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

Abstract

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.

Scaling Laws for Optimal Power-Law Fluid Flow within Converging-Diverging Dendritic Networks of Tubes and Rectangular Channels

“…Expanding upon Murray's application of the principle of minimum work in circular tubes [25], researchers have investigated flow dynamics in tree-like branching networks. For instance, Revellin et al [26] extended Murray's law to analyze non-Newtonian power-law fluid flow in two channels, revealing a constant diameter ratio (D (k+1) /D k = 2 −1/3 ) for the optimal flow irrespective of the fluid's power-law index n. Advancements by Garg et al [1,27,28] delved into networks with varying branching numbers and power-law indices, examining circular and elliptical cross-sections under volume and surface area constraints for both laminar and turbulent flows. Notably, in the laminar flow regime, Garg et al [1,27] discovered that under volume constraints, the optimal radius/diameter/length relationship remains consistent regardless of the fluid's shear-thinning or shear-thickening behavior, denoted as (D (k+1) /D k ) * = N −1/3 .…”

“…However, under surface area constraints, this relationship becomes significantly influenced by the power-law index, expressed as (D (k+1) /D k ) * = N −(n+1)/(3n+2) . For turbulent flow, Garg et al [28] identified scaling pat- terns for smooth and rough tube networks, respectively described as…”

Rheological Investigations of Water-based and Solvent-based Lacquer Varieties and Their Significance in Leather Coating Characteristics

“…During application, the lacquer flows through channels, tubes, and sometimes into porous materials like wood. These pores can be visualized as a simplified network of tiny channels, resembling a tree structure [13][14][15][16][17]. As the lacquer enters these pores, its flow properties, particularly viscosity and surface tension, significantly impact how it fills the network and interacts with the pore walls.…”

Rheological Study: Sodium Hexametaphosphate's Influence on Emulsification and Stability of Nitrocellulose-Butanol Colloid Lacquers

“…Further when the lacquer is applied to a very porous material, like some types of wood, the lacquer interact with the network of tiny pores within the material. These pores could be viewed as a simplified version of a tree-like network [11][12][13][14][15]. As the lacquer penetrates the pores, its flow properties (viscosity, surface tension) would influence how it fills the network and interacts with the pore walls.…”