A note on the coating of an inclined plane in the presence of soluble surfactant

We consider the flow of a thin liquid film coating an inclined plane in the presence of a soluble surfactant. A two-dimensional three-equation model is derived using lubrication theory in the rapid diffusion limit and then used to investigate the stability of the fluid height and the surfactant surface and bulk concentrations. We present solutions for an insoluble surfactant system, which are then contrasted with those obtained for a system containing a soluble surfactant; both transient growth and fully nonlinear two-dimensional simulation results are discussed. Our results indicate that the characteristics of the fingering phenomena which accompany the flow are altered by the effects of solubility. In particular, we find that these effects destabilise the system further over an intermediate range of surfactant solubility.

Graphical abstractFingering instability in surfactant-laden thin films flowing down an inclined plane. Panels (a), (d) and (g): film thickness; (b), (e) and (h): interfacial concentration; (c), (f) and (i): bulk concentration.Figure optionsDownload full-size imageDownload as PowerPoint slide