Scaling Laws for Optimised Power-Law Fluid Flow in Self-Similar Tree-like Branching Networks

The power-law fluid flow in tree-like self similar branching networks prevalent throughout the natural world, but also finds numerous applications in technology. We investigate an analysis of optimal power-law fluid flow conditions and the optimal structures within tree-like branching networks, focusing on maximizing flow conductance under the constraint of the network tube’s volume and the surface-area. The study considered fully developed laminar power-law fluid flow regimes without considering any losses in the network system. A key observation was the sensitivity of the dimensionless effective flow conductance to the network's geometrical parameters. We found that the maximum flow conductance occurs when a dimensionless radius ratio $\beta^*$ satisfies the equation $\beta^* = N^{-1/3}$, and $\beta^* = N^{-(n+1)/(3n+2)}$ under constrained tube-volume and surface-area, respectively. Here, $N$ represents the bifurcation number of branches splitting at each junction and $n$ is the fluid power-law index. We further find that this optimal condition occur at the equipartition of pressure drop across each branching level. We validated our results with various experimental results and theory under limiting conditions as well as Hess–Murray's law is justified and extended for the shear-thinning and shear-thickening fluid flows for arbitrary number of branches splitting $N$. Further, in this study we also derive the relationships between geometrical and flow characteristics of the parent and daughter tubes as well as the generalised scaling laws at the optimal conditions for the other important parameters such as tube-wall stresses, length ratios, mean velocities, tube-volume, and surface-area of the tube distributing within the networks. We find that the fluid power-law index $n$ does not influence the constrained tube-volume scaling at the optimal conditions however the scaling laws varies with $n$ under constrained tube's surface-area. These findings offer valuable design principles for the development of efficient transport and flow systems.