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/degenerateparabolic, 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.
© 2007 Cambridge University Press
Levy, R, Shearer, M, Witelski, TP. Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves. Eur J App Math. 2007;18(6): 679-708.
First published in the European Journal of Applied Mathematics, vol. 18, no. 6 (December 2007), by Cambridge University Press.