Scaling Laws for Optimal Herschel–Bulkley Yield Stress Fluid Flow in Self-Similar Tree-like Branching Networks

This study analyzes optimal flow conditions and structures in tree-like branching networks for yield stress Herschel–Bulkley fluids. We focus on maximizing flow conductance under volume constraints, assuming fully developed laminar flow in circular tubes. We propose a conjecture that if the tube-wall stress, remains the same in the network for all branches, then an optimal solution exists. We find that optimal network geometry depends on the number of branch splits $N$, and independent of the power-law index $n$ and the yield stress $\tau_y$. This optimal condition leads to an equal pressure drop across each branching level. Our results are validated with existing theory and extended to encompass shear-thinning and shear-thickening behaviors for any number of splits $N$ with and without yield stress. Additionally, we derive relationships between geometrical and flow characteristics for parent and daughter tubes, including wall stresses, length ratios. These findings provide valuable design principles for efficient transport systems involving yield stress fluids.