Article

# The Viscous Froth Model: steady states and the high-velocity limit

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences (Impact Factor: 2.19). 06/2009; 465(2108). DOI: 10.1098/rspa.2009.0057

Source: OAI

**ABSTRACT**

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these. Financial support from (D.W.) the European Space Agency (MAP AO-99-108: C14914/02/NL/SH, MAP AO-99-075: C14308/00/NL/SH) (G.M.), the Wales Institute of Mathematical and Computational Sciences and (S.J.C.) EPSRC (EP/D048397/1, EP/D071127/1) is gratefully acknowledged.

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**ABSTRACT:**The viscous froth model is used to study the evolution of a long and initially straight soap film which is sheared by movin its endpoint at a constant velocity in a direction perpendicular to the initial film orientation. Film elements are thereb set into motion as a result of the shear, and the film curves. The simple scenario described here enables an analysis of th transport of curvature along the film, which is important in foam rheology, in particular for energy-relaxing ‘topologica transformations’. Curvature is shown to be transported diffusively along films, with an effective diffusivity scaling as th ratio of film tension to the viscous froth drag coefficient. Computed (finite-length) film shapes at different times are foun 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 endpoin is dragged, and this angle decays roughly exponentially in time. The computed results are described in terms of a simple asymptoti solution corresponding to an infinite film that initially contains a right-angled corner.