Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-17T17:03:51.608Z Has data issue: false hasContentIssue false
  • English
  • Français

On the depinning of a drop of partially wetting liquid on a rotating cylinder

Published online by Cambridge University Press:  26 January 2011

UWE THIELE
UWE THIELE*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK
*
†Email address for correspondence: u.thiele@lboro.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the analogy of the behaviour of films and drops of liquid on a rotating horizontal cylinder on the one hand and substrates with regular one-dimensional wettability patterns on the other hand. On the basis of the similarity between the respective governing long-wave equations, we show that a drop of partially wetting liquid on a rotating cylinder undergoes a depinning transition when the rotation speed is increased. The transition occurs via a sniper bifurcation, as in a recently described scenario for drops depinning on heterogeneous substrates.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Lubrication theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 121 - 136
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Acrivos, A. & Jin, B. 2004 Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions. J. Engng Math. 50, 99–120.CrossRefGoogle Scholar
Ashmore, J., Hosoi, A. E. & Stone, H. A. 2003 The effect of surface tension on rimming flows in a partially filled rotating cylinder. J. Fluid Mech. 479, 65–98.CrossRefGoogle Scholar
Bauer, C. & Dietrich, S. 2000 Phase diagram for morphological transitions of wetting films on chemically structured substrates. Phys. Rev. E 61, 1664–1669.CrossRefGoogle ScholarPubMed
Beltrame, P., Hänggi, P. & Thiele, U. 2009 Depinning of three-dimensional drops from wettability defects. Europhys. Lett. 86, 24006.Google Scholar
Beltrame, P., Knobloch, E., Hänggi, P. & Thiele, U. 2010 Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates. Phys. Rev. E (in press).CrossRefGoogle Scholar
Beltrame, P. & Thiele, U. 2010 Time integration and steady-state continuation method for lubrication equations. SIAM J. Appl. Dyn. Syst. 9, 484–518.CrossRefGoogle Scholar
Benilov, E. S., Benilov, M. S. & Kopteva, N. 2008 Steady rimming flows with surface tension. J. Fluid Mech. 597, 91–118.CrossRefGoogle Scholar
Benilov, E. S. & O'Brien, S. B. G. 2005 Inertial instability of a liquid film inside a rotating horizontal cylinder. Phys. Fluids 17, 052106.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Brandon, S. & Marmur, A. 1996 Simulation of contact angle hysteresis on chemically heterogeneous surfaces. J. Colloid Interface Sci. 183, 351–355.CrossRefGoogle ScholarPubMed
Brusch, L., Kühne, H., Thiele, U. & Bär, M. 2002 Dewetting of thin films on heterogeneous substrates: pinning vs. coarsening. Phys. Rev. E 66, 011602.CrossRefGoogle Scholar
Doedel, E. J., Paffenroth, R. C., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Oldeman, B. E., Sandstede, B. & Wang, X. J. 1997 AUTO2000: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University.Google Scholar
Duffy, B. R. & Wilson, S. K. 2009 Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder. Eur. Phys. J. Spec. Top. 166, 147–150.CrossRefGoogle Scholar
Evans, P. L., Schwartz, L. W. & Roy, R. V. 2004 Steady and unsteady solutions for coating flow on a rotating horizontal cylinder: two-dimensional theoretical and numerical modeling. Phys. Fluids 16, 2742–2756.CrossRefGoogle Scholar
Evans, P. L., Schwartz, L. W. & Roy, R. V. 2005 Three-dimensional solutions for coating flow on a rotating horizontal cylinder: theory and experiment. Phys. Fluids 17, 072102.CrossRefGoogle Scholar
Gau, H., Herminghaus, S., Lenz, P. & Lipowsky, R. 1999 Liquid morphologies on structured surfaces: from microchannels to microchips. Science 283, 46–49.CrossRefGoogle ScholarPubMed
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Hinch, E. J. & Kelmanson, M. A. 2003 On the decay and drift of free-surface perturbations in viscous thin-film flow exterior to a rotating cylinder. Proc. R. Soc. Lond. A 459, 1193–1213.CrossRefGoogle Scholar
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11, 97–106.CrossRefGoogle Scholar
Hunt, R. 2008 Numerical solution of the free-surface viscous flow on a horizontal rotating elliptical cylinder. Numer. Meth. Part Differ. Equ. 24, 1094–1114.Google Scholar
Israelachvili, J. N. 1992 Intermolecular and Surface Forces. Academic Press.Google Scholar
John, K., Hänggi, P. & Thiele, U. 2008 Ratchet-driven fluid transport in bounded two-layer films of immiscible liquids. Soft Matter 4, 1183–1195.Google Scholar
Karabut, E. A. 2007 Two regimes of liquid film flow on a rotating cylinder. J. Appl. Mech. Tech. Phys. 48, 55–64.CrossRefGoogle Scholar
Kargupta, K. & Sharma, A. 2003 Mesopatterning of thin liquid films by templating on chemically patterned complex substrates. Langmuir 19, 5153–5163.CrossRefGoogle Scholar
Kelmanson, M. A. 2009 On inertial effects in the Moffatt–Pukhnachov coating-flow problem. J. Fluid Mech. 633, 327–353.Google Scholar
Konnur, R., Kargupta, K. & Sharma, A. 2000 Instability and morphology of thin liquid films on chemically heterogeneous substrates. Phys. Rev. Lett. 84, 931–934.CrossRefGoogle ScholarPubMed
Lenz, P. & Lipowsky, R. 2000 Stability of droplets and channels on homogeneous and structured surfaces. Eur. Phys. J. E 1, 249–262.CrossRefGoogle Scholar
Merkt, D., Pototsky, A., Bestehorn, M. & Thiele, U. 2005 Long-wave theory of bounded two-layer films with a free liquid–liquid interface: short- and long-time evolution. Phys. Fluids 17, 064104.CrossRefGoogle Scholar
Moffatt, H. K. 1977 Behavior of a viscous film on outer surface of a rotating cylinder. J. Mec. 16, 651–673.Google Scholar
Noakes, C. J., King, J. R. & Riley, D. S. 2005 On three-dimensional stability of a uniform, rigidly rotating film on a rotating cylinder. Q. J. Mech. Appl. Math. 58, 229–256.CrossRefGoogle Scholar
Noakes, C. J., King, J. R. & Riley, D. S. 2006 On the development of rational approximations incorporating inertial effects in coating and rimming flows: a multiple-scales approach. Q. J. Mech. Appl. Math. 59, 163–190.Google Scholar
O'Brien, S. B. G. 2002 Linear stability of rimming flow. Q. Appl. Math. 60, 201–211.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Oron, A. & Rosenau, P. 1992 Formation of patterns induced by thermocapillarity and gravity. J. Phys. II France 2, 131–146.Google Scholar
Pedley, T. J. 1967 The stability of rotating flows with a cylindrical free surface. J. Fluid Mech. 30, 127–147.CrossRefGoogle Scholar
Peterson, R. C., Jimack, P. K. & Kelmanson, M. A. 2001 On the stability of viscous free-surface flow supported by a rotating cylinder. Proc. R. Soc. Lond. A 457, 1427–1445.CrossRefGoogle Scholar
Phillips, O. M. 1960 Centrifugal waves. J. Fluid Mech. 7, 340–352.CrossRefGoogle Scholar
Pismen, L. M. 2001 Nonlocal diffuse interface theory of thin films and the moving contact line. Phys. Rev. E 64, 021603.Google Scholar
Pismen, L. M. & Thiele, U. 2006 Asymptotic theory for a moving droplet driven by a wettability gradient. Phys. Fluids 18, 042104.CrossRefGoogle Scholar
Pukhnachev, V. V. 1977 The liquid film motion on the rotating cylinder surface at the presence of gravity. Prikl. Mekh. Teor. Fiz. 3, 77–88.Google Scholar
Pukhnachov, V. V. 2005 On the equation of a rotating film. Siber. Math. J. 46, 913–924.Google Scholar
Quéré, D., Azzopardi, M. J. & Delattre, L. 1998 Drops at rest on a tilted plane. Langmuir 14, 2213–2216.Google Scholar
Reisfeld, B. & Bankoff, S. G. 1992 Nonisothermal flow of a liquid-film on a horizontal cylinder. J. Fluid Mech. 236, 167–196.CrossRefGoogle Scholar
Rockford, L., Liu, Y., Mansky, P., Russell, T. P., Yoon, M. & Mochrie, S. G. J. 1999 Polymers on nanoperiodic, heterogeneous surfaces. Phys. Rev. Lett. 82, 2602–2605.Google Scholar
Ruschak, K. J. & Scriven, L. E. 1976 Rimming flow of liquid in a rotating horizontal cylinder. J. Fluid Mech. 76, 113–125.CrossRefGoogle Scholar
Schwartz, L. W. & Garoff, S. 1985 Contact-angle hysteresis on heterogeneous surfaces. Langmuir 1, 219–230.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Addison-Wesley.Google Scholar
Thiele, U. 2010 Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth. J. Phys.: Condens. Matter 22, 084019.Google ScholarPubMed
Thiele, U., Brusch, L., Bestehorn, M. & Bär, M. 2003 Modelling thin-film dewetting on structured substrates and templates: bifurcation analysis and numerical simulations. Eur. Phys. J. E 11, 255–271.Google Scholar
Thiele, U. & Knobloch, E. 2004 Thin liquid films on a slightly inclined heated plate. Physica D 190, 213–248.CrossRefGoogle Scholar
Thiele, U. & Knobloch, E. 2006 a Driven drops on heterogeneous substrates: onset of sliding motion. Phys. Rev. Lett. 97, 204501.CrossRefGoogle ScholarPubMed
Thiele, U. & Knobloch, E. 2006 b On the depinning of a driven drop on a heterogeneous substrate. New J. Phys. 8 (313), 1–37.CrossRefGoogle Scholar
Thiele, U., Velarde, M. G., Neuffer, K., Bestehorn, M. & Pomeau, Y. 2001 Sliding drops in the diffuse interface model coupled to hydrodynamics. Phys. Rev. E 64, 061601.CrossRefGoogle ScholarPubMed
Thoroddsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in a partially-filled horizontally rotating cylinder. Exp. Fluids 23, 1–13.CrossRefGoogle Scholar
Weidner, D. E., Schwartz, L. W. & Eres, M. H. 1997 Simulation of coating layer evolution and drop formation on horizontal cylinders. J. Colloid Interface Sci. 187, 243–258.Google Scholar
Yih, C. S. & Kingman, J. F. C. 1960 Instability of a rotating liquid film with a free surface. Proc. R. Soc. Lond. A 258, 63–89.Google Scholar