# Shear localisation with 2D Viscous Froth and its relation to the Continuum Model 

Joseph D. Barry, Denis Weaire, Stefan Hutzler 

School of Physics, Trinity College Dublin, Ireland 

Received: date / Revised version: date 

Abstract Simulations of monodisperse and polydis- perse ( µ$_{2}$ ( A ) = 0 . 13 ± 0 . 002) 2D foam samples undergo- ing simple shear are performed using the 2D Viscous Froth (VF) Model. These simulations clearly demon- strate shear localisation. The dependence of localisation length on the product λV (shearing velocity V times external wall friction coefficient λ ) is examined and is shown to agree qualitatively with other published ex- perimental data. A wide range of localisation lengths is found at low λV , an effect which is attributed to the existence of distinct yield and limit stresses. The gen- eral Continuum Model is extended to incorporate such an effect and its parameters are subsequently related to those of the VF Model. A Herschel-Bulkley exponent of a = 0 . 3 is shown to accurately describe the observed behaviour. The localisation length is found to be inde- pendent of λV for monodisperse foam samples. 

Paper presented at 5th Annual European Rheology Conference (AERC), April 15-17, 2009, Cardiff, United Kingdom. 

## 1 Introduction 

Foam is defined as a two-phase system in which a dis- persed phase of gas is enclosed by a continuous phase of liquid [29]. In aqueous foams, the dispersed phase is typically air, and the liquid phase water with an added surfactant. To simplify the task of understanding the rheology of these systems, it has become popular to con- centrate on two-dimensional (2D) foams, which consist of a single planar layer of bubbles. It is the rheology of these systems which is under scrutiny in this paper, using the 2D VF Model. 

We endeavour to understand the mechanisms which cause localisation of shear at a moving boundary, as re- ported in a number of recent experiments (see below). 

