We study a mesoscopic model for the flow of amorphous solids. The model is based on the key features identified at the microscopic level, namely peri- ods of elastic deformation interspersed with localised rearrangements of parti- cles that induce long-range elastic deformation. These long-range deformations are derived following a continuum mechanics approach, in the presence of solid boundaries, and are included in full in the model. Indeed, they mediate spatial cooperativity in the flow, whereby a localised rearrangement may lead a distant region to yield. In particular, we simulate a channel flow and find manifestations of spatial cooperativity that are consistent with published experimental obser- vations for concentrated emulsions in microchannels. Two categories of effects are distinguished. On the one hand, the coupling of regions subject to different shear rates, for instance,leads to finite shear rate fluctuations in the seemingly un- sheared plug in the centre of the channel. On the other hand, there is convinc- ing experimental evidence of a specific rheology near rough walls. We discuss diverse possible physical origins for this effect, and we suggest that it may be associated with the bumps of particles into surface asperities as they slide along the wall.
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