Adaptive Hagen-Poiseuille flows on graphs

We derive a class of equations describing low Reynolds number steady flows of incompressible and viscous fluids in networks made of straight channels, with several sources and sinks, and adaptive conductivities. The flow is controlled by the fluxes at sources and sinks. The network is represented by a graph and the adaptive conductivities describe the transverse channel elasticities, mirroring several network structures found in physics and biology. Minimising the dissipated energy per unit time, we have found an explicit form for the adaptation equations and, asymptotically in time, a steady state tree geometry for the graph connecting sources and sinks is reached. A phase transition tuned by an order parameter for the adapted steady sate graph has been found.