Murray's law revisited: Quémada's fluid model and fractal trees

Abstract

In 1926, Murray proposed the first law for the optimal design of blood\nvessels. He minimized the power dissipation arising from the trade-off between\nfluid circulation and blood maintenance. The law, based on a constant fluid\nviscosity, states that in the optimal configuration the fluid flow rate inside\nthe vessel is proportional to the cube of the vessel radius, implying that wall\nshear stress is not dependent on the vessel radius. Murray's law has been found\nto be true in blood macrocirculation, but not in microcirculation. In 2005,\nAlarc\\'on et al took into account the non monotonous dependence of viscosity on\nvessel radius - F{\\aa}hr{\\ae}us-Lindqvist effect - due to phase separation\neffect of blood. They were able to predict correctly the behavior of wall shear\nstresses in microcirculation. One last crucial step remains however: to account\nfor the dependence of blood viscosity on shear rates. In this work, we\ninvestigate how viscosity dependence on shear rate affects Murray's law. We\nextended Murray's optimal design to the whole range of Qu\\'emada's fluids, that\nmodels pseudo-plastic fluids such as blood. Our study shows that Murray's\noriginal law is not restricted to Newtonian fluids, it is actually universal\nfor all Qu\\'emada's fluid as long as there is no phase separation effect. When\nphase separation effect occurs, then we derive an extended version of Murray's\nlaw. Our analyses are very general and apply to most of fluids with shear\ndependent rheology. Finally, we study how these extended laws affect the\noptimal geometries of fractal trees to mimic an idealized arterial network.\n