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0 . 06 using the definition eq. 3), consistent with experi- mental observation. In these simulations of simple shear, where there is no wall drag, either wall may be regarded as the one that moves. Recent results by Wyn [33] sug- gest that as the second moment of the bubble area dis- tribution µ$_{2}$ ( A ) is increased in such simulations to val- ues approaching 0.2 or higher, shear-banding can occur in regions away from the moving boundary. The width of these shear bands has a square-root dependence on µ$_{2}$ ( A ). This type of behaviour has yet to be observed in experiments. In this paper, we only examine the rheol- ogy of foam samples of disorder µ$_{2}$ ( A ) = 0 . 13 but will probe higher values of disorder in future work in order to search for similar effects. 

With the aid of the Surface Evolver software [4], quasi-static simulations are increasingly easy to imple- ment and, with modern computing, are certainly fast. But are they suitable for rheology? In quasi-statics, the foam is relaxed to equilibrium at each step. There is therefore no relevant time scale present and so no con- cept of shear-rate. It makes no sense to consider Herschel- Bulkley type relations or to discuss the dependence of localisation on boundary velocity. 

What then, are the alternatives to quasi-static sim- ulations? Bubble models [9], where a foam is modeled as a collection of interacting disks appear to represent at some level the dynamics of 2D foams. Langlois et al. [24] report a Herschel-Bulkley exponent of a = 0 . 54 ( µ$_{2}$ ( A ) ≈ 0 . 03). In addition, shear localisation is ob- 

served when wall drag is present. For dry foams though, where low liquid fraction causes bubbles to become more polygonal in shape, this approach is no longer accurate [12]. 

In this paper, we adopt the 2D Viscous Froth (VF) Model [11, 21] as a more realistic model for dry 2D foam dynamics. We have performed an extensive study of shear localisation with the VF Model in a straight ge- ometry which shows realistic dynamics and a rich variety of behaviour, particularly at low λV . 

For a summary of the experimental and theoretical work presented in this section, see [27]. 

## 2 The 2D Viscous Froth Model and its implementation 

The model describes the motion of a soap film in the 2D systems described above, with wall drag [21]. In the present case, bubble areas are kept constant. The foam is considered to be sufficiently dry (liquid fraction less than 0.01) so that a soap film may be accurately described by a curved line and the junctions are represented by points. In the present simulations, a soap film is approximated as a system of connected straight line segments. The motion of a point s joining these segments is given via the equation 

<img src='content_image/416740.jpg'>

