Adaptive Hagen–Poiseuille flows on graphs

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