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If shear localisation is present in a foam, there exists at least one point y$_{B}$ which lies on the boundary between flowing and stationary regions. In our VF simulations, this corresponds to the point at which the velocity profile intercepts the v = 0 axis (see, for example Fig. 5). At this point, the system can co-exist at the same value of stress in both static and flowing regions, as indicated by the filled dots in Fig. 8. The stress at y$_{B}$ can take on any value between σ$_{l}$ and σ$_{y}$ as the foam is sheared. From eq. 18 we see that this leads to the inequality 

As V is increased, on average we expect the viscous stress c$_{v}$ ˙ ϵ ( y ) a between 0 and y$_{B}$ to cause the stress in the flowing region to lie closer to σ$_{y}$ so that the stress overshoot is less evident. However, at low V , the effect is obvious (see Fig. 10) and may have important effects. 

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The differential equation given by eq. 11 may be solved numerically, yielding velocity profile solutions of the kind we envisage, which are valid if they satisfy the inequality given by eq. 19. As the local strain rate is defined as 

in this case [27], the quantity ˙ ϵ ( y$_{B}$ ) can be directly mea- sured from the calculated velocity profiles, provided they intersect the v = 0 axis at some finite value. 

The results of these calculations can be seen in Fig. 9, where we have solved the model numerically for the values a = 0 . 5, b = c$_{d}$ = c$_{v}$ = Δ = 1. The upper bound l $^{+}$( V ) corresponds to where the shear stress σ ( y$_{B}$ ) = σ$_{l}$ , where the viscous stress c$_{v}$ ˙ ϵ ( y$_{B}$ ) a = 0 and the ana- lytic solution for localisation length given by eq. 16. The lower bound l $^{−}$( V ) corresponds to where the shear stress σ ( y$_{B}$ ) = σ$_{y}$ and where the viscous stress c$_{v}$ ˙ ϵ ( y$_{B}$ ) a = Δ . Thus, for a given V , l $^{−}$( V ) ≤ l ( V ) ≤ l $^{+}$( V ) gives the range of allowed solutions, indicated by the shaded re- gion in Fig. 9. 

We note that Fig. 9 is qualitatively similar to Fig. 7, with a large range of possible localisation lengths at low V and convergent behaviour at high V . Remarkably, the model predicts that for low V , the localisation length can take any value 0 < l < ∞ . This prediction, of course, holds only in the presence of viscous drag. 

An important question to be answered is how does one define the critical velocity V$_{c}$ below which the foam can take on a wide range of localisation lengths? If one 

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