Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-n7x5d Total loading time: 0.298 Render date: 2021-12-07T13:53:40.572Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Steady and quasi-steady thin viscous flows near the edge of a solid surface

Published online by Cambridge University Press:  07 May 2010

G. I. BARENBLATT
Affiliation:
Department of Mathematics and E.O. Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720-3840, USA
M. BERTSCH
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy, and Istituto per le Applicazioni del Calcolo “M. Picone”, C.N.R., Via dei Taurini 19, 00185 Roma, Italy
L. GIACOMELLI
Affiliation:
Dipartimento Me.Mo.Mat., Università di Roma “La Sapienza”, Via Scarpa 16, 00161 Roma, Italy email: giacomelli@dmmm.uniroma1.it

Abstract

A new approach is proposed for the description of thin viscous flows near the edges of a solid surface. For a steady flow, the lubrication approximation and the no-slip condition are assumed to be valid on most of the surface, except for relatively small neighbourhoods of the edges, where a universality principle is postulated: the behaviour of the liquid in these regions is universally determined by flux, external conditions and material properties. The resulting mathematical model is formulated as an ordinary differential equation involving the height of the liquid film and the flux as unknowns, and analytical results are outlined. The form of the universal functions which describe the behaviour in the edge regions is also discussed, obtaining conditions of compatibility with lubrication theory for small fluxes. Finally, an ordinary differential equation is introduced for the description of intermediate asymptotic profiles of a liquid film which flows off a bounded solid surface.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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]Ansini, L. & Giacomelli, L. (2004) Doubly nonlinear thin-film equations in one space dimension. Arch. Rat. Mech. Anal. 173, 89131.CrossRefGoogle Scholar
[2]Barenblatt, G. I., Beretta, E. & Bertsch, M. (1997) The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation. Proc. Acad. Sci. USA 94, 1002410030.CrossRefGoogle ScholarPubMed
[3]Bernis, F., Hulshof, J. & King, J. R. (2000) Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity 13, 413439.CrossRefGoogle Scholar
[4]Bielarz, C. & Kalliadasis, S. (2003) Time-dependent free-surface thin film flows over topography. Phys. Fluids 15, 25122524.CrossRefGoogle Scholar
[5]Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. (2009) Wetting and spreading. Rev. Mod. Phys. 81, 739805.CrossRefGoogle Scholar
[6]Bowen, M., Hulshof, J. & King, J. R. (2001) Anomalous exponents and dipole solutions for the thin film equation. SIAM J. Appl. Math. 62, 149179.Google Scholar
[7]Bowen, M. & King, J. R. (2001) Asymptotic behaviour of the thin film equation in bounded domains. Eur. J. Appl. Math. 12, 135157.CrossRefGoogle Scholar
[8]Bowen, M. & King, J. K. (2001) Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Math. 12, 321356.Google Scholar
[9]Bowen, M. & Witelski, T. P. (2006) The linear limit of the dipole problem for the thin film equation. SIAM J. Appl. Math. 66, 17271748.CrossRefGoogle Scholar
[10]Buckingham, R., Shearer, M. & Bertozzi, A. (2002) Thin film traveling waves and the Navier slip condition. SIAM J. Appl. Math. 63, 722744.Google Scholar
[11]Craster, R. V. & Matar, O. K. (2009) Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.CrossRefGoogle Scholar
[12]de Gennes, P. G. (1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827863.CrossRefGoogle Scholar
[13]Dussan V., E. B., & Davis, S. H. (1974) On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 7195.CrossRefGoogle Scholar
[14]Giacomelli, L. (2008) Nonlinear higher-order boundary value problems describing thin viscous flows near edges. J. Math. Anal. Appl. 345, 632649.CrossRefGoogle Scholar
[15]Giacomelli, L., Knuepfer, H. & Otto, F. (2008) Smooth zero-contact-angle solutions to the thin-film equation around the steady state. J. Differ. Equ. 245, 14541506.CrossRefGoogle Scholar
[16]Giacomelli, L. & Otto, F. (2003) Rigorous lubrication approximation. Interfaces Free Bound. 5, 483529.CrossRefGoogle Scholar
[17]Greenspan, H. P. (1978) On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125143.CrossRefGoogle Scholar
[18]Grün, G. (2004) Droplet spreading under weak slippage — existence for the Cauchy problem, Comm. Partial Differ. Equ. 29, 16971744.CrossRefGoogle Scholar
[19]Huh, C. & Scriven, L. E. (1971) Hydrodynamic model of a steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.CrossRefGoogle Scholar
[20]Kalliadasis, S., Bielarz, C. & Homsy, G. M. (2000) Steady free-surface thin film flows over topography. Phys. Fluids 12, 18891898.CrossRefGoogle Scholar
[21]King, J. R. (2007) Microscale sensitivity in moving-boundary problems for the thin-film equation. Control methods in PDE-dynamical systems. Contemp. Math. 426, 269292.CrossRefGoogle Scholar
[22]Lagerstrom, P. A. (1988) Matched asymptotic expansions. Ideas and techniques. Appl. Math. Sci., 76. Springer-Verlag, New York.CrossRefGoogle Scholar
[23]Laugesen, R. S. & Pugh, M. C. (2000) Linear stability of steady states for thin film and Cahn-Hilliard type equations. Arch. Ration. Mech. Anal. 154, 351.CrossRefGoogle Scholar
[24]Laugesen, R. S. & Pugh, M. C. (2000) Properties of steady states for thin film equations. Eur. J. Appl. Math. 11, 293351.CrossRefGoogle Scholar
[25]Laugesen, R. S. & Pugh, M. C. (2002) Energy levels of steady states for thin-film-type equations. J. Differ. Equ. 182, 377415.CrossRefGoogle Scholar
[26]Laugesen, R. S. & Pugh, M. C. (2002) Heteroclinic orbits, mobility parameters and stability for thin film type equations. Electron. J. Differ. Equ. 2002 No. 95, 129.Google Scholar
[27]Moosavi, A., Rauscher, M. & Dietrich, S. (2009) Dynamics of nanodroplets on topographically structured substrates. J. Phys.: Condens. Matter 21, 464120.Google ScholarPubMed
[28]Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931980.CrossRefGoogle Scholar
[29]Shikhmurzaev, Y. D. (2008) Capillary Flows With Forming Interfaces, Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[30]Thompson, P. A. & Troian, S. M. (1997) A general boundary condition for liquid flow at solid surfaces. Nature 389, 360362.CrossRefGoogle Scholar
[31]van den Berg, J. B., Bowen, M., King, J. R. & El-Sheikh, M. M. A. (2004) The self-similar solution for draining in the thin film equation. Eur. J. Appl. Math. 15, 329346.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Steady and quasi-steady thin viscous flows near the edge of a solid surface
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Steady and quasi-steady thin viscous flows near the edge of a solid surface
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Steady and quasi-steady thin viscous flows near the edge of a solid surface
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *