Dynamics of a falling film with solutal Marangoni effect

We investigate the dynamics of a thin liquid film on an inclined planar substrate in the presence of an insoluble surfactant on its free surface. We consider both the linear and nonlinear regimes. The linear regime is examined through the Orr-Sommerfeld eigenvalue problem of the full Navier-Stokes and concentration equations and wall and free-surface boundary conditions. The nonlinear regime is investigated through two different models. The first one is obtained from the classical long-wave expansion and the second one through an integral-boundary-layer approximation combined with a simple Galerkin projection. Although accurate close to the instability threshold, the first model fails to describe the dynamics of the system far from criticality. On the other hand, the second model not only captures accurately the behavior close to the instability threshold, but is also valid far from criticality. Analytical and numerical results on the role of the surfactant on the free-surface dynamics are presented. In the linear regime, the Marangoni stresses induced by the surfactant reduce the domain of instability for the base flow. In the nonlinear regime, the system evolves into solitary pulses for both the free surface and surfactant concentration. The amplitude and velocity of these pulses decrease as the Marangoni effect becomes stronger.