Properties of Hagen-Poiseuille flow in channel networks

We derive the main properties of adaptive Hagen-Poiseuille flows in elastic microchannel networks similar to biological veins found in organisms. We demonstrate that adaptive Hagen-Poiseuille flows effectively simulate key features of \textit{Physarum polycephalum} networks, replicating physiological out-of-equilibrium phenomena such as peristalsis and shuttle streaming, which are associated with the mechanism of nutrient transport in \textit{Physarum}. A new topological steady state has been identified for asynchronous adaptation, supporting out-of-equilibrium laminar fluxes. Adaptive Hagen-Poiseuille flows exhibit saturation effects on the fluxes in contractile veins, as observed in both animal and artificial contractile veins. These results suggest that the non-equilibrium effects observed in \textit{Physarum} have a hydrodynamic origin