Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T17:48:10.348Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Ablowitz, M. J. & Fokas, A. S. 2003 Complex Variables: Introduction and Applications. Cambridge University Press.CrossRefGoogle Scholar
Alben, S., Shelley, M. & Zhang, J. 2002 Drag reduction through self-similar bending of a flexible body. Nature 420 (6915), 479–481.CrossRefGoogle ScholarPubMed
Attia, R., Pregibon, D. C., Doyle, P. S., Viovy, J. L. & Bartolo, D. 2009 Soft microflow sensors. Lab on a Chip 9 (9), 1213–1218.CrossRefGoogle ScholarPubMed
Autrusson, N., Guglielmini, L., Lecuyer, S., Rusconi, R. & Stone, H. A. 2011 The shape of an elastic filament in a two-dimensional corner flow. Phys. Fluids 23 (6), 063602.CrossRefGoogle Scholar
Berthet, H. 2012 Single and collective fibre dynamics in confined microflows. PhD thesis, ESPCI.Google Scholar
Cosentino Lagomarsino, M., Pagonabarraga, I. & Lowe, C. 2005 Hydrodynamic induced deformation and orientation of a microscopic elastic filament. Phys. Rev. Lett. 94 (14), 1–4.CrossRefGoogle ScholarPubMed
Day, R. F. & Stone, H. A. 2000 Lubrication analysis and boundary integral simulations of a viscous micropump. J. Fluid Mech. 416, 197–216.CrossRefGoogle Scholar
Dendukuri, D., Gu, S. S., Pregibon, D. C., Hatton, T. A. & Doyle, P. S. 2007 Stop-flow lithography in a microfluidic device. Lab on a Chip 7 (7), 818–828.CrossRefGoogle Scholar
Dendukuri, D., Panda, P., Haghgooie, R., Kim, J. M., Hatton, T. A. & Doyle, P. S. 2008 Modelling of oxygen-inhibited free radical photopolymerization in a PDMS microfluidic device. Macromolecules 41 (22), 8547–8556.CrossRefGoogle Scholar
DiLuzio, W. R., Turner, L., Mayer, M., Garstecki, P., Weibel, D. B., Berg, H. C. & Whitesides, G. M. 2005 Escherichia coli swim on the right-hand side. Nature 435 (7046), 1274.CrossRefGoogle ScholarPubMed
Gervais, T., El-Ali, J., Günther, A. & Jensen, K. F. 2006 Flow-induced deformation of shallow microfluidic channels. Lab on a Chip 6 (4), 500–507.CrossRefGoogle ScholarPubMed
Guglielmini, L., Kushwaha, A., Shaqfeh, E. S. G. & Stone, H. A. 2012 Buckling transitions of an elastic filament in a viscous stagnation point flow. Phys. Fluids 24 (12), 123601.CrossRefGoogle Scholar
Joung, C. G., Phan-Thien, N. & Fan, X. J. 2001 Direct simulation of flexible fibres. J. Non-Newtonian Fluid Mech. 99, 1–36.CrossRefGoogle Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Dover Publications.CrossRefGoogle Scholar
Païdoussis, M. P. 2004 Fluid–Structure Interactions, vol. 1–2. Academic Press.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2011 Shear flow past slender elastic rods attached to a plane. International Journal of Solids and Structures 48 (1), 137–143.CrossRefGoogle Scholar
Qian, B., Powers, T. & Breuer, K. 2008 Shape transition and propulsive force of an elastic rod rotating in a viscous fluid. Phys. Rev. Lett. 100 (7), 078101.CrossRefGoogle Scholar
Rusconi, R., Lecuyer, S., Guglielmini, L. & Stone, H. A. 2010 Laminar flow around corners triggers the formation of biofilm streamers. J. Roy. Soc. Int. 7 (50), 1293–1299.CrossRefGoogle ScholarPubMed
Semin, B., Hulin, J. P. & Auradou, H. 2009 Influence of flow confinement on the drag force on a static cylinder. Phys. Fluids 21 (10), 103604.CrossRefGoogle Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. Wiley.Google Scholar
Squires, T. & Quake, S. 2005 Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977–1026.CrossRefGoogle Scholar
Stockie, J. M. & Green, S. I. 1998 Simulating the motion of flexible pulp fibres using the immersed boundary method. J. Comp. Phys. 147 (1), 147–165.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Ann. Rev. Fluid Mech. 36 (1), 381–411.CrossRefGoogle Scholar
Thompson, B. W. 1968 Secondary flow in a Hele–Shaw cell. J. Fluid Mech. 31 (2), 379–395.CrossRefGoogle Scholar
Tuck, E. O. 1964 A systematic asymptotic expansion procedure for slender ships. J. Ship Res. 8 (1), 639–668.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
Wandersman, E., Quennouz, N., Fermigier, M., Lindner, A. & du Roure, O. 2010 Buckled in translation. Soft Matt. 6 (22), 5715.CrossRefGoogle Scholar
Wiggins, C. & Goldstein, R. 1998 Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80 (17), 3879–3882.CrossRefGoogle Scholar
Young, Y.-N., Downs, M. & Jacobs, C. R. 2012 Dynamics of the primary cilium in shear flow. Biophys. J. 103 (4), 629–639.CrossRefGoogle ScholarPubMed
Yu, T. S., Lauga, E. & Hosoi, A. E. 2006 Experimental investigations of elastic tail propulsion at low Reynolds number. Phys. Fluids 18 (9), 091701.CrossRefGoogle Scholar