Tree Networks for Flows in Composite Porous Media

This article reports a numerical study of the geometric minimization of the resistance to Darcy flow between a finite-size volume and one point. The volume is two dimensional and contains materials with several permeabilities. The optimization starts with the smallest volume subsystem, and proceeds toward larger subsystems (assemblies) until the given volume is covered. It is shown that at every scale the geometric shape of the subsystem can be optimized. This principle allows us to construct the volume-to-point flow path by using assemblies of previously optimized building blocks, hence the "constructal" name for the associated theory of shape and structure formation in natural flow systems. The optimized flow architecture is such that the regions of relatively high permeability form a tree network that is completely deterministic. Every feature of this architecture is the result of a single optimization principle: the geometric minimization of flow resistance subject to constraints.