The Constructal Law of Structure Formation in Natural Systems With Internal Flows

This paper outlines a series of recent developments that provide a theoretical basis for the formation of shape and structure in nonequilibrium (flow) systems subjected to overall constraints. It is shown that geometrical form can be deduced from a single principle: the geometric minimization of resistance to flow. This is illustrated in two ways: by minimizing the flow resistance between a finite volume and one point, and by minimizing the time of travel between a finite area and one point. The discovery is that any volume element can have its shape optimized such that its flow resistance is minimal. This principle applies at any volume scale. The given volume is covered in successive steps of optimization and construction. Optimally shaped elements are grouped into a “construct,” and then the shape of the construct is optimized. The more visible portion of the optimized volume-to-point flow path that emerges is a tree network that is completely deterministic. This solution has a definite time direction: from small to large, hence the name “constructal.” Small size and slow and shapeless flow (diffusion) come first, and larger sizes and organized flows (streams) come later.