Stephenson David, Lockerby Duncan A
School of Engineering , University of Warwick , Coventry CV4 7AL, UK.
Proc Math Phys Eng Sci. 2016 Jul;472(2191):20160451. doi: 10.1098/rspa.2016.0451.
When applied to a branching network, Murray's law states that the optimal branching of vascular networks is achieved when the cube of the parent channel radius is equal to the sum of the cubes of the daughter channel radii. It is considered integral to understanding biological networks and for the biomimetic design of artificial fluidic systems. However, despite its ubiquity, we demonstrate that Murray's law is only optimal (i.e. maximizes flow conductance per unit volume) for symmetric branching, where the local optimization of each individual channel corresponds to the global optimum of the network as a whole. In this paper, we present a generalized law that is valid for asymmetric branching, for any cross-sectional shape, and for a range of fluidic models. We verify our analytical solutions with the numerical optimization of a bifurcating fluidic network for the examples of laminar, turbulent and non-Newtonian fluid flows.
当应用于分支网络时，默里定律指出，当母通道半径的立方等于子通道半径的立方之和时，血管网络实现了最优分支。它被认为是理解生物网络以及人工流体系统仿生设计不可或缺的一部分。然而，尽管它无处不在，但我们证明，默里定律仅在对称分支情况下是最优的（即单位体积的流动传导率最大化），在对称分支中，每个单独通道的局部优化对应于整个网络的全局最优。在本文中，我们提出了一个广义定律，该定律适用于非对称分支、任何横截面形状以及一系列流体模型。我们通过对分叉流体网络进行数值优化，以层流、湍流和非牛顿流体流动为例，验证了我们的解析解。
