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Bending of elastic fibres in viscous flows: the influence of confinement‡

Published online by Cambridge University Press:  27 February 2013

Jason S. Wexler ,
Philippe H. Trinh ,
Helene Berthet ,
Nawal Quennouz ,
Olivia du Roure ,
Herbert E. Huppert ,
Anke Lindner  and
Howard A. Stone
Jason S. Wexler*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA PMMH, ESPCI, CNRS UMR 7636, Université Pierre et Marie Curie, Université Paris Diderot, 10 rue Vauquelin, 75005 Paris, France
Philippe H. Trinh
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Helene Berthet
Affiliation:
PMMH, ESPCI, CNRS UMR 7636, Université Pierre et Marie Curie, Université Paris Diderot, 10 rue Vauquelin, 75005 Paris, France
Nawal Quennouz
Affiliation:
PMMH, ESPCI, CNRS UMR 7636, Université Pierre et Marie Curie, Université Paris Diderot, 10 rue Vauquelin, 75005 Paris, France
Olivia du Roure
Affiliation:
PMMH, ESPCI, CNRS UMR 7636, Université Pierre et Marie Curie, Université Paris Diderot, 10 rue Vauquelin, 75005 Paris, France
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK School of Mathematics, University of New South Wales, Kensington, NSW 2052, Australia
Anke Lindner
Affiliation:
PMMH, ESPCI, CNRS UMR 7636, Université Pierre et Marie Curie, Université Paris Diderot, 10 rue Vauquelin, 75005 Paris, France
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: jwexler@princeton.edu, anke.lindner@espci.fr, hastone@princeton.edu
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Abstract

We present a mathematical model and corresponding series of microfluidic experiments examining the flow of a viscous fluid past an elastic fibre in a three-dimensional channel. The fibre’s axis lies perpendicular to the direction of flow and its base is clamped to one wall of the channel; the sidewalls of the channel are close to the fibre, confining the flow. Experiments show that there is a linear relationship between deflection and flow rate for highly confined fibres at low flow rates, which inspires an asymptotic treatment of the problem in this regime. The three-dimensional problem is reduced to a two-dimensional model, consisting of Hele-Shaw flow past a barrier, with boundary conditions at the barrier that allow for the effects of flexibility and three-dimensional leakage. The analysis yields insight into the competing effects of flexion and leakage, and an analytical solution is derived for the leading-order pressure field corresponding to a slit that partially blocks a two-dimensional channel. The predictions of our model show favourable agreement with experimental results, allowing measurement of the fibre’s elasticity and the flow rate in the channel.

JFM classification

Aerodynamics: Flow-structure interactions Low-Reynolds-number flows: Hele-Shaw flows Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 720 , 10 April 2013 , pp. 517 - 544
Copyright
©2013 Cambridge University Press 

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Footnotes

‡

The original version of this article was published with A. Lindner’s name incorrectly spelled. A notice detailing this has been published and the error rectified in the online PDF and HTML copies.

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