Fluid flow in a porous tree-shaped network: optimal design and extension of Hess–Murray’s law

Abstract

This paper aims to contribute to the ongoing research on tree-shaped flow structures. Here, we focuses on porous-walled tree-shaped networks, namely the laminar fluid flow. Analytical models are developed for pressure distribution along the porous tree-network and for the hydraulic resistance of the network in terms of geometry of successive vessel segments, number of branches, branching levels and intrinsic permeability of walls. We also rely on constructal design to find important insights regarding the allometric relationships between the sizes of successive vessel segments of tree networks. Among other results, we show that the flow distribution depends on the aspect ratio of the branching vessels as well as on the wall permeability of vessels. Maximum physical efficiency to connect successive vessel segments is homothetic with a size ratio of 2^−1/3 (Hess–Murray law) only for impermeable tree-networks. Our results indicate that for porous vessels, this homothetic ratio increases with the intrinsic permeability of the vessel wall. This result may help to understand the occurrence of different optimal relationships between the vessel diameters such as in the branching hierarchy of the conductive and respiratory zones of the lungs.