Nonlocal effects in the continuum theory of shear localisation in 2d foams

Goyon et al. [J. Goyon, A. Colin, G. Ovarlez, A. Ajdari and L. Bocquet, Nature 454 (2008) p. 84] have shown that nonlocal effects in the rheology of foams may be accounted for by a modification of the standard (Herschel–Bulkley) model. Here we consider the effects of this modification on the continuum theory of 2d shear localisation. We compute results for various examples, showing that the localisation length is increased, and explore the limiting cases of zero and infinite nonlocality length ξ. Velocity profiles are shown to take an exponential form in the case where ξ is large. As the formulation of the nonlocal continuum model presented in this article is general, it may also be directly applicable to other complex fluids.