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Applied Mathematics Research eXpress (2006) Vol. 2006 : article ID 15487, 21 pages, doi:10.1155/AMRX/2006/15487
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Growing surfactant waves in thin liquid films driven by gravity

Thomas P. Witelski, Michael Shearer and Rachel Levy

Department of Mathematics and Center for Nonlinear and Complex Systems, Duke University Durham, NC 27708-0320, USA E-mail address: witelski{at}math.duke.edu
Department of Mathematics and Center for Research in Scientific Computation N.C. State University, Raleigh, NC 27695, USA E-mail address: shearer{at}ncsu.edu
Department of Mathematics, Duke University Durham, NC 27708-0320, USA; Department of Mathematics, Harvey Mudd College 1250 N. Dartmouth Avenue, Claremont, CA 91711, USA E-mail address: levy{at}hmc.edu

The dynamics of a gravity-driven thin film flow with insoluble surfactant are described in the lubrication approximation by a coupled system of nonlinear PDEs. When the total quantity of surfactant is fixed, a traveling wave solution exists. For the case of constant flux of surfactant from an upstream reservoir, global traveling waves no longer exist as the surfactant accumulates at the leading edge of the thin film profile. The dynamics can be described using matched asymptotic expansions for t -> {infty}. The solution is constructed from quasistatically evolving traveling waves. The rate of growth of the surfactant profile is shown to be $$O\left(\sqrt{t}\right)$$ and is supported by numerical simulations.



References

  1. Bertozzi A. L. The mathematics of moving contact lines in thin liquid films. Notices of the American Mathematical Society (1998) 45(6):689–697.
  2. Bertozzi A. L., Münch A., Shearer M. Undercompressive shocks in thin film flows. Physica D: Nonlinear Phenomena (1999) 134(4):431–464.[CrossRef]
  3. Borgas M. S., Grotberg J. B. Monolayer flow on a thin film. Journal of Fluid Mechanics (1988) 193:151–170.[CrossRef][Web of Science]
  4. Buckmaster J. Viscous sheets advancing over dry beds. Journal of Fluid Mechanics (1977) 81(4):735–756.[CrossRef][Web of Science]
  5. Dussaud A. D., Matar O. K., Troian S. M. Spreading characteristics of an insoluble surfactant film on a thin liquid layer: comparison between theory and experiment. Journal of Fluid Mechanics (2005) 544:23–51.[CrossRef][Web of Science]
  6. Edmonstone B. D., Matar O. K., Craster R. V. Flow of surfactant-laden thin films down an inclined plane. Journal of Engineering Mathematics (2004) 50(2-3):141–156.[CrossRef][Web of Science]
  7. Edmonstone B. D., Matar O. K., Craster R. V. Surfactant-induced fingering phenomena in thin film flow down an inclined plane. Physica D: Nonlinear Phenomena (2005) 209(1–4):62–79.[CrossRef][Web of Science]
  8. Flitton J. C., King J. R. Surface-tension-driven dewetting of Newtonian and power-law fluids. Journal of Engineering Mathematics (2004) 50(2-3):241–266.[CrossRef][Web of Science]
  9. Garcke H., Wieland S. Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM Journal on Mathematical Analysis (2006) 37(6):2025–2048.[CrossRef][Web of Science]
  10. Gilding B. H., Kersner R. Travelling Waves in Nonlinear Diffusion-Convection Reaction. In: Progress in Nonlinear Differential Equations and Their Applications (2004) 60. Basel: Birkhäuser. x+209.
  11. Holmes M. H. Introduction to Perturbation Methods. In: Texts in Applied Mathematics (1995) 20. New York: Springer. ix+337.
  12. Huppert H. Flow and instability of a viscous current down a slope. Nature (1982) 300(5891):427–429.[CrossRef]
  13. Jensen O. E., Grotberg J. B. Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. Journal of Fluid Mechanics (1992) 240:259–288.[CrossRef][Web of Science]
  14. Jensen O. E., Grotberg J. B. The spreading of heat or soluble surfactant along a thin liquid film. Physics of Fluids A (1993) 5(1):58–68.[CrossRef]
  15. Kevorkian J., Cole J. D. Multiple Scale and Singular Perturbation Methods. In: Applied Mathematical Sciences (1996) 114. New York: Springer. viii+632.
  16. Levy R., Shearer M. The motion of a thin film driven by surfactant and gravity. SIAM Journal of Applied Mathematics (2006) 66(5):1588–1609.[CrossRef]
  17. Levy R., Shearer M., Witelski T. P. Traveling waves in thin liquid films driven by gravity and a finite volume of surfactant. in preparation.
  18. Logan J. D. Transport Modeling in Hydrogeochemical Systems. In: Interdisciplinary Applied Mathematics (2001) 15. New York: Springer. xiv+223.
  19. Lu H.-W., Glasner K., Bertozzi A. L., Kim C.-J. A diffuse interface model for electrowetting droplets in a Hele-Shaw cell. submitted to Journal of Fluid Mechanics.
  20. Matar O. K., Troian S. M. Growth of non-modal transient structures during the spreading of surfactant coated films. Physics of Fluids (1998) 10(5):1234–1236.[CrossRef][Web of Science]
  21. Matar O. K., Troian S. M. The development of transient fingering patterns during the spreading of surfactant coated films. Physics of Fluids (1999) 11(11):3232–3246.[CrossRef][Web of Science]
  22. Moriarty J. A., Schwartz L. W., Tuck E. O. Unsteady spreading of thin liquid films with small surface tension. Physics of Fluids A (1991) 3(5):733–742.[CrossRef]
  23. Münch A. Pinch-off transition in Marangoni-driven thin films. Physical Review Letters (2003) 91(1). Article ID 016105.
  24. Münch A., Wagner B., Witelski T. P. Lubrication models with small to large slip lengths. Journal of Engineering Mathematics (2005) 53(3-4):359–383.[CrossRef][Web of Science]
  25. Myers T. G. Thin films with high surface tension. SIAM Review (1998) 40(3):441–462.[CrossRef][Web of Science]
  26. Oron A., Davis S. H., Bankoff S. G. Long-scale evolution of thin liquid films. Reviews of Modern Physics (1997) 69(3):931–980.[CrossRef][Web of Science]
  27. Renardy M. A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Analysis. Theory, Methods & Applications (1996) 27(3):287–296.[CrossRef][Web of Science]
  28. Renardy M. On an equation describing the spreading of surfactants on thin films. Nonlinear Analysis. Theory, Methods & Applications (1996) 26(7):1207–1219.[CrossRef][Web of Science]
  29. Renardy M. A degenerate parabolic-hyperbolic system modeling the spreading of surfactants. SIAM Journal on Mathematical Analysis (1997) 28(5):1048–1063.[CrossRef][Web of Science]
  30. Schwartz L. W., Roy R. V. Theoretical and numerical results for spin coating of viscous liquids. Physics of Fluids (2004) 16(3):569–584.[CrossRef][Web of Science]
  31. Tuck E. O., Schwartz L. W. A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Review (1990) 32(3):453–469.[CrossRef][Web of Science]
  32. Volpert A. I., Volpert V. A., Volpert V. A. Traveling Wave Solutions of Parabolic Systems. In: Translations of Mathematical Monographs (1994) 140. Rhode Island: American Mathematical Society. xii+448.

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