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

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