Adaptive Hagen-Poiseuille flows on graphs

TL;DR

This paper models low Reynolds number flows in networks with adaptive conductivities, deriving equations that lead to steady state tree structures and revealing a phase transition influenced by an order parameter.

Contribution

It introduces a novel class of equations for adaptive Hagen-Poiseuille flows on graphs, linking flow dynamics with network adaptation and phase transitions.

Findings

Explicit adaptation equations derived

Steady state tree geometries identified

A phase transition controlled by an order parameter observed

Abstract

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.