Optimal flow and scaling laws for power-law fluids in elliptical cross-sectional self-similar tree-like networks

Abstract

Tree-like self-similar branching networks with power-law fluid flow in elliptical cross-sectional tubes are ubiquitous in nature and engineered systems. This study optimizes flow conductance within these networks under tube volume and tube surface-area constraints for fully developed laminar power-law fluid flow in elliptical cross-sectional tubes. We identify key network parameters influencing flow conductance and find that efficient flow occurs when a specific ratio of the semi-major or semi-minor axis lengths is achieved. This ratio depends on the number of daughter branches splitting at each junction (bifurcation number N) and the fluid’s power-law index n. This study extends Hess–Murray’s law to non-Newtonian fluids (thinning and thickening fluids) with arbitrary branch numbers for elliptical cross-sectional tubes. We find that the maximum flow conductance occurs when a non-dimensional semi-major or semi-minor axis length ratio β∗ satisfies β∗=N−1/3 and β∗=N−(n+1)/(3n+2) under constrained-volume and constrained tube’s surface-area, respectively. We also analyze the spatial variation of shear stress within elliptical tube cross sections across generations. The stress field is found to be independent of rheological parameters and solely governed by pressure gradient and geometry. Under the volume constraint, stress distributions at optimal conditions are identical across generations, while under the surface-area constraint, the stress magnitude at optimal conditions increases with generation level as |τ|k+1/|τ|k∝Nn/(3n+2). These results provide insights into near-wall transport, wall stress anisotropy, and flow resistance. When the semi-major and semi-minor axis is equal, our findings are validated through experiments and theory under the limiting case of circular tube fractal networks. These insights provide important design principles for developing efficient and optimal transport and flow systems inspired by nature’s and engineered intricate networks.