Scaling Laws for Optimal Turbulent Flow in Tree-Like Networks with Smooth and Rough Tubes and Power-Law Fluids

In this study, we develop a comprehensive analytical framework to derive the optimal scaling laws for turbulent flows within tree-like self-similar branching networks, integrating a non-Newtonian power-law fluid model with index $n$. Our analysis encompasses turbulent flows occurring in both smooth and rough tubes under constraints of network's tube-volume and tube surface area. We introduce the non-dimensional conductance parameter $E$ to quantify flow conditions, investigating its variations with diameter ratio $\beta$, length ratio $\gamma$, branch splitting $N$, and branching generation levels $m$. Our findings reveal a decrease in $E$ with increasing $\gamma$, $N$, and $m$, highlighting the influence of these parameters on flow conductance. Under volume constraint, we identify optimal flow conditions for both smooth and rough tube networks, characterized by distinct scaling laws as $ D_{k+1}/D_{k} = \beta^* = N^{-(10n+1)/(24n+3)} $, and $D_{k+1}/D_{k} = N^{-3/7} $ (or flow rate proportional to $D_k^{(24n+3)/(10n+1)}$ and $D_k^{7/3}$ ), respectively, where $D_k$ is tube-diameter and $\dot{m}_k$ is the mass flow-rate in a branch at the $k_{th}$ level . Notably, the scaling in the rough tube network remains independent of the power-law index $n$, unlike the smooth tube network where it depends on $n$. Similarly, under surface-area constraint, we observe distinct optimal flow conditions for smooth and rough tube networks as with different scaling laws as $D_{k+1}/D_{k} = \beta^* = N^{-(10n+1)/(21n+2)} $, and $D_{k+1}/D_{k} = N^{-1/2} $ (or flow rate proportional to $D_k^{(21n+2)/(10n+1)}$ and $D_k^{2}$ ), respectively, again smooth tube network showing dependency on the power-law index $n$. Moreover, we uncover a trend where the scaling exponent slope decreases with increasing $n$ in volume constraint networks, while the opposite holds true for surface-area constraint networks. In conclusion, our research significantly extends the applicability of Murray's Law, offering valuable insights into the design and optimization of branching networks under various constraints and fluid properties. By incorporating non-Newtonian fluid behavior and considering tube-wall characteristics, our findings contribute to enhancing the efficiency and performance of diverse engineering systems involving fluid flow.