A thin layer of fluid flowing down a solid planar surface has a free surface height described by a nonlinear PDE derived via the lubrication approximation from the Navier Stokes equations. For thin films, surface tension plays an important role both in providing a significant driving force and in smoothing the free surface. Surfactant molecules on the free surface tend to reduce surface tension, setting up gradients that modify the shape of the free surface. In earlier work [12, 13J a traveling wave was found in which the free surface undergoes three sharp transitions, or internal layers, and the surfactant is distributed over a bounded region. This triple-step traveling wave satisfies a system of PDE, a hyperbolic conservation law for the free surface height, and a degenerate parabolic equation describing the surfactant distribution. As such, the traveling wave is overcornpressive. An examination of the linearized equations indicates the direction and growth rates of one-dimensional waves generated by small perturbations in various parts of the wave. Numerical simulations of the nonlinear equations offer further evidence of stability to one-dimensional perturbations.
©2009 American Mathematical Society
Peterson, E, Shearer, M, Witelski, TP, Levy, R. Stability of traveling waves in thin liquid films driven by gravity and surfactant. In: Tadmor E, Liu JG, Tzavaras A, editors. Hyperbolic Problems: Theory, Numerics and Applications. Proceedings of Symposia in Applied Mathematics. Vol. 67. Pt. 2. Providence: American Mathematical Society, 2009.