Application: Embedded Irrigation Networks

Abstract

This chapter is devoted to a discussion of physical models of lungs, blood circulation and other natural or artificial networks irrigating a volume from a source. We shall address here the main objection to the model treated throughout these notes: they do not consider the fact that irrigation networks are embedded in a body. We have reduced the vessels or roads or tubes to infinitely thin paths, but they must in fact be thick enough. The Gilbert model reduces the three parameters of a tube, namely its length, diameter, and flow to only two, the length and the flow. Yet once we fix the flow to be conveyed in a tube we can make the tube very thick so that the flow becomes slow and has little kinetic, energy. If instead the tube is very thin the fluid velocity must be large and therefore the dissipated power high. Thus, embedded models have apparently one more degree of freedom, the diameter of the tubes and this parameter changes the energy. By making the tube diameters tend to infinity we can even let the dissipated energy tend to zero. Of course embedded irrigation circuits cannot occupy the whole space and their volume must be constrained, unless we only grow lentils. Section 14.1 describes the physical principles and parameters of tube models. Section 14.2 shows that under a volume constraint the tube diameter can be eliminated in the expression of the dissipated power. This elimination is done by simple Lagrangian calculus and shows the dissipated power to be a Gilbert energy. The expression of this energy also leads to the interesting conclusion that, under the Poiseuille’s law of laminar flow in the tubes, the α exponent is simply critical.