Effect of surfactants on inverted thin film flow

We investigate the stability of a thin film containing insoluble surfactant that is flowing along the underside of an inclined plane, sloped at an angle $\theta$, taking the small $\theta$ limit so that the incline is near horizontal. Two cases are examined; as well as a single layered film, a two-layer film is considered whereby each liquid layer contains surfactant. The waveless solution in both cases is perturbed and a linear stability analysis conducted, with a discussion on the corresponding growth rates of the perturbations. This work is then compared to an experimental investigation, whereby the underside of a slide-fed coating die is considered as the inclined plane, and a series of liquid threads are formed with a uniform spacing. This thread spacing is compared to the most unstable mode of the preceding stability analysis, shown to improve upon the classical Rayleigh-Taylor wavelength which considers a constant surface tension.