Optimal power-law fluid flow in tree-like branching networks with self-similar and uniform roughness models

This study presents an analytical model for the flow of a power-law non-Newtonian fluid through a roughened tree-like branching network under volume and surface area constraints. We assume steady-state, axisymmetric, and laminar flow with non-slip boundary conditions along the network walls. We investigate and compare two different roughness models. In the first model, the roughness length scale is self-similar and aligns with the branching network pattern, while in the second model, the surface roughness length scale is uniform. We find that in the case of the self-similar roughness model, the effective conductance remains the same as that of the smooth network. However, in the case of the uniform roughness model, the effective conductance presents an overall decrease. We argue that the uniform roughness model is a more realistic one. Furthermore, the optimal effective conductance, Eopt, and the critical diameter ratio βc, are analyzed as functions of network geometry and fluid rheology. Under both volume and surface area constraints, increasing geometrical parameters such as the number of daughter branches and network generations, generally reduced Eopt, especially for shear-thickening fluids, while shear-thinning fluids were less affected. In macroscopic networks, where roughness is relatively small, the effect of roughness on Eopt is negligible; however, in microscopic networks, where roughness approaches the scale of the diameters of the smallest branches, it leads to pronounced conductance reduction. Furthermore, networks under surface area constraint show significantly lower Eopt values compared to volume-constrained systems. Moreover, we find that the uniform surface roughness model predicts scaling laws for optimal flow (at βc) that vary with all geometrical and rheological parameters. Finally, for macroscopic networks under the uniform roughness assumption, an approximation for βc was derived using linearization with respect to the roughness intensity parameter, and it was found to be in good agreement with the full model equations.