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Generalizing Murray's law : an optimization principle for fluidic networks of arbitrary shape and scale
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Stephenson, David, Patronis, Alexander, Holland, David M. and Lockerby, Duncan A. (2015) Generalizing Murray's law : an optimization principle for fluidic networks of arbitrary shape and scale. Journal of Applied Physics, 118 (17). pp. 1-9. 174302. doi:10.1063/1.4935288 ISSN 0021-8979.
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Official URL: http://dx.doi.org/10.1063/1.4935288
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−2/3Γ=N−2/3 for large-scale channels, and Γ=N−4/5Γ=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.
Item Type: | Journal Article | ||||||||
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Subjects: | R Medicine > RC Internal medicine | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering | ||||||||
Library of Congress Subject Headings (LCSH): | Blood flow | ||||||||
Journal or Publication Title: | Journal of Applied Physics | ||||||||
Publisher: | American Institute of Physics | ||||||||
ISSN: | 0021-8979 | ||||||||
Official Date: | 6 November 2015 | ||||||||
Dates: |
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Volume: | 118 | ||||||||
Number: | 17 | ||||||||
Number of Pages: | 9 | ||||||||
Page Range: | pp. 1-9 | ||||||||
Article Number: | 174302 | ||||||||
DOI: | 10.1063/1.4935288 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 16 March 2016 | ||||||||
Date of first compliant Open Access: | 16 March 2016 | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
Grant number: | EP/I011927/1 (EPSRC), EP/K038664/1 (EPSRC) |
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