Scaling laws for optimal Herschel–Bulkley yield stress fluid flow in self-similar tree-like branching networks

Abstract Fractal networks fluid flows have attracted significant research interest, yet most studies often assume a constant viscosity or power-law fluid flows. This research explores yield stress fluid flows in fractal tube networks, which are commonly found in engineered microfluidic devices and various industrial processes. We analyze optimal flow conditions and structures in tree-like branching networks using Herschel-Bulkley fluid model to understand yield stress materials. We focus on maximizing flow conductance under volume constraints, assuming steady, incompressible, 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 and we derive the theoretical formulations for it. We find that the flow conductance is extremely sensitive to the geometry of the network. The effective conductance initially rises as the daughter-parent radius ratio increases, but eventually, it begins to decline. The peak conductance occurs at a specific radius ratio. 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 τ y . This optimal condition leads to an equal pressure drop as well as equal tube-volume across each branching generation 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.