Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-20T16:31:09.178Z Has data issue: false hasContentIssue false
  • English
  • Français

The elastocapillary Landau–Levich problem

Published online by Cambridge University Press:  16 October 2013

Harish N. Dixit  and
G. M. Homsy
Harish N. Dixit*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
G. M. Homsy
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
*
†Present address: Department of Mechanical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram, Andhra Pradesh, India - 502205. Email address for correspondence: hdixit@iith.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the classical Landau–Levich dip-coating problem for the case in which the interface possesses both elasticity and surface tension. The aim of the study is to develop a complete asymptotic theory of the elastocapillary Landau–Levich problem in the limit of small flow speeds. As such, the paper also extends our previous study on purely elastic Landau–Levich flow (Dixit & Homsy J. Fluid Mech., vol. 732, 2013, pp. 5–28) to include the effect of surface tension. The elasticity of the interface is described by the Helfrich model and surface tension is modelled in the usual way. We define an elastocapillary number, $\epsilon $, which represents the relative strength of elasticity to surface tension. Based on the size of $\epsilon $, we can define three different regimes of interest. In each of these regimes, we carry out asymptotic expansions in the small capillary (or elasticity) numbers, which represents the balance of viscous forces to surface tension (or elasticity).

In the weak elasticity regime, the film thickness is a small correction to the classical Landau–Levich law and can be written as

$$\begin{eqnarray*}{\tilde {h} }_{\infty , c} = (0. 9458- 0. 0839~\mathscr{E}){l}_{c} C{a}^{2/ 3} , \quad \epsilon \ll 1,\end{eqnarray*}$$
where ${l}_{c} $ is the capillary length, $Ca$ is the capillary number and $\mathscr{E}= \epsilon / C{a}^{2/ 3} $. In the elastocapillary regime, the film thickness is a function of $\epsilon $ through the power-law relationship
$$\begin{eqnarray*}{\tilde {h} }_{\infty , ec} = {\bar {h} }_{\infty , e} L\hspace{0.167em} f(\epsilon )C{a}^{4/ 7} , \quad \epsilon \sim O(1),\end{eqnarray*}$$
 where ${\bar {h} }_{\infty , e} $ is a numerical coefficient obtained in our previous study, $L$ is the elastocapillary length, and $f(\epsilon )$ represents the functional dependence of film thickness on the elastocapillary parameter.

JFM classification

Materials Processing Flows: Coating Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Capillary flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 1 - 28
Copyright
©2013 Cambridge University Press 

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

Bico, J., Roman, B., Moulin, L. & Boudaoud, A. 2004 Elastocapillary coalescence in wet hair. Nature 432, 690.CrossRefGoogle ScholarPubMed
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Das, S., Marchand, A., Andreotti, B. & Snoeijer, J. H. 2011 Elastic deformation due to tangential capillary forces. Phys. Fluids 23, 072006.Google Scholar
Dixit, H. N. & Homsy, G. M. 2013 The elastic Landau–Levich problem. J. Fluid Mech. 732, 5–28.Google Scholar
Duprat, C., Aristoff, J. M. & Stone, H. A. 2011 Dynamics of elastocapillary rise. J. Fluid Mech. 679, 641–654.Google Scholar
Kaoui, B., Ristow, G. H., Cantat, I., Misbah, C. & Zimmermann, W. 2008 Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow. Phys. Rev. E 77, 021903.Google Scholar
Kim, H.-Y. & Mahadevan, L. 2006 Capillary rise between elastic sheets. J. Fluid Mech. 548, 141–150.Google Scholar
Ouriemi, M. & Homsy, G. M. 2013 Experimental study of the effect of surface-adsorbed hydrophobic particles on the Landau–Levich law. Phys. Fluids 25, 082111.CrossRefGoogle Scholar
Park, C.-W. 1991 Effects of insoluble surfactants on dip coating. J. Colloid Interface Sci. 146, 382–394.CrossRefGoogle Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108, 074502.Google Scholar
Planchette, C., Lorenceau, E. & Biance, A.-L. 2012 Surface wave on a particle raft. Soft Matt. 8, 2444–2451.Google Scholar
Py, C., Reverdy, P., Doppler, L., Bico, J., Roman, B. & Baroud, C. N. 2007 Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Phys. Rev. Lett. 98, 156103.Google Scholar
Seifert, U. 1995 The concept of effective tension for fluctuating vesicles. Z. Phys. B 97, 299–309.Google Scholar
Van Dyke, M. D. 1975 Perturbation Methods in Fluid Mechanics. Parabolic Press.Google Scholar
Vella, D., Aussillous, P. & Mahadevan, L. 2004 Elasticity of an interfacial particle raft. Europhys. Lett. 68, 212–218.Google Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16, 209–221.CrossRefGoogle Scholar