In this paper, we investigate the flow of discontinuous shear-thickening fluids, which exhibit S-shaped rheology, in confined and disordered geometries such as constrictions in channels or porous media. We demonstrate that such fluids tend to limit the flow rate at a given pressure gradient. This limitation arises due to the emergence of highly viscous structures. In channel geometries with obstacles, these structures form in the most constricted regions. In porous media, they occur similarly in the narrowest pore throats. However, we show that their locations are strongly correlated in the direction transverse to the flow, effectively forming a viscous barrier. We further characterize the emergence of such barriers, whose lengths diverge as the pressure difference approaches a critical value. This behavior exhibits scaling properties reminiscent to those encountered in other critical systems, such as percolation, depinning transitions, clogging, or yield-stress fluids in porous media.

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• Flows in porous media
• Complex fluids
• Non-Newtonian fluids