Laws of non-symmetric optimal flow structures, from the macro to the micro scale

Abstract

Many natural systems and engineering processes occur in which a fluid invades a territory from one entry point (invasion), or conversely is expelled from the territory through an outlet (drainage). In any such situation an evolutionary flow structure develops that bridges the gap between the micro-scale (diffusion dominant) and the macroscale (convection dominant). The respiratory and circulatory systems of animals are clear examples of complex flow trees in which both the invasion and drainage processes occur. These flow trees display successive bifurcations (almost always non-symmetric) which allow them to cover and serve the entire territory to be bathed. Although they are complex, it is possible to understand its internal structuring in the light of Constructal Law. A scaling law for optimal diameters of symmetric bifurcations was proposed by Murray (1926), while Bejan and coworkers (2000-2006) added a new scaling law for channel lengths, and based scaling laws of tree shaped structures on theoretical grounds. In this work we use the Constructal Law to study the internal structure and scaling laws of nonsymmetric flow structures, and show how the results might help understand some flow patterns found in Nature. We show that the global flow resistances depend on the parameter ξ=D2/D1=L2/L1 defining the degree of asymmetry between branches 1 and 2 in a bifurcation. We also present a more accurate and general form, of Murray’s law, as a result of the application of the Constructal law to branching flow structures. We end with a brief analysis of the use of these results in the analysis of flow structures of the human respiratory and circulatory systems.