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 constraint. We assume steady-state, axisymmetric, and laminar flow with non-slip boundary conditions along the network walls. We investigate two different roughness models (1) a self-similar roughness length scale aligned with the branching network pattern, and (2) a uniform roughness length scale and compare the results. We find that in case of the self-similar roughness model, the effective conductance remains the same as that of the smooth channels forming the network. However, in 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. Further, the optimal effective conductance and the critical geometrical scaling parameter, such as diameter ratio, are analyzed as functions of network geometry and fluid rheology with power-law index. Under both volume and surface area constraint, increasing geometrical parameters such as daughter branches and network generations generally reduced optimal effective conductance, especially for shear-thickening fluids, while shear-thinning fluids were less affected. In macroscopic networks where roughness is relatively small, its effect on the optimal effective conductance is negligible; however, in microscopic networks, where roughness approaches the scale of the smallest channels, it leads to pronounced conductance reduction. Further constrained surface area shows significantly lower optimal effective conductance values compared to volume-constrained systems. Furthermore, we find that uniform surface roughness model predicts the scaling laws for optimal flow varying with all geometrical and rheology parameters. Moreover, in the case of self-similar roughness, the scaling laws for optimal flows are the same as those for the smooth network. These are the function of the branch-spitting number N, for volume constraint, and for both N and n, for surface area constraint. For macroscopic networks under the uniform roughness assumption, an approximation for the critical geometrical scaling parameter was derived using linearization with respect to the roughness intensity parameter, and it was found in good agreement with the full model equations.