Stress distribution and scaling laws for optimized fluid flow in tree-like networks with triangular cross-sections
Ashish Garg · Physica Scripta · 2025
Abstract This study presents an analytical framework for modeling Newtonian fluid flow in an equilateral triangular channel network with a self-similar tree-like structure. The network is characterized by varying bifurcation levels N and generation stages m , with optimization based on two primary constraints: volume and surface area limitation. The study assumes fully developed laminar flow, neglecting secondary flow effects and junction losses. The analysis evaluates non-dimensional flow resistance R and conductance E , considering their dependence on the channel side ratio β , length ratio γ , bifurcation number N , and branching levels m . The results indicate that flow conductance decreases with increasing generation levels, with different scaling laws under volume and surface-area constraints. For volume constraints, the optimal width ratio follows β * = N −1/3 , leading to Q k ∝ a k 3 , whereas for surface-area constraints, the scaling follows β * = N −2/5 , yielding Q k ∝ a k 5 / 2 , where a is the length of the side of triangular channel. Optimal scaling laws and stress fields are presented across branching generations, showing how triangular geometries affect transport efficiency and wall stress localization. Shear stress variation in triangular channel networks is governed solely by geometry and pressure gradient. Under a volume constraint, stress distributions remain self-similar across generations, while under a surface area constraint, stress magnitudes increase proportionally to N 1/5 . This work provides the first comprehensive analysis of optimal flow in such geometries, extending the well-known Murray’s Law to triangular conduits. These results inform the design of energy-efficient fluid networks in compact and fabrication-constrained domains such as lab-on-chip devices, cooling channels, and porous scaffolds.