Radial Spreading of a Surfactant on a Thin Liquid Film

When a surfactant is placed on a layer of fluid, it reduces surface tension locally, creating a surface stress imbalance that sets the fluid in motion. The lubrication approximation is applied to axisymmetric spreading, yielding a coupled system of nonlinear partial differential equation for the height of the fluid free surface and the distribution of the surfactant. For a simplified system ignoring the effects of gravity and capillarity, as well as diffusion of surfactant molecules, the location of the surfactant can be tracked numerically. The free surface height converges quickly to a similarity form [Jensen, “Self-similar, surfactant-driven flows.” Physics of Fluids 6 (1994): 1084–94] away from the origin. Near the origin, a self-similar solution is identified, but it differs qualitatively from long-time numerical solutions. Including nonself-similar terms in an expansion around the origin corrects this inconsistency.