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

Power-law fluid flows in the converging-diverging tubes and rectangular channel are prevalent in engineered microfluidic devices, many industrial processes and heat transfer applications. We analyzed optimal flow conditions and network structures for power-law fluids in linear, parabolic, hyperbolic, hyperbolic cosine and sinusoidal converging-diverging dendritic networks of tubes and rectangular channels, and aiming to maximize flow conductance under volume and surface-area constraints. This model shed light on strategies to achieve efficient fluid transport within these complex dendritic networks. Our study focused on steady, incompressible, 2D planar and axisymmetric laminar flow without considering network losses. We found that the flow conductance is highly sensitive to network geometry. The maximum conductance occurs when a specific radius/channel-height ratio $\beta$ is achieved. This value depends on the constraint as well as on the vessel geometry such as tube or rectangular channel. However independent of the kind of the converging-diverging profile along the length of the vessel. We found that the scaling, i.e., $\beta_{\max}^* = \beta_{\min}^* = N^{-1/3}$ for constrained tube volume and $\beta_{\max}^* = \beta_{\min}^*= N^{-(n+1)/(3n+2)}$ for constrained surface area for all converging-diverging tube-networks profile remains the same as found by \citet{garg2024scaling} for the power-law fluid flow in a uniform tube. Here, $\beta_{\max}^*,~ \beta_{\min}^*$ are the radius ratios of daughter-parent pair at the maximum divergent part 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. Further, we found that the optimal flow scaling for the height ratio in the rectangular channel, i.e., $\beta_{\max}^* = \beta_{\min}^* = N^{-1/2} \alpha^{-1/2}$ for constrained tube volume and $\beta_{\max}^* = \beta_{\min}^*= N^{-1/2} \alpha^{-n/(2n+2)}$ for constrained surface area for all converging-diverging channel-networks, respectively, where $\alpha$ is the channel-width ratio between parent and daughter branches. We validated our results with experiments, existing theory for limiting conditions, and extended Hess-Murray's law to encompass shear-thinning and shear-thickening fluids for any branching number $N$.