DETERMINISTIC TREE NETWORKS FOR FLUID-FLOW - GEOMETRY FOR MINIMAL FLOW RESISTANCE BETWEEN A VOLUME AND ONE-POINT

The function of many natural flow systems is to connect by a fluid flo w a finite-size volume and one point. This paper outlines a strategy f or constructing the architecture of the volume-to-point path such that the flow resistance is minimal (constructal theory(1)). The given vol ume 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 o ptimization principle applies at every volume scale. The smallest volu me element contains a fluid saturated porous medium with Darcy flow, w hich is collected by and channeled through a high permeability path (e .g., fissure) to one point on the element boundary. The geometric opti mization is repeated for larger volume elements, which are constructs (assemblies) of optimized smaller volumes. The flow integrated over ea ch new assembly is channeled through a high-permeability path to a poi nt on the side of the assembly. One remarkable feature of the emerging minimal-resistance flow path is that the high-permeability channels o f the various volume elements form a tree network which is completely deterministic. The interstices of the network are filled with low perm eability porous medium. The method is extended to applications where t he high-permeability paths are empty spaces (e.g., parallel-plate chan nels). It is shown that when the total void volume is constrained it c an be distributed optimally among the volume elements to further decre ase the overall flow resistance.