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

Published online by Cambridge University Press:  01 December 2007

RACHEL LEVY ,
MICHAEL SHEARER  and
THOMAS P. WITELSKI
RACHEL LEVY
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
MICHAEL SHEARER
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA Department of Mathematics and Center for Research in Scientific Computation, N.C. State University Raleigh, NC 27695, USA
THOMAS P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA
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Abstract

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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Type
Papers
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European Journal of Applied Mathematics , Volume 18 , Issue 6 , December 2007 , pp. 679 - 708
Copyright
Copyright © Cambridge University Press 2007

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