Diffusion of curvature on a sheared semi-infinite film

Abstract

The viscous froth model is used to study the evolution of a long and initially straight soap film which is sheared by moving its endpoint at a constant velocity in a direction perpendicular to the initial film orientation. Film elements are thereby set into motion as a result of the shear, and the film curves. The simple scenario described here enables an analysis of the transport of curvature along the film, which is important in foam rheology, in particular for energy-relaxing ‘topological transformations’. Curvature is shown to be transported diffusively along films, with an effective diffusivity scaling as the ratio of film tension to the viscous froth drag coefficient. Computed (finite-length) film shapes at different times are found to approximate well to the semi-infinite film and are observed to collapse with distances rescaled by the square root of time. The tangent to the film at the endpoint reorients so as to make a very small angle with the line along which the film endpoint is dragged, and this angle decays roughly exponentially in time. The computed results are described in terms of a simple asymptotic solution corresponding to an infinite film that initially contains a right-angled corner.