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which can be solved using the boundary conditions v (0) = V and v ( L ) = 0. These are equivalent to the boundary conditions imposed in our VF simulations; see eq. 5 (al- beit that here the distance y is measured downward from the shearing boundary). 

Upon inspection, it is clear that that a velocity profile of the following form satisfies eq. 11, and exhibits flow localisation: 

where y$_{0}$ and n may be obtained by inserting eq. 12 into eq. 11 and equating prefactors and exponents [27]. This gives 

and 

Eq. 12 clearly satisfies the first boundary condition, v (0) = V of eq. 11. The second boundary condition is satisfied if we take our sample size L → ∞ [27]. This approach is valid so long as the size of the sample is much greater than the localisation length ( L ≫ l ) which may be defined as 

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an alternative definition to that of the previous section; see eq. 8. Inserting eq. 12 into eq. 15 leads to the follow- ing expression for localisation length as a function of the shearing velocity V , 

Its relation to the definition of localisation length l$_{int}$ (see eq. 8) is 

as given by [27]. For a < b (as is the case for our VF simulations; see Sec. 6), localisation length therefore de- creases as the shearing velocity V is increased . 

The possibility of having a range of localisation lengths at low V (as in Fig. 7) can be accounted for by extending the Continuum Model to incorporate what we will refer to as a stress overshoot. This we will now proceed to do. 

In a recent paper [28], Weaire et al. introduced the idea of distinct yield σ$_{y}$ and limit σ$_{l}$ stresses as a possi- ble mechanism for localisation in the absence of viscous drag. An illustration of the typical stress vs strain pic- ture is shown in Fig. 8. The constitutive stress relation thus becomes 

where σ$_{l}$ denotes the limit stress. When the magnitude of the stress overshoot, Δ = σ$_{y}$ − σ$_{l}$ is set to zero we recover the original Herschel-Bulkley relation (see eq. 1). 

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