Non-locality and viscous drag effects on the shear localisation in soft-glassy materials

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 σ_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., 2008). The effects of cooperativity are parametrized with a steady-state diffusion-relaxation equation for the fluidity field f=γ ˙/σ, defined as the ratio between shear rate γ ˙ and shear stress σ. For particular rheological flow-curves (Bingham fluids), the problem is tackled analytically: we explore the two regimes σ≫σ_Y and σ≈σ_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., 2011) are also discussed.