Abstract

A computational fluid dynamics model is developed to study the dynamics of meniscus formation and capillary flow between vertical parallel plates. An arbitrary Lagrangian–Eulerian approach is employed to predict and reconstruct the shape of the meniscus with no need to employ implicit interface tracking schemes. The developed model is validated by comparing the equilibrium capillary height and meniscus shape with those predicted by available theoretical models. The model was used to predict the capillary flow of water in hydrophilic (silver) and hydrophobic (Teflon) vertical channels with wall spacings ranging from 0.5 mm to 3 mm. It is shown that the computational model accurately predicts the capillary flow regardless of the channel width, whereas the theoretical models fail at relatively large wall spacings. The model captures several important hydrodynamic phenomena that cannot be accounted for in the theoretical models including the presence of developing flow in the entrance region, time-dependent formation of the meniscus, and the inertial effects of the liquid in the reservoir. The sharp interface tracking technique enables direct access to the flow variables and transport fluxes at the meniscus with no need to use averaging techniques.

References

1.
Shikhmurzaev
,
Y. D.
,
2007
,
Capillary Flows With Forming Interfaces
,
CRC Press
,
Boca Raton, FL
.
2.
Hernández-Baltazar
,
E.
, and
Gracia-Fadrique
,
J.
,
2005
, “
Elliptic Solution to the Young-Laplace Differential Equation
,”
J. Colloid Interface Sci.
,
287
(
1
), pp.
213
216
.10.1016/j.jcis.2005.01.102
3.
Lucas
,
V. R.
,
1918
, “
Cleber Das Zeitgesetz Des Kapillaren P, Ufstiegs Yon Fliissigkeiten
,”
Kolloid-Z.
,
23
(
1
), pp.
15
22
.10.1007/BF01461107
4.
Washburn
,
E. W.
,
1921
, “
The Dynamics of Capillary Flow
,”
Phys. Rev.
,
17
(
3
), pp.
273
283
.10.1103/PhysRev.17.273
5.
Fisher
,
L. R.
, and
Lark
,
P. D.
,
1979
, “
An Experimental Study of the Washburn Equation for Liquid Flow in Very Fine Capillaries
,”
J. Colloid Interface Sci.
,
69
(
3
), pp.
486
492
.10.1016/0021-9797(79)90138-3
6.
Rideal
,
E. K.
,
1922
, “
CVIII. On the Flow of Liquids Under Capillary Pressure
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
44
(
264
), pp.
1152
1159
.10.1080/14786441008634082
7.
Bosanquet
,
C. H.
,
1923
, “
LV. On the Flow of Liquids Into Capillary Tubes
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
45
(
267
), pp.
525
531
.10.1080/14786442308634144
8.
Fries
,
N.
, and
Dreyer
,
M.
,
2008
, “
An Analytic Solution of Capillary Rise Restrained by Gravity
,”
J. Colloid Interface Sci.
,
320
(
1
), pp.
259
263
.10.1016/j.jcis.2008.01.009
9.
Levine
,
S.
,
Reed
,
P.
,
Watson
,
E. J.
, and
Neale
,
G.
,
1976
, “
A Theory of the Rate of Rise of a Liquid in a Capillary
,”
Colloid Interface Sci.
, 3, pp.
403
419
.
10.
Ichikawa
,
N.
,
Hosokawa
,
K.
, and
Maeda
,
R.
,
2004
, “
Interface Motion of Capillary-Driven Flow in Rectangular Microchannel
,”
J. Colloid Interface Sci.
,
280
(
1
), pp.
155
164
.10.1016/j.jcis.2004.07.017
11.
Waghmare
,
P. R.
, and
Mitra
,
S. K.
,
2012
, “
A Comprehensive Theoretical Model of Capillary Transport in Rectangular Microchannels
,”
Microfluid. Nanofluid.
,
12
(
1–4
), pp.
53
63
.10.1007/s10404-011-0848-8
12.
Thammanna Gurumurthy
,
V.
,
Roisman
,
I. V.
,
Tropea
,
C.
, and
Garoff
,
S.
,
2018
, “
Spontaneous Rise in Open Rectangular Channels Under Gravity
,”
J. Colloid Interface Sci.
,
527
, pp.
151
158
.10.1016/j.jcis.2018.05.042
13.
Wu
,
Z.
,
Huang
,
Y.
,
Chen
,
X.
, and
Zhang
,
X.
,
2018
, “
Capillary-Driven Flows Along Curved Interior Corners
,”
Int. J. Multiphase Flow
,
109
, pp.
14
25
.10.1016/j.ijmultiphaseflow.2018.04.004
14.
Dreyer
,
M.
,
Delgado
,
A.
, and
Path
,
H. J.
,
1994
, “
Capillary Rise of Liquid Between Parallel Plates Under Microgravity
,”
J. Colloid Interface Sci.
,
163
(
1
), pp.
158
168
.10.1006/jcis.1994.1092
15.
Wolf
,
F. G.
,
dos Santos
,
L. O. E.
, and
Philippi
,
P. C.
,
2010
, “
Capillary Rise Between Parallel Plates Under Dynamic Conditions
,”
J. Colloid Interface Sci.
,
344
(
1
), pp.
171
179
.10.1016/j.jcis.2009.12.023
16.
Wu
,
P.
,
Zhang
,
H.
,
Nikolov
,
A.
, and
Wasan
,
D.
,
2016
, “
Rise of the Main Meniscus in Rectangular Capillaries: Experiments and Modeling
,”
J. Colloid Interface Sci.
,
461
, pp.
195
202
.10.1016/j.jcis.2015.08.071
17.
Waghmare
,
P. R.
, and
Mitra
,
S. K.
,
2010
, “
On the Derivation of Pressure Field Distribution at the Entrance of a Rectangular Capillary
,”
ASME J. Fluids Eng.
,
132
(
5
), p.
054502
.10.1115/1.4001641
18.
Xiao
,
Y.
,
Yang
,
F.
, and
Pitchumani
,
R.
,
2006
, “
A Generalized Analysis of Capillary Flows in Channels
,”
J. Colloid Interface Sci.
,
298
(
2
), pp.
880
888
.10.1016/j.jcis.2006.01.005
19.
Bullard
,
J. W.
, and
Garboczi
,
E. J.
,
2009
, “
Capillary Rise Between Planar Surfaces
,”
Phys. Rev. E
,
79
(
1
), pp.
4
10
.10.1103/PhysRevE.79.011604
20.
Hirt
,
C. W.
, and
Nichols
,
B. D.
,
1981
, “
Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries
,”
J. Comput. Phys.
, 39(1), pp.
201
225
.
21.
Youngs
,
D. L.
,
1982
, “
Time-Dependent Multi-Material Flow With Large Fluid Distortion
,” Numerical Methods in Fluid Dynamics,
Academic Press
,
Cambridge, MA
, pp. 273–285
.
22.
Gaulke
,
D.
, and
Dreyer
,
M. E.
,
2015
, “
CFD Simulation of Capillary Transport of Liquid Between Parallel Perforated Plates Using Flow3D
,”
Microgravity Sci. Technol.
,
27
(
4
), pp.
261
271
.10.1007/s12217-015-9449-6
23.
Schönfeld
,
F.
, and
Hardt
,
S.
,
2009
, “
Dynamic Contact Angles in CFD Simulations
,”
Comput. Fluids
,
38
(
4
), pp.
757
764
.10.1016/j.compfluid.2008.05.007
24.
Ashish Saha
,
A.
, and
Mitra
,
S. K.
,
2009
, “
Effect of Dynamic Contact Angle in a Volume of Fluid (VOF) Model for a Microfluidic Capillary Flow
,”
J. Colloid Interface Sci.
,
339
(
2
), pp.
461
480
.10.1016/j.jcis.2009.07.071
25.
Saha
,
A. A.
, and
Mitra
,
S. K.
,
2009
, “
Numerical Study of Capillary Flow in Microchannels With Alternate Hydrophilic-Hydrophobic Bottom Wall
,”
ASME J. Fluids Eng.
, 131(6), p.
061202
.10.1115/1.3129130
26.
Gründing
,
D.
,
Smuda
,
M.
,
Antritter
,
T.
,
Fricke
,
M.
,
Rettenmaier
,
D.
,
Kummer
,
F.
,
Stephan
,
P.
,
Marschall
,
H.
, and
Bothe
,
D.
,
2019
, “Capillary Rise—A Computational Benchmark for Wetting Processes,” arXiv preprint
arXiv:1907.05054
.https://arxiv.org/abs/1907.05054
27.
Gründing
,
D.
,
2020
, “
An Enhanced Model for the Capillary Rise Problem
,”
Int. J. Multiphase Flow
,
128
, p.
103210
.10.1016/j.ijmultiphaseflow.2020.103210
28.
Hirt
,
C. W.
,
Amsden
,
A. A.
, and
Cook
,
J. L.
,
1974
, “
An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds
,”
J. Comput. Phys.
, 14(3), pp.
227
253
.
29.
Souli
,
M.
, and
Benson
,
D. J.
,
2013
, Arbitrary Lagrangian-Eulerian and Fluid-Structure Interaction Numerical Simulation,
Wiley
,
Hoboken, NJ
.
30.
Hu
,
H. H.
,
Patankar
,
N. A.
, and
Zhu
,
M. Y.
,
2001
, “
Direct Numerical Simulations of Fluid-Solid Systems Using the Arbitrary Lagrangian-Eulerian Technique
,”
J. Comput. Phys.
,
169
(
2
), pp.
427
462
.10.1006/jcph.2000.6592
31.
Stepanov
,
V. G.
,
1977
, “
Wetting Contact Angles for Some Systems
,”
J. Eng. Phys. Thermophys.
,
32
(
6
), pp.
1000
1003
.
32.
Goswami
,
S.
,
Klaus
,
S.
, and
Benziger
,
J.
,
2008
, “
Wetting and Absorption of Water Drops on Nafion Films
,”
Langmuir
,
24
(
16
), pp.
8627
8633
.10.1021/la800799a
33.
Huh
,
C.
, and
Scriven
,
L. E.
,
1971
, “
Hydrodynamic Model of Steady Movement of a Solid/Liquid/Fluid Contact Line
,”
J. Colloid Interface Sci.
,
35
(
1
), pp.
85
101
.10.1016/0021-9797(71)90188-3
34.
Lee
,
S. L.
, and
Lee
,
H. D.
,
2007
, “
Evolution of Liquid Meniscus Shape in a Capillary Tube
,”
ASME J. Fluids Eng.
,
129
(
8
), pp.
957
965
.10.1115/1.2746898
35.
Patankar
,
S. V.
,
1980
, “
Numerical Heat Transfer and Fluid Flow
,”
Hemisphere, Washington, DC
, p.
210
.
36.
Gaydos
,
J.
,
1998
, “
The Laplace Equation of Capillarity
,”
Stud. Interface Sci.
, 6, pp.
1
59
.
37.
Duarte
,
A. A.
,
Strier
,
D. E.
, and
Zanette
,
D. H.
,
1996
, “
The Rise of a Liquid in a Capillary Tube Revisited: A Hydrodynamical Approach
,”
Am. J. Phys.
,
64
(
4
), pp.
413
418
.10.1119/1.18256
38.
Quéré
,
D.
,
1997
, “
Inertial Capillarity
,”
Europhys. Lett.
,
39
(
5
), pp.
533
538
.10.1209/epl/i1997-00389-2
39.
Quéré
,
D.
,
Raphaël
,
É.
, and
Ollitrault
,
J. Y.
,
1999
, “
Rebounds in a Capillary Tube
,”
Langmuir
,
15
(
10
), pp.
3679
3682
.10.1021/la9801615
40.
Hamraoui
,
A.
, and
Nylander
,
T.
,
2002
, “
Analytical Approach for the Lucas-Washburn Equation
,”
J. Colloid Interface Sci.
,
250
(
2
), pp.
415
421
.10.1006/jcis.2002.8288
41.
Masoodi
,
R.
,
Languri
,
E.
, and
Ostadhossein
,
A.
,
2013
, “
Dynamics of Liquid Rise in a Vertical Capillary Tube
,”
J. Colloid Interface Sci.
,
389
(
1
), pp.
268
272
.10.1016/j.jcis.2012.09.004
You do not currently have access to this content.