Inertialess multilayer film flow with surfactant: Stability and traveling waves

Multilayer film flow down an inclined plane in the presence of an insoluble surfactant is investigated with particular emphasis on determining flow stability and investigating the possibility of traveling-wave solutions. The investigation is conducted for two or three layers under conditions of Stokes flow and, separately, on the basis of a long-wave assumption. A normal mode linear stability analysis for Stokes flow shows that adding surfactant to one of the film surfaces can destabilize an otherwise stable flow configuration. For the long-wave system, periodic traveling-wave branches are detected and traced, revealing solutions with pulselike solitary waves on each film surface traveling in phase with each other, traveling waves with capillary ridge structures, and solutions with two of the film surfaces almost in contact. Time-periodic traveling-wave solutions are also found. The stability of the traveling waves is determined by solving initial-value problems and by computing eigenvalue spectra. Boundary element simulations for Stokes flow confirm the existence of traveling waves outside the long-wave regime.