Skip to main content
×
×
Home

Inertial rise of a meniscus on a vertical cylinder

  • Doireann O’Kiely (a1), Jonathan P. Whiteley (a2), James M. Oliver (a1) and Dominic Vella (a1)
Abstract

We consider the inertia-dominated rise of a meniscus around a vertical circular cylinder. Previous experiments and scaling analysis suggest that the height of the meniscus, $h_{m}$ , grows with the time following the initiation of rise, $t$ , like $h_{m}\propto t^{1/2}$ . This is in contrast to the rise on a vertical plate, which obeys the classic capillary–inertia scaling $h_{m}\propto t^{2/3}$ . We highlight a subtlety in the scaling analysis that yielded $h_{m}\propto t^{1/2}$ and investigate the consequences of this subtlety. We develop a potential flow model of the dynamic problem, which we solve using the finite element method. Our numerical results agree well with previous experiments but suggest that the correct early time behaviour is, in fact, $h_{m}\propto t^{2/3}$ . Furthermore, we show that at intermediate times the dynamic rise of the meniscus is governed by two parameters: the contact angle and the cylinder radius measured relative to the capillary length scale, $t^{2/3}$ . This result allows us to collapse previous experimental results with different cylinder radii (but similar static contact angles) onto a single master curve.

Copyright
Corresponding author
Email address for correspondence: dominic.vella@maths.ox.ac.uk
References
Hide All
Ablett, R. 1923 An investigation of the angle of contact between paraffin wax and water. Phil. Mag. 46, 244256.
Billingham, J. & King, A. C. 1995 The interaction of a moving fluid fluid interface with a flat plate. J. Fluid Mech. 296, 325357.
Bush, J. W. M. & Hu, D. L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339369.
Clanet, C. & Quéré, D. 2002 Onset of menisci. J. Fluid Mech. 460, 131149.
Duchemin, L., Eggers, J. & Josserand, C. 2003 Inviscid coalescence of drops. J. Fluid Mech. 487, 167178.
Duchemin, L. & Vandenberghe, N. 2014 Impact dynamics for a floating elastic membrane. J. Fluid Mech. 756, 544554.
Eriksson, K., Estep, D., Hansbo, P. & Johnson, C. 1996 Computational Differential Equations. Cambridge University Press.
Finn, R. 1986 Equilibrium Capillary Surfaces. Springer.
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2003 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.
Howison, S. D., Morgan, J. D. & Ockendon, J. R. 1997 A class of codimension-two free boundary problems. SIAM Rev. 39 (2), 221253.
Hu, D. L. & Bush, J. W. M. 2005 Meniscus-climbing insects. Nature 437, 733736.
James, D. F. 1974 The meniscus on the outside of a small circular cylinder. J. Fluid Mech. 63 (4), 657664.
Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Maths 43 (2), 268277.
King, J. R., Ockendon, J. R. & Ockendon, H. 1999 The Laplace–Young equation near a corner. Q. J. Mech. Appl. Maths 52, 7397.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, Course of Theoretical Physics, vol. 6. Butterworth-Heinemann.
Leppinen, D. & Lister, J. R. 2003 Capillary pinch-off in inviscid fluids. Phys. Fluids 15, 568578.
Lo, L. L. 1983 The meniscus on a needle – a lesson in matching. J. Fluid Mech. 132 (1), 6578.
Ponomarenko, A., Quéré, D. & Clanet, C. 2011 A universal law for capillary rise in corners. J. Fluid Mech. 666, 146154.
Quéré, D. 1997 Inertial capillarity. Europhys. Lett. 39, 533538.
Reyssat, M., Courbin, L., Reyssat, E. & Stone, H. A. 2008 Imbibition in geometries with axial variations. J. Fluid Mech. 615, 335344.
Sierou, A. & Lister, J. R. 2004 Self-similar recoil of inviscid drops. Phys. Fluids 16, 13791394.
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269292.
Thompson, A. B. & Billingham, J. 2012 Inviscid coalescence in the presence of a surrounding fluid. IMA J. Appl. Maths 77 (5), 678696.
Vella, D., Lee, D.-G. & Kim, H.-Y. 2006 Sinking of a horizontal cylinder. Langmuir 22, 29722974.
Vella, D. & Li, J. 2010 The impulsive motion of a small cylinder at an interface. Phys. Fluids 22, 052104.
Vella, D. & Metcalfe, P. D. 2007 Surface tension dominated impact. Phys. Fluids 19, 072108.
Washburn, E. W. 1921 The dynamics of capillary flow. Phys. Rev. 17, 273283.
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. 95, 6587.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed