Constructal dendritic geometry and the existence of asymmetric bifurcation

This paper considers the fundamental question of whether the optimal geometry of dendritic tree networks is symmetric or asymmetric. Asymmetry of dendritic networks is a result of the relation between flow resistance and flow fraction at each bifurcation node. Asymmetry of bifurcation (asymmetry of Y-shaped assemblies) appears when the flow fraction at each bifurcation node is not equal to one-half. Asymmetric bifurcation provides lower flow resistance than symmetric bifurcation. Murray’s law of bifurcation (Di+1∕Di=2−1∕3) is valid only when the flow fraction at every bifurcation node in dendritic networks is one-half and the networks are symmetric. General rules to construct asymmetric trees are developed and reported in this paper. It is shown that even through pressure drops across round and square cross-sectional shape channels are different; their flow resistances can be expressed by similar relations.