Journal
JOURNAL OF APPLIED PHYSICS
Volume 118, Issue 17, Pages -Publisher
AIP Publishing
DOI: 10.1063/1.4935288
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Funding
- EPSRC [EP/I011927/1, EP/K038664/1]
- EPSRC [EP/K038664/1] Funding Source: UKRI
- Engineering and Physical Sciences Research Council [EP/I011927/1, EP/K038664/1] Funding Source: researchfish
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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 C, for symmetric branching into N daughter channels of any constant cross-sectional shape, is Gamma = N-2/3 for large-scale channels, and Gamma = 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. (C) 2015 AIP Publishing LLC.
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