Generalizing Murray's law: An optimization principle for fluidic networks of arbitrary shape and scale
Generalizing Murray's law: An optimization principle for fluidic networks of arbitrary shape and scale
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DOI：
10.1063/1.4935288
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发表时间：
2015-11
影响因子：
3.2
通讯作者：
David Stephenson;A. Patronis;D. M. Holland;D. Lockerby
中科院分区：
文献类型：
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作者：
David Stephenson;A. Patronis;D. M. Holland;D. Lockerby
默里定律指出，体积流量与圆柱形通道中半径的立方成比例，该圆柱形通道被优化以需要最小的功来驱动和维持流体。然而，将这一原理应用于微/纳米制造网络的仿生设计需要优化具有任意横截面形状（不仅仅是圆形）的通道，并且比Murray最初的假设更小。我们提出了一个广义的对称分支的法律，（a）是有效的任何横截面形状，提供的形状是恒定的通过网络;（B）是有效的滑移流和活塞流发生在非常小的尺度;（c）是有效的网络具有恒定的深度，这往往是一个实验室上的芯片制造程序的要求。通过考虑广义定律的限制，我们证明了对于对称分支成任意恒定横截面形状的N个子通道的最佳子-母面积比Γ，对于大尺度通道为Γ=N−2/3，对于特征长度尺度远小于滑移长度的通道为Γ =N−4/5。我们的分析结果进行了验证，通过比较与一个两级网络模型的流量数据从各种来源，包括Navier-Stokes滑移计算，动力学理论数据，和随机粒子模拟的基础上的数值优化。
Murray's law states that the volumetric flow rate is proportional to the cube of the radius in a cylindrical channel optimized to require the minimum work to drive and maintain the fluid. However, application of this principle to the biomimetic design of micro/nano fabricated networks requires optimization of channels with arbitrary cross-sectional shape (not just circular) and smaller than is valid for Murray's original assumptions. We present a generalized law for symmetric branching that (a) is valid for any cross-sectional shape, providing that the shape is constant through the network; (b) is valid for slip flow and plug flow occurring at very small scales; and (c) is valid for networks with a constant depth, which is often a requirement for lab-on-a-chip fabrication procedures. By considering limits of the generalized law, we show that the optimum daughter-parent area ratio Γ, for symmetric branching into N daughter channels of any constant cross-sectional shape, is Γ=N−2/3 for large-scale channels, and Γ=N−4/5 for channels with a characteristic length scale much smaller than the slip length. Our analytical results are verified by comparison with a numerical optimization of a two-level network model based on flow rate data obtained from a variety of sources, including Navier-Stokes slip calculations, kinetic theory data, and stochastic particle simulations.