Scaling laws for optimal power-law fluid flow within converging–diverging dendritic networks of tubes and rectangular channels

Abstract

Flows in dendritic–fractal networks have garnered extensive research attention, but most studies assume a constant tube or channel cross section. In many applications, the cross section of the tube or channel changes as the flow progresses through it, such as the blood flow through the arterial system, which is a prime example of a deformable or non-uniform tree-like network. Heating, ventilation, and air conditioning ductwork also exemplify a tree-like network with varying cross sections. This research investigates power-law fluid flows in the converging–diverging tubes and rectangular channels, prevalent in engineered microfluidic devices, many industrial processes, and heat transfer applications. Power-law fluid flows through linear, parabolic, hyperbolic, hyperbolic cosine, and sinusoidal converging–diverging dendritic networks of tubes and rectangular channels are studied. The flow is assumed to be steady, incompressible, two-dimensional planar, and axisymmetric laminar flow without considering network losses. A theoretical model has been derived to evaluate the flow conductance under network volume and surface-area constraints. The flow conductance is highly sensitive to network geometry. The effective conductance of all networks increases with increasing daughter-to-parent radius ratio before eventually declining. The maximum conductance occurs when a specific radius or channel-height daughter–parent ratio β* is achieved. This value depends on the constraint and vessel geometry, such as tubes or rectangular channels. The optimal flow conditions for maximum conductance in a constrained tube volume network, βmax*=βmin*=N−1/3, while for a constrained tube's surface-area network, βmax*=βmin*=N−(n+1)/(3n+2). This scaling applies to all converging–diverging tube network profiles. Here, βmax*, βmin* are the radius ratios of the daughter–parent pair at the maximum divergent or minimum convergent part of the vessel. N represents the number of branches splitting at each junction, and n is the power-law index of the fluid. Furthermore, the optimal flow scaling for the height ratio in the rectangular channel, βmax*=βmin*=N−1/2α−1/2 for constrained channel volume and βmax*=βmin*=N−1/2α−n/(2n+2) for constrained surface area for all converging–diverging channel networks, respectively, where α is the channel-width ratio between parent and daughter branches. Additionally, at optimal conditions in both the channels and tube network, pressure drops are equally partitioned across each branching level. The results in this work are validated with experiments and existing theories for limiting conditions. This research expands existing design principles for efficient flow systems, previously in the literature developed for uniform vessels, to encompass non-uniform converging–diverging vessels. Additionally, it provides a valuable framework for studying non-Newtonian flows within complex, non-uniform tree-like networks.