On the role of variable surface viscosity in free viscous films

The stability of a thin liquid film bounded by two free surfaces is examined in the presence of insoluble surface active agents. The surface active agents not only cause gradients in surface tension, but could also render surface viscosity to be significant, and variable, as a function of their concentration. A set of three coupled nonlinear evolution equations are derived for the film height, concentration of surface active agents, and the horizontal liquid velocity which governs the dynamics of the free film. Suitable phenomenological models for variation of surface tension and surface viscosity with the concentration of surface active agents are incorporated in the interfacial stress boundary condition. Linear stability analysis reveals the effect of various non-dimensional parameters, specifically the retarding nature of surface viscosity and Marangoni effects on film rupture. An analysis of the `jamming' limit of surfactant concentration reveals that $\Gamma_0^{(nl)}<3 \mathbb{D}/M$ is a sufficient criteria for instability of the system, where $\Gamma_0^{(nl)}$ is a normalized initial surface particle concentration, $\mathbb{D}$ is the disjoining pressure number and $\textit{M}$ is the Marangoni number. Nonlinear simulations suggest film profiles at rupture are also qualitatively different from those reported in earlier studies, revealing that free films in the jamming limit are remarkably stable and their free surfaces behave like immobile interfaces. Furthermore, self-similar exponents extracted from our nonlinear simulations are used to explain topological differences between zero and constant surface viscosity in the vicinity of rupture.