The stress singularity in surfactant-driven thin-film flows. Part 1. Viscous effects

O. E. Jensen, D. Halpern

    Research output: Contribution to journalArticlepeer-review

    Abstract

    The leading edge of a localized, insoluble surfactant monolayer, advancing under the action of surface-tension gradients over the free surface of a thin, viscous, fluid layer, behaves locally like a rigid plate. Since lubrication theory fails to capture the integrable stress singularity at the monolayer tip, so overestimating the monolayer length, we investigate the quasi-steady two-dimensional Stokes flow near the tip. assuming that surface tension or gravity keeps the free surface locally flat. Wiener-Hopf and matched-eigenfunction methods are used to compute the 'stick-slip' flow when the singularity is present; a boundary-element method is used to explore the nonlinear regularizing effects of weak 'contaminant' surfactant or surface diffusion. In the limit in which gravity strongly suppresses film deformations, a spreading monolayer drives an unsteady return flow (governed by a nonlinear diffusion equation) beneath most of the monolayer, and a series of weak vortices in the fluid ahead of the tip. As contaminant or surface diffusion increase in strength, they smooth the tip singularity over short lengthscales. eliminate the local stress maximum and ultimately destroy the vortices. The theory is readily extended to cases in which the film deforms freely over long lengthscales. Limitations of conventional thin-film approximations are discussed.
    Original languageEnglish
    Pages (from-to)273-300
    Number of pages27
    JournalJournal of Fluid Mechanics
    Volume372
    DOIs
    Publication statusPublished - 10 Oct 1998

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