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Effect of disjoining pressure in a thin film equation with non-uniform forcing

Published online by Cambridge University Press:  02 August 2013

D. E. MOULTON  and
J. LEGA
D. E. MOULTON
Affiliation:
OCCAM, Mathematical Institute, University of Oxford, UK email: moulton@maths.ox.ac.uk Department of Mathematics, University of Arizona, Tucson, AZ, USA email: lega@math.arizona.edu
J. LEGA
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ, USA email: lega@math.arizona.edu
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Abstract

We explore the effect of disjoining pressure on a thin film equation in the presence of a non-uniform body force, motivated by a model describing the reverse draining of a magnetic film. To this end, we use a combination of numerical investigations and analytical considerations. The disjoining pressure has a regularizing influence on the evolution of the system and appears to select a single steady-state solution for fixed height boundary conditions; this is in contrast with the existence of a continuum of locally attracting solutions that exist in the absence of disjoining pressure for the same boundary conditions. We numerically implement matched asymptotic expansions to construct equilibrium solutions and also investigate how they behave as the disjoining pressure is sent to zero. Finally, we consider the effect of the competition between forcing and disjoining pressure on the coarsening dynamics of the thin film for fixed contact angle boundary conditions.

Keywords

evolution equationlubrication approximationmagnetic filmmatched asymptotic expansioncoarsening
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 887 - 920
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Diff. Equ. 83, 179–206.Google Scholar
[2]Bertozzi, A. L., Grün, G. & Witelski, T. P. (2001) Dewetting films: Bifurcations and concentrations. Nonlinearity 14, 1569–1592.Google Scholar
[3]Braun, R. J., Snow, S. A. & Naire, S. (2002) Models for gravitationally-driven free-film drainage. J. Eng. Math. 43, 281–314.Google Scholar
[4]Braun, R. J., Snow, S. A. & Pernisz, U. C. (1999) Gravitational drainage of a tangentially-immobile thick film. J. Colloid Interface Sci. 219, 225–240.Google Scholar
[5]Broyden, C. G.et al. (1965) A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19 (92), 577–593.Google Scholar
[6]Burelbach, J. P., Bankoff, S. G. & Davis, S. H. (1988) Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195, 463–494.Google Scholar
[7]Craster, R. V. & Matar, O. K. (2009) Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.Google Scholar
[8]de Gennes, P. G. (1985) Wetting: Statics and dynamics. Rev. Mod. Phys. 57, 827–880.Google Scholar
[9]Edwards, D. A., Brenner, H. & Wasan, D. T. (1991) Interfacial Transport Processes and Rheology, Butterworth-Heinemann, New York.Google Scholar
[10]Elias, F., Bacri, J.-C., Flament, C., Janiaud, E., Talbot, D., Drenckhan, W., Hutzler, S. & Weaire, D. (2005) Magnetic soap films and magnetic soap foams. Colloids Surf A 263, 65–75.Google Scholar
[11]Glasner, K. B. & Witelski, T. P. (2005) Collision versus collapse of droplets in coarsening of dewetting thin films. Phys. D 209, 80–104.Google Scholar
[12]Grün, G. & Rumpf, M. (2001) Simulation of singularities and instabilities arising in thin film flow. Eur. J. Appl. Math. 12, 293–320.Google Scholar
[13]Haskett, R. P., Witelski, T. P. & Sur, J. (2005) Localized Marangoni forcing in driven thin films. Phys. D 209, 117–134.Google Scholar
[14]Israelachvili, J. N. (1992) Intermolecular and Surface Forces, 2nd ed., Academic Press, New York.Google Scholar
[15]Jones, T. B. (1995) Electromechanics of Particles, Cambridge University Press, New York.Google Scholar
[16]Kalliadasis, S., Bielarz, C. & Homsy, G. M. (2000) Steady free-surface thin film flows over topography. Phys. Fluids 12 (8), 1889–1898.Google Scholar
[17]Kalliadasys, S., Kiyashko, A. & Demekhin, E. A. (2003) Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475, 377–408.Google Scholar
[18]Kargupta, K., Konnur, R. & Sharma, A. (2000) Instability and pattern formation in thin liquid films on chemically heterogeneous substrates. Langmuir 16, 10243–10253.Google Scholar
[19]Mitlin, V. S. (1993) Dewetting of solid surface: Analogy with spinodal decomposition. J. Colloid Interface Sci. 156, 491–497.Google Scholar
[20]Moulton, D. E. & Lega, J. (2009) Reverse draining of a magnetic soap film - analysis and simulation of a thin film equation with non-uniform forcing. Phys. D 238, 2153–2165.Google Scholar
[21]Moulton, D. E. & Pelesko, J. A. (2010) Reverse draining of a magnetic soap film. Phys. Rev. E 81, 046320.Google Scholar
[22]Oron, A. & Bankoff, S. G. (2001) Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys. Fluids 13, 1107–1117.Google Scholar
[23]Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.Google Scholar
[24]Oron, A. & Rosenau, P. (1992) Formation of patterns induced by thermocapillarity and gravity. J. Phys. II France 2, 131–146.Google Scholar
[25]Pismen, L. M. & Thiele, U. (2006) Asymptotic theory for a moving droplet driven by a wettability gradient. Phys. Fluids 18, 042104.Google Scholar
[26]Savva, N. & Kalliadasis, S. (2009) Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 092102.Google Scholar
[27]Scheid, B., Oron, A., Colinet, P., Thiele, U. & Legros, J. C. (2002) Nonlinear evolution of nonuniformly heated falling liquid films. Phys. Fluids 14, 4130–4151.Google Scholar
[28]Schwartz, L. W. & Eley, R. R. (1998) Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202, 173–188.Google Scholar
[29]Schwartz, L. W. & Roy, R. V. (1999) Modeling draining flow in mobile and immobile films. J. Colloid Interface Sci. 218, 309–323.Google Scholar
[30]Skotheim, J. M., Thiele, U. & Scheid, B. (2003) On the instability of a falling film due to localized heating. J. Fluid Mech. 475, 1–19.Google Scholar
[31]Teletzke, G. F., Davis, H. T. & Scriven, L. E. (1987) How liquids spread on solids. Chem. Eng. Comm. 155, 41–81.Google Scholar
[32]Thiele, U. (2011) On the depinning of a drop of partially wetting liquid on a rotating cylinder. J. Fluid Mech. 671, 121–136.Google Scholar
[33]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
[34]Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. (2009) Viscous electrified film flow over step topography. SIAM J. Appl. Math. 70, 845–865.Google Scholar
[35]Tseluiko, D. & Papageorgiou, D. T. (2006) Wave evolution on electrified falling films. J. Fluid Mech. 556, 361–386.Google Scholar
[36]Witelski, T. P. & Bernoff, A. J. (1999) Stability of self-similar solutions for van der waals driven thin film rupture. Phys. Fluids 11, 2443–2445.Google Scholar
[37]Zhang, Y. (2009) Counting the stationary states and the convergence to equilibrium for the 1-d thin film equation. Nonlinear Anal. 71, 1425–1437.Google Scholar