Stress Distribution and Scaling Laws for Optimized Fluid Flow in Tree-Like Networks with Triangular Cross-Sections

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 \(\mathcal{R}\) and conductance \(E\), considering their dependence on the channel side ratio \(\beta\), length ratio \(\gamma\), 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 \( \beta^* = N^{-1/3} \), leading to \( Q_k \propto a_k^3 \), whereas for surface-area constraints, the scaling follows \( \beta^* = N^{-2/5} \), yielding \( Q_k \propto a_k^{5/2} \), where $a$ is the length of the side of triangular channel. Stress distribution in the conduit exhibits symmetry, with higher stresses near the walls. As generation levels increase, stress magnitudes reduce more significantly under surface-area constraints compared to volume constraints. This study excludes junction flow resistance and secondary flows, which can further refine the model. The findings extend Murray’s Law to equilateral triangular networks, offering insights into optimal flow structures and paving the way for applications in fluidic engineering.