The steepest S curve of spreading and collecting flows: Discovering the invading tree, not assuming it

Spreading and collecting flows are united by the flow design known as the S-curve: when plotted versus time, the size of the domain that is filled or emptied has a history that is shaped as an S. Here, we show that the fastest spreading or collecting (i.e., the steepest S curve) is discovered by allowing the tree architecture to morph freely, toward greater access over time, in accord with the constructal law of design in nature. The angles between the lines of the invading flow architecture can be selected such that the overall flow proceeds the fastest, covering the greatest territory at any moment. The design is a sequence of two distinct phenomena: “invasion” by channels and branches that grow fast, and “consolidation” by slow diffusion perpendicular to the channels. Invasion and consolidation collaborate hand-in-glove to facilitate the spreading or collecting over the available finite area or volume.