Optimal Flow and Scaling Laws for Power-Law Fluids in Elliptical Cross-Section Self-Similar Tree-Like Networks

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's 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 the 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 the 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 $\beta^*$ satisfies $\beta^* = N^{-1/3}$, and $\beta^* = N^{-(n+1)/(3n+2)}$ under constrained-volume and constrained tube's surface-area, respectively. When semi-major and semi-minor axis are equal, our findings are validated through experiments, and theory under limiting case of circular tube. These insights provide important design principles for developing efficient and optimal transport and flow systems inspired by nature's and engineered intricate networks.