Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increaes at t "SUP alpha" , then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t "SUP 1+ alpha)/3" (t "SUP (1+ alpha)/4" ). A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exist ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For alpha 1/2 (alpha 1), the diffusive lengthscale at the shcok grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds at t "SUP 1/2" . The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. (Authors)