Binan Gu (WPI): Dynamics On and Of Weighted Metric Graphs: Reactive Flow in Fractures and Waves at Channel Junctions

Fluid, heat, and waves often move through networks of narrow pathways. Metric graphs model these as one-dimensional edges, with PDEs governing the dynamics along each edge and junction conditions coupling them. The framework links local physical laws to the response of the whole network, but it represents physical geometry only through edge lengths, edge weights that may evolve in time, and junction conditions. This raises a central modelling question: how do geometry and junction coupling shape the response of a network, and how can reduced PDE models capture that response? I will address this in two settings: reactive pore networks and branching shallow-water channels.

The first example originates from aquifer thermal energy storage, where thermal and geochemical processes change pore geometry and the conductance of a rock fracture network. A spectral reduction on each edge near chemical equilibrium substantially lowers computational cost, enabling a controlled study of bottleneck length and heat exchange with the surrounding rock. A scaling informed by geometry and heat exchange captures the conductance response mainly for shorter bottlenecks, with larger deviations for longer ones. The second example concerns shallow-water waves in branching channels. A quantum graph with width-weighted Neumann–Kirchhoff junction conditions predicts reflection and transmission from channel widths alone, independent of wavelength and branching angle. Comparing with two-dimensional simulations, we find this holds in the long-wave limit. At shorter wavelengths, the finite-width junction introduces a dependence on angle and wavelength that the graph cannot represent.