Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T17:01:12.201Z Has data issue: false hasContentIssue false
  • English
  • Français

Compound capillary rise

Published online by Cambridge University Press:  23 August 2012

Mark M. Weislogel
Mark M. Weislogel*
Affiliation:
Department of Mechanical and Materials Engineering, Portland State University, PO Box 751, Portland, OR 97207, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Irregular conduits, complex surfaces, and porous media often manifest more than one geometric wetting condition for spontaneous capillary flows. As a result, different regions of the flow exhibit different rates of flow, all the while sharing common dynamical capillary pressure boundary conditions. The classic problem of sudden capillary rise in tubes with interior corners is revisited from this perspective and solved numerically in the self-similar visco-capillary limit à la Lucas–Washburn. Useful closed-form analytical solutions are obtained in asymptotic limits appropriate for many practical flows in conduits containing one or more interior corner. The critically wetted corners imbibe fluid away from the bulk capillary rise, shortening the viscous column length and slightly increasing the overall flow rate. The extent of the corner flow is small for many closed conduits, but becomes significant for flows along open channels and the method is extended to approximate hemiwicking flows across triangular grooved surfaces. It is shown that an accurate application of the method depends on an accurate a priori assessment of the competing viscous cross-section length scales, and the expedient Laplacian scaling method is applied herein toward this effect.

JFM classification

Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Channel flow Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 709 , 25 October 2012 , pp. 622 - 647
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Concus, P. & Finn, R. 1969 On the behaviour of a capillary free surface in a wedge. Proc. Natl Acad. Sci. 63, 292–299.CrossRefGoogle Scholar
2. Dong, M. & Chatzis, I. 1995 The imbibition and flow of a wetting liquid along the corners of a square capillary tube. J. Colloid Interface Sci. 172 (2), 278–288.CrossRefGoogle Scholar
3. Finn, R. & Neel, R. W. 1999 C-singular solutions of the capillary problem. J. Reine Angew. Math. 512, 1–25.CrossRefGoogle Scholar
4. Ichikawa, N., Hosokawa, K. & Maeda, R. 2004 Interface motion of a capillary-driven flow in rectangular microchannel. Colloid Interface Sci. 280, 155–164.CrossRefGoogle ScholarPubMed
5. Kistler, S. F. 1993 Hydrodynamics of Wetting, Surfactant Science Series , vol. 49, pp. 311–430. Marcel Dekker.Google Scholar
6. de Lazzer, A., Langbein, D., Dreyer, M. & Rath, H. J. 1996 Mean curvature of liquid surfaces in cylindrical containers of arbitrary cross-section. Microgravity Sci. Technol. 9 (3), 208–219.Google Scholar
7. Liu, W., Li, Y., Cai, Y. & Sekulic, D. P. 2011 Capillary rise of liquids over a microstructured solid surface. Langmuir 27 (23), 14260–14266.CrossRefGoogle Scholar
8. Lucas, R. 1918 Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten. Kolloid Z. 23, 15–22.CrossRefGoogle Scholar
9. Ponomarenko, A., Clanet, C. & Quéré, D. 2011 Capillary rise in wedges. J. Fluid Mech. 666, 146–154.CrossRefGoogle Scholar
10. Quéré, D. 1997 Inertia capillarity. Euro. Phys. 533.Google Scholar
11. Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38, 71–99.CrossRefGoogle Scholar
12. Romero, L. A. & Yost, F. G. 1996 Flow in an open channel capillary. J. Fluid Mech. 322, 109–129.CrossRefGoogle Scholar
13. Rye, R. R., Mann, J. A. Jr & Yost, F. G. 1996 The flow of liquids in surface grooves. Langmuir 12, 555–565.CrossRefGoogle Scholar
14. Stange, M., Dreyer, M. E. & Rath, H. J. 2003 Capillary driven flow in circular cylindrical tubes. Phys. Fluids 15.CrossRefGoogle Scholar
15. Washburn, E. W. 1921 The dynamics of capillary flow. Phys. Rev. 3 (2 XVII), 273–283.CrossRefGoogle Scholar
16. Weislogel, M. M. 2001 Capillary flow in containers of polygonal section. AIAA J. 39 (12), 2320–2326.CrossRefGoogle Scholar
17. Weislogel, M. M., Baker, J. A. & Jenson, R. M. 2011 Quasi-steady capillarity driven flow. J. Fluid Mech. 685, 271–305.CrossRefGoogle Scholar
18. Weislogel, M. M., Chen, Y & Bolledulla, D 2008 A better non-dimensionalization scheme for slender laminar flows: the Laplacian operator scaling method. Phys. Fluids 20 (2), 163–170.CrossRefGoogle Scholar
19. Weislogel, M. M. & Lichter, S. 1998 Capillary flow in an interior corner. J. Fluid Mech. 373, 349–378.CrossRefGoogle Scholar
20. Wollman, D. 2012 Capillarity-driven droplet ejection. Master’s thesis, Portland State University.Google Scholar