spontaneous imbibition. Curr. Opin. Colloid Interface Sci. 6(4), 321–337 (2001)CrossRefGoogle Scholar
  3. 3.
    A.T. Jafry, H. Lim, S.I. Kang, J.W. Suk, J. Lee, comparative study of paper-based microfluidic devices with respect to channel geometry. Colloid Surf. A 492, 190–198 (2016)CrossRefGoogle Scholar
  4. 4.
    A.T. Jafry, H. Lim, W.K. Sung, J. Lee, Flexible time-temperature indicator: a versatile platform for laminated paper-based analytical devices. Microfluid. Nanofluid. 21(3), 57 (2017)CrossRefGoogle Scholar
  5. 5.
    R. Lucas, Rate of capillary ascension of liquids. Kolloid Z 23(15), 15–22 (1918)CrossRefGoogle Scholar
  6. 6.
    E.W. Washburn, The dynamics of capillary flow. Phys. Rev. 17(3), 273 (1921)CrossRefGoogle Scholar
  7. 7.
    D. Quéré, Inertial capillarity. Europhys. Lett. 39(5), 533–538 (1997)CrossRefGoogle Scholar
  8. 8.
    N. Fries, M. Dreyer, An analytic solution of capillary rise restrained by gravity. J. Colloid Interface Sci. 320(1), 259–263 (2008)CrossRefGoogle Scholar
  9. 9.
    S. Das, P.R. Waghmare, S.K. Mitra, Early regimes of capillary filling. Phys. Rev. E 86(6), 067301 (2012)CrossRefGoogle Scholar
  10. 10.
    P.L.L. Walls, G. Dequidt, J.C. Bird, Capillary displacement of viscous liquids. Langmuir 32(13), 3186–3190 (2016)CrossRefGoogle Scholar
  11. 11.
    B.V. Zhmud, F. Tiberg, K. Hallstensson, Dynamics of capillary rise. J. Colloid Interface Sci. 228(2), 263–269 (2000)CrossRefGoogle Scholar
  12. 12.
    W.B. Young, Analysis of capillary flows in non-uniform cross-sectional capillaries. Colloid Surf. A 234(1–3), 123–128 (2004)CrossRefGoogle Scholar
  13. 13.
    K.A. Polzin, E.Y. Choueiri, A similarity parameter for capillary flows. J. Phys. D Appl. Phys. 36(24), 3156–3167 (2003)CrossRefGoogle Scholar
  14. 14.
    M. Reyssat, L. Courbin, E. Reyssat, H.A. Stone, Imbibition in geometries with axial variations. J. Fluid Mech. 615, 335–344 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Erickson, D. Li, C.B. Park, Numerical simulations of capillary-driven flows in nonuniform cross-sectional capillaries. J. Colloid Interface Sci. 250(2), 422–430 (2002)CrossRefGoogle Scholar
  16. 16.
    M. Reyssat, L.Y. Sangne, E.A. van Nierop, H.A. Stone, Imbibition in layered systems of packed beads. EPL 86(5), 56002 (2009)CrossRefGoogle Scholar
  17. 17.
    A. Hamraoui, T. Nylander, Analytical approach for the Lucas-Washburn equation. J. Colloid Interface Sci. 250(2), 415–421 (2002)CrossRefGoogle Scholar
  18. 18.
    S. Gruener, T. Hofmann, D. Wallacher, A.V. Kityk, P. Huber, Capillary rise of water in hydrophilic nanopores. Phys. Rev. E 79(6), 067301 (2009)CrossRefGoogle Scholar
  19. 19.
    J.I. Siddique, D.M. Anderson, A. Bondarev, Capillary rise of a liquid into a deformable porous material. Phys. Fluids 21(1), 013106 (2009)CrossRefGoogle Scholar
  20. 20.
    R. Fazio, S. Iacono, An analytical and numerical study of liquid dynamics in a one-dimensional capillary under entrapped gas action. Math. Methods Appl. Sci. 37(18), 2923–2933 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Lim, A. Tripathi, J. Lee, Dynamics of a capillary invasion in a closed-end capillary. Langmuir 30(31), 9390–9396 (2014)CrossRefGoogle Scholar
  22. 22.
    Y.J. Gao, Z.R. Luo, L.L. Huang, H. Mao, C.G. Huang, K. Lin, Phase field crystal study of nano-crack growth and branch in materials. Model. Simul. Mat. Sci. Eng. 24(5), 055010 (2016)CrossRefGoogle Scholar
  23. 23.
    C.F. Wei, M.T. Tyree, E. Steudle, Direct measurement of xylem pressure in leaves of intact maize plants. A test of the cohesion-tension theory taking hydraulic architecture into consideration. Plant Physiol. 121(4), 1191–1205 (1999)CrossRefGoogle Scholar
  24. 24.
    G. P. Association, Physical properties of glycerine and its solutions (Glycerine Producers’ Association, New York, 1963)Google Scholar
  25. 25.
    R.L. Hoffman, A study of the advancing interface. I. Interface shape in liquid—gas systems. J. Colloid Interface Sci. 50(2), 228–241 (1975)CrossRefGoogle Scholar
  26. 26.
    M. Darzi, C. Park, Optical distortion correction of a liquid-gas interface and contact angle in cylindrical tubes. Phys. Fluids 29(5), 052004 (2017)CrossRefGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringSungkyunkwan UniversitySuwonRepublic of Korea

Personalised recommendations