  We study the Couette flow of a quasi-2d soft-glassy material in a Hele-Shaw
geometry. The material is chosen to be above the jamming point, where a yield
stress $\sigma_Y$ emerges, below which the material deforms elastically and
above which it flows like a complex fluid according to a Herschel-Bulkley (HB)
rheology. Simultaneously, the effect of the confining plates is modelled as an
effective linear friction law, while the walls aside the Hele-Shaw cell are
sufficiently close to each other to allow visible cooperativity effects in the
velocity profiles (Goyon et al., Nature 454, 84-87 (2008)). The effects of
cooperativity are parametrized with a steady-state diffusion-relaxation
equation for the fluidity field $f = \dot{\gamma}/\sigma$, defined as the ratio
between shear rate $\dot{\gamma}$ and shear stress $\sigma$. For particular
rheological flow-curves (Bingham fluids), the problem is tackled analytically:
we explore the two regimes $\sigma \gg \sigma_Y$ and $\sigma \approx \sigma_Y$
and quantify the effect of the extra localisation induced by the wall friction.
Other rheo-thinning fluids are explored with the help of numerical simulations
based on lattice Boltzmann models, revealing a robustness of the analytical
findings. Synergies and comparisons with other existing works in the literature
(Barry et al., Phil. Mag. Lett. 91, 432-440 (2011)) are also discussed.

Soft Condensed Matter