Flow of Thin Liquid Films with Surfactant: Analysis, Numerics, and Experiment

When surfactant is deposited on a thin layer of fluid the liquid is instantaneously set into motion. This striking effect is caused by a surface force, the Marangoni force, induced by a surface tension gradient produced by the local presence of surfactant. We investigate the motion of the fluid, and associated spreading of surfactant, in two scenarios: spreading on a horizontal solid substrate, where surface tension is the sole driving force, and on an inclined substrate, where gravity provides an additional force. The governing equations in both cases are derived from the Navier Stokes equations applying the lubrication approximation. The resulting fourth order system of nonlinear PDE consists of two equations: one for the height of the fluid free surface and the other for the distribution of surfactant.;On a horizontal substrate, we introduce a droplet of insoluble surfactant on a film with initially uniform height. Neglecting the physical and smoothing effects of gravity, capillarity, and surface diffusion the development of a numerical method is complicated by the loss of smoothness at the leading edge of surfactant. We address this issue by transforming the spatial variable to a fixed domain and using the jump conditions of the simplified system as boundary conditions. These numerical results are then compared to a known similarity scaling and solution developed by Jensen and Grotberg [34, 35] for the region of the solution near the leading edge of the surfactant. We further this investigation by examining the solution near the center of the droplet. Using a phase plane analysis we determine that a similarity solution does exist for this region of the solution. However, this solution contradicts the behavior observed in the numerical simulations and we turn to an asymptotic analysis to determine the structure of the solution, which does not have self-similarity but agrees with numerical simulations. We compare the spreading behavior of the thin film and surfactant to the results of an innovative experiment in which the location of the surfactant molecules and the deformation of the free surface of the film are recorded simultaneously.;We compare numerical simulations (now including the physical parameters) to experimental results for the location and spreading of the maximum film height and leading edge of surfactant, as well as the shape of the film height and surfactant profiles. We find agreement between the model and experiment for the spreading of both the deformation of the film and the surfactant concentration. While we are able to align the experimental and numerical height profiles, the lack of agreement between experiment and simulations with regard to the surfactant profiles brings into question the model for surfactant distribution.;On an inclined plane, we investigate the stability of a triple-step traveling wave. We use the dispersion relation of the one-dimensional system (without gravity, capillarity or surface diffusion) to determine that the solution is stable to small perturbations. When these physical effects are included we use the Evans function and stability indicator function to determine the analysis is consistent with the solution being stable. In two dimensions we find the inclusion of surfactant merely introduces a smoothing effect which results in the stability determined by properties of the height equation, analogous to the work of Bertozzi and Brenner [6].;Finally, we explore the spreading behavior of a droplet of surfactant placed on an initially uniform film on an inclined plane. The film develops into two waves and we examine the dynamics of this spreading.