Darcy's law of yield stress fluids on a treelike network
arXiv:2208.06048 · doi:10.1103/PhysRevE.108.L023102
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.
arXiv comment5 pages,3 figures + 1 Supplementary material
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