Deterministic Tree Networks for Fluid Flow: Geometry for Minimal Flow Resistance Between a Volume and One Point

Abstract

The function of many natural flow systems is to connect by a fluid flow a finite-size volume and one point. This paper outlines a strategy for constructing the architecture of the volume-to-point path such that the flow resistance is minimal (constructal theory1). The given volume is viewed as an assembly of volume elements of various sizes. The main discovery is that the shape of each element can be optimized such that the elemental volume-to-point flow resistance is minimal. This optimization principle applies at every volume scale. The smallest volume element contains a fluid saturated porous medium with Darcy flow, which is collected by and channeled through a high permeability path (e.g., fissure) to one point on the element boundary. The geometric optimization is repeated for larger volume elements, which are constructs (assemblies) of optimized smaller volumes. The flow integrated over each new assembly is channeled through a high-permeability path to a point on the side of the assembly. One remarkable feature of the emerging minimal-resistance flow path is that the high-permeability channels of the various volume elements form a tree network which is completely deterministic. The interstices of the network are filled with low permeability porous medium. The method is extended to applications where the high-permeability paths are empty spaces (e.g., parallel-plate channels). It is shown that when the total void volume is constrained it can be distributed optimally among the volume elements to further decrease the overall flow resistance.