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    Escher, Joachim Hillairet, Matthieu Laurençot, Philippe and Walker, Christoph 2015. Traveling Waves for a Thin Film with Gravity and Insoluble Surfactant. SIAM Journal on Applied Dynamical Systems, Vol. 14, Issue. 4, p. 1991.

    Ghazaryan, Anna Lafortune, Stéphane and Manukian, Vahagn 2015. Stability of front solutions in a model for a surfactant driven flow on an inclined plane. Physica D: Nonlinear Phenomena, Vol. 307, p. 1.

    Swanson, Ellen R. Strickland, Stephen L. Shearer, Michael and Daniels, Karen E. 2015. Surfactant spreading on a thin liquid film: reconciling models and experiments. Journal of Engineering Mathematics, Vol. 94, Issue. 1, p. 63.

    King, John R. and Taranets, Roman M. 2013. Asymmetric travelling waves for the thin film equation. Journal of Mathematical Analysis and Applications, Vol. 404, Issue. 2, p. 399.

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    Mavromoustaki, A. Matar, O.K. and Craster, R.V. 2012. Dynamics of a climbing surfactant-laden film II: Stability. Journal of Colloid and Interface Science, Vol. 371, Issue. 1, p. 121.

    Schwartz, L. W. and Davidson, M. R. 2011. Mathematical modeling and numerical simulation of wave-front flow on a vertical wall with surfactant effects. Journal of Engineering Mathematics, Vol. 70, Issue. 1-3, p. 307.

    Fallest, David W Lichtenberger, Adele M Fox, Christopher J and Daniels, Karen E 2010. Fluorescent visualization of a spreading surfactant. New Journal of Physics, Vol. 12, Issue. 7, p. 073029.

    Craster, R. V. and Matar, O. K. 2009. Dynamics and stability of thin liquid films. Reviews of Modern Physics, Vol. 81, Issue. 3, p. 1131.

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  • European Journal of Applied Mathematics, Volume 18, Issue 6
  • December 2007, pp. 679-708

Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves

  • RACHEL LEVY (a1) (a2), MICHAEL SHEARER (a1) (a3) and THOMAS P. WITELSKI (a1)
  • DOI:
  • Published online: 01 December 2007

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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