Active hydraulics laws from frustration principles

Viscous flows are laminar and deterministic. Robust linear laws accurately predict their streamlines in structures 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, and therefore non-linear. As a consequence, their emergent patterns in channel networks are out of reach of available theories, and lack quantitative experiments. Here, we lay out the basic laws of active hydraulics. We show that active hydraulic flows are non-deterministic and yield degenerate streamline patterns ruled by frustration at nodes with an odd coordination number. More precisely, colloidal-roller experiments in trivalent networks reveal how active-hydraulic flows realize dynamical spin ices. The resulting streamline patterns split into two distinct 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 as a double spin model. A series of mappings on loop O(n) models then allow us to exactly predict the geometry of the degenerate streamlines. We expect our fundamental understanding to provide robust design rules for active microfluidic devices, and to offer unanticipated avenues to understand the motion of living cells and organisms in complex habitats.