Optimal Flow and Scaling Laws for Power-Law Fluids in Elliptical Cross-Section 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'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.

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

“…Additionally, maintaining the total geometrical volume of the tube within constraints is necessary to prevent the structure from becoming infinitely large. Moreover, given the expansive and limited nature of space, different parts of the system compete for available space and hence the constraint on the surface-area of the tube becomes crucial [20]. Therefore, it is essential to assess the flow resistance of the network in comparison to that of a single channel with the same length and volume or surface-area.…”

“…Further applying the principle of mass conservation across a branching network and assuming that pressure losses at junctions (due to bends and contractions) are negligible compared to those along the channel lengths, the Murray's law was extended for laminar flow of non-Newtonian power-law fluids for blood in two branching channels [17,18]. These laws were further extended with arbitrary branch splitting N and power-law index n in circular and elliptical cross-sections under volume and surface-area constrained by Garg et al [19], and Garg et al [20], respectively, where they found that the optimal radius/diameter/semi-major axis length relation does not change with shear thinning/thickening power-law behaviour of fluids under volume constraint, however strongly depends on the power-law index n under surface-area constraint. Lee et al [21] introduced a theory for capillary microchannel flows within tree-like branches [21,22].…”

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 .…”

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.…”