Darcy's law of yield stress fluids on a treelike network

TL;DR

This paper derives an exact solution for the flow of yield stress fluids in a treelike porous medium, revealing a universal non-linear Darcy law linked to the properties of directed polymers on a Cayley tree.

Contribution

It introduces an exact analytical approach to model yield stress fluid flow in a treelike structure using a mapping to directed polymers, providing universal predictions.

Findings

Confirmed non-linear Darcy law for yield stress fluids.

Derived explicit pressure-dependence via low-energy path density.

Validated predictions with extensive numerical simulations.

Abstract

Understanding the flow of yield stress fluids in porous media is a major challenge. In particular, experiments and extensive numerical simulations report a non-linear Darcy law as a function of the pressure gradient. In this letter, we consider a tree-like porous structure for which the problem of the flow can be resolved exactly thanks to a mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Our results confirm the non-linear behavior of the flow and expresses its full pressure-dependence via the density of low-energy paths of DP restricted to vanishing overlap. These universal predictions are confirmed by extensive numerical simulations.