RESUMO
The instability of a falling liquid film of an aqueous surfactant solution along a vertical slope with surfactant adsorption-desorption at its open surface originating surface stresses (Marangoni effect) is investigated. The diffusion of surfactant to the film surface from the bulk and desorption of surfactant to the gas phase are taken into account. The Navier-Stokes and Fick equations are reduced to a system of simpler hence, analytically and numerically, more tractable nonlinear evolution equations albeit with nine dimensionless parameters. The linear stability analysis yields a dispersion equation that is numerically solved and eigenvalues are obtained for various values of significant dimensionless parameters. A very rich picture of instabilities appears. In addition to the earlier known (Kapitza) hydrodynamic mode there are up to four new (Marangoni-driven) diffusion modes. Two modes travel with the liquid velocity on the film surface and the other two travel on their own downstream and upstream, respectively. One diffusion mode could be identified, in the reference frame moving with the liquid on the film surface, as a monotonic instability mode hence leading to a patterned film surface. All other modes are oscillatory ones. Resonance of modes is also predicted for suitable combinations of the parameters of the problem. The mode observed depends upon the surface stress (in terms of a dimensionless Marangoni number), the particular choice of the adsorption-desorption kinetics, and the surface tension state equation at the open surface of the film.