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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.


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