The primary instability of falling films in the presence of soluble surfactants

We investigate the linear stability of a film flowing down a solid substrate in the presence of soluble surfactants. The Navier-Stokes equations for the liquid motion are considered, together with advection-diffusion equations for the concentrations of the species involved, which include monomers dissolved in the bulk and adsorbed at the liquid-air and at the liquid-substrate interfaces. The adsorption-desorption kinetics of the surfactant at both interfaces is explicitly accounted for. An Orr-Sommerfeld eigenvalue problem is formulated, and solved analytically in the limit of long-wave disturbances and numerically for arbitrary wavelength using a finite element method. An extensive parametric study is performed to reveal the role of surfactant solubility and adsorption-desorption kinetics. The results quantify the stabilizing effect of soluble surfactants due to the presence of Marangoni stresses, and indicate that moderately soluble surfactants may be more effective than insoluble ones. Disturbances of finite wavelength are stabilized by more than an order of magnitude, and their detailed behaviour depends in a non-monotonic way on the amount of surfactant and on its solubility and kinetics. The above predictions provide insights for the interpretation of recent experimental findings on the primary instability and on the ensuing unstable dynamics of liquid films doped with soluble surfactants