Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-23T18:45:35.107Z Has data issue: false hasContentIssue false
  • English
  • Français

The sedimentation of flexible filaments

Published online by Cambridge University Press:  29 October 2013

Lei Li ,
Harishankar Manikantan ,
David Saintillan  and
Saverio E. Spagnolie
Lei Li
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Harishankar Manikantan
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
David Saintillan
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
Saverio E. Spagnolie*
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, WI 53706, USA
*
†Email address for correspondence: spagnolie@math.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favourably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulations.

JFM classification

Instability: Instability Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Slender-body theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 705 - 736
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.)

References

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, 063602.CrossRefGoogle Scholar
Batchelor, G. 1970 Slender body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419–440.CrossRefGoogle Scholar
Becker, L. E. & Shelley, M. J. 2001 Instability of elastic filaments in shear flow yields first-normal-stress differences. Phys. Rev. Lett. 87, 198301.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, vol. 1. Springer.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. I. Fluid Mechanics. Wiley Interscience.Google Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.Google Scholar
Cosentino Lagomarsino, M., Pagonabarraga, I. & Lowe, C. P. 2005 Hydrodynamic induced deformation and orientation of a microscopic elastic filament. Phys. Rev. Lett. 94, 148104.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791–810.Google Scholar
Ehrlich, H. P., Grislis, G. & Hunt, T. K. 1977 Evidence for the movement of the microtubules in wound contraction. Am. J. Surg. 133, 706–709.CrossRefGoogle Scholar
Evans, A. A., Spagnolie, S. E., Bartolo, D. & Lauga, E. 2013 Elastocapillary self-folding: buckling, wrinkling, and collapse of floating filaments. Soft Matt. 9, 1711–1720.CrossRefGoogle Scholar
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371–394.Google Scholar
Fulford, G. R. & Blake, J. R. 1986 Mucociliary transport in the lung. J. Theor. Biol. 121, 381–402.Google Scholar
Gaffney, E. A., Gadelha, H., Smith, D. J., Blake, J. R. & Kirkman-Brown, J. C. 2011 Mammalian sperm motility: observation and theory. Annu. Rev. Fluid Mech. 43, 501–528.Google Scholar
Gardel, M. L., Nakamura, F., Hartwig, J. H., Crocker, J. C., Stossel, T. P. & Weitz, D. A. 1995 Prestressed f-actin networks cross-linked by hinged filamins replicate mechanical properties of cells. Proc. Natl Acad. Sci. USA 103, 1762–1767.Google Scholar
Götz, T. 2000 Interactions of fibres and flow: asymptotics, theory and numerics. PhD thesis, University of Kaiserslautern, Germany.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.Google Scholar
Guglielmini, L., Kushwaha, A., Shaqfeh, E. & Stone, H. 2012 Buckling transitions of an elastic filament in a viscous stagnation point flow. Phys. Fluids 24, 123601.Google Scholar
Gustavsson, K. & Tornberg, A.-K. 2009 Gravity induced sedimentation of slender fibres. Phys. Fluids 21, 123301.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice Hall.Google Scholar
Harasim, M., Wunderlich, B., Peleg, O., Kröger, M. & Bausch, A. R. 2013 Direct observation of the dynamics of semiflexible polymers in shear flow. Phys. Rev. Lett. 110, 108302.CrossRefGoogle ScholarPubMed
Hinch, E. J. 1976 The distortion of a flexible inextensible thread in a shearing flow. J. Fluid Mech. 74, 317–333.Google Scholar
Jayaraman, G., Ramachandran, S., Ghose, S., Laskar, A., Bhamla, M., Kumar, P. & Adhikari, R. 2012 Autonomous motility of active filaments due to spontaneous flow-symmetry breaking. Phys. Rev. Lett. 109, 158302.CrossRefGoogle ScholarPubMed
Johnson, R. 1980 An improved slender-body theory for Stokes-flow. J. Fluid Mech. 99, 411–431.CrossRefGoogle Scholar
Jung, S., Spagnolie, S. E., Parikh, K., Shelley, M. & Tornberg, A.-K. 2006 Periodic sedimentation in a Stokesian fluid. Phys. Rev. E 74, 035302.Google Scholar
Kantsler, V. & Goldstein, R. E. 2012 Fluctuations, dynamics, and the stretch-coil transition of single actin filaments in extensional flows. Phys. Rev. Lett. 108, 038103.Google Scholar
Keller, J. B. & Rubinow, S. I. 1976 Slender-body theory for slow viscous flow. J. Fluid Mech. 75 (04), 705–714.Google Scholar
Kim, S. & Karrila, S. 1991 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Koch, D. L. & Shaqfeh, E. S. G. 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521–542.Google Scholar
Lauga, E. 2007 Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E 75, 041916.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.CrossRefGoogle Scholar
Llopis, I., Pagonabarraga, I., Cosentino Lagomarsino, M. & Lowe, C. P. 2007 Sedimentation of pairs of hydrodynamically interacting semiflexible filaments. Phys. Rev. E 76, 061901.CrossRefGoogle ScholarPubMed
Love, A. E. H. 1892 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.Google Scholar
Manikantan, H. & Saintillan, D. 2013 Subdiffusive transport of fluctuating elastic filaments in cellular flows. Phys. Fluids 25 (7), 073603.CrossRefGoogle Scholar
Metzger, B., Guazzelli, E. & Butler, J. E. 2005 Large-scale streamers in the sedimentation of a dilute fibre suspension. Phys. Rev. Lett. 95, 164506.Google Scholar
Munk, T., Hallatschek, O., Wiggins, C. H. & Frey, E. 2006 Dynamics of semiflexible polymers in a flow field. Phys. Rev. E 74, 041911.CrossRefGoogle Scholar
Pan, L., Morozov, C., Wagner, C. & Arratia, P. 2013 Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110, 174502.CrossRefGoogle ScholarPubMed
Reinsch, S. & Gönczy, P. 1998 Mechanisms of nuclear positioning. J. Cell Sci. 111, 2283–2295.Google Scholar
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006 The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation. J. Fluid Mech. 553, 347–388.CrossRefGoogle Scholar
Schlagberger, X. & Netz, R. R. 2005 Orientation of elastic rods in homogeneous Stokes flow. Europhys. Lett. 70, 129–135.CrossRefGoogle Scholar
Seifert, U., Wintz, W. & Nelson, P. 1996 Straightening of thermal fluctuations in semiflexible polymers by applied tension. Phys. Rev. Lett. 77, 5389–5392.CrossRefGoogle ScholarPubMed
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.Google Scholar
Shinar, T., Mana, M., Piano, F. & Shelley, M. J. 2011 A model of cytoplasmically-driven microtubule-based motion in the single-celled C. elegans embryo. Proc. Natl Acad. Sci. USA 108, 10508–10513.Google Scholar
Spagnolie, S. E. & Lauga, E. 2010 The optimal elastic flagellum. Phys. Fluids 22, 031901.CrossRefGoogle Scholar
Steinhauser, D., Köster, S. & Pfohl, T. 2012 Mobility gradient induces cross-streamline migration of semiflexible polymers. ACS Macro Lett. 1, 541–545.Google Scholar
Thomases, B., Shelley, M. J. & Thiffeault, J.-L. 2011 A Stokesian viscoelastic flow: transition to oscillations and mixing. Physica D 240, 1602–1614.Google Scholar
Tilney, L. G., Tilney, M. S. & DeRosier, D. J. 1992 Actin filaments, stereocilia, and hair cells: How cells count and measure. Annu. Rev. Cell Biol. 8, 257–274.CrossRefGoogle ScholarPubMed
Tornberg, A. K. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibres in Stokes flows. J. Comput. Phys. 196, 8–40.CrossRefGoogle Scholar
Tozzi, E. J., Scott, C. T., Vahey, D. & Klingenberg, D. J. 2011 Settling dynamics of asymmetric rigid fibres. Phys. Fluids 23, 033301.Google Scholar
Wandersman, E., Quennouz, N., Fermigier, M., Lindner, A. & du Roure, O. 2010 Buckled in translation. Soft Matt. 6, 5715–5719.CrossRefGoogle Scholar
Wang, J., Tozzi, E. J., Graham, M. D. & Klingenberg, D. J. 2012 Flipping, scooping, and spinning: drift of rigid curved nonchiral fibres in simple shear flow. Phys. Fluids 24, 123304.CrossRefGoogle Scholar
Wexler, J. S., Trinh, P. H., Berthet, H., Quennouz, N., du Roure, O., Huppert, H. E., Lindner, A. & Stone, H. A. 2013 Bending of elastic fibres in viscous flows: the influence of confinement. J. Fluid Mech. 720, 517–544.CrossRefGoogle Scholar
Wiggins, C. H. & Goldstein, R. E. 1998 Flexible and propulsive dynamics of elastic at low Reynolds number. Phys. Rev. Lett. 80, 3879–3882.Google Scholar
Xu, X. & Nadim, A. 1994 Deformation and orientation of an elastic slender body sedimenting in a viscous liquid. Phys. Fluids 6, 2889–2894.Google Scholar
Young, Y. N. 2009 Hydrodynamic interactions between two semiflexible inextensible filaments in Stokes flow. Phys. Rev. E 79, 046317.Google Scholar
Young, Y.-N. & Shelley, M. J. 2007 Stretch-coil transition and transport of fibres in cellular flows. Phys. Rev. Lett. 99, 058303.Google Scholar
Supplementary material: PDF

Li et al. supplementary material

Supplementary movies captions

Download Li et al. supplementary material(PDF)
PDF 86.8 KB

Li et al. supplementary movie

Trajectories of weakly flexible filaments

Download Li et al. supplementary movie(Video)
Video 860.7 KB

Li et al. supplementary movie

Particle clouds

Download Li et al. supplementary movie(Video)
Video 1.1 MB

Li et al. supplementary movie

Buckling of flexible filaments

Download Li et al. supplementary movie(Video)
Video 3 MB