Abstract
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.
Highlights
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 C, for symmetric branching into N daughter channels of any constant cross-sectional shape, is C 1⁄4 NÀ2=3 for large-scale channels, and C 1⁄4 NÀ4=5 for channels with a characteristic length scale much smaller than the slip length
In 1926, Murray1 posited that there were two competing factors contributing to the energy cost of blood flow through the arterial system: (1) the energy required to drive the flow, which increases as the vessel radius decreases; and (2) the energy required to metabolically maintain the fluid, which increases with increasing vessel radius
Summary
In 1926, Murray posited that there were two competing factors contributing to the energy cost of blood flow through the arterial system: (1) the energy required to drive the flow, which increases as the vessel radius decreases; and (2) the energy required to metabolically maintain the fluid, which increases with increasing vessel radius. Since the original derivation of Murray’s law, it has been noted that the application of other optimization principles (not just that of minimum work) result in Eq (7): minimizing the total mass of the network, minimizing volume for a constant pressure drop and flow rate, keeping the shear stress constant in all channels, or minimizing flow resistance for a constant volume.. Despite many developments to Murray’s law, there are three major barriers that prevent it from being relevant to the design of many artificial fluidic networks: (1) it is not demonstrably applicable to cross sections of any arbitrary shape; (2) it is not applicable at the micro/nanoscale, where a fluid can no longer be accurately described as a continuous material; and (3) it is not applicable to networks which maintain a constant depth through branching, wherein the cross-sectional shape changes between the parent and daughter channels
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