Active hydraulics laws from frustration principles

Viscous flows are laminar and deterministic. Robust linear laws accurately predict their streamlines in geometries as complex as blood vessels, porous media and pipe networks. However, biological and synthetic active fluids defy these fundamental laws. Irrespective of their microscopic origin, confined active flows are intrinsically bistable, making it challenging to predict flows in active fluid networks. Although early theories attempted to tackle this problem, quantitative experiments to validate their relevance to active hydraulics are lacking. Here, we present a series of laws that accurately predict the geometry of active flows in trivalent networks. Experiments with colloidal rollers reveal that active hydraulic flows realize dynamical spin ices: they are frustrated, non-deterministic and yield degenerate streamline patterns. These patterns split into two geometric classes of self-similar loops, which reflect the fractionalization of topological defects at the subchannel scales. Informed by our measurements, we formulate the laws of active hydraulics in trivalent networks as a double spin model. We then use these laws to predict the random geometry of the degenerate streamlines. We expect our fundamental understanding to provide robust design rules for active microfluidic devices and to offer avenues to investigate the motion of living cells and organisms in complex habitats.