Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-19T18:26:09.488Z Has data issue: false hasContentIssue false
  • English
  • Français

The instability of a sedimenting suspension of weakly flexible fibres

Published online by Cambridge University Press:  09 September 2014

Harishankar Manikantan ,
Lei Li ,
Saverio E. Spagnolie  and
David Saintillan
Harishankar Manikantan
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, San Diego, CA 92093, USA
Lei Li
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Saverio E. Spagnolie
Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, WI 53706, USA
David Saintillan*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, San Diego, CA 92093, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Suspensions of sedimenting slender fibres in a viscous fluid are known to be unstable to fluctuations of concentration. In this paper we develop a theory for the role of fibre flexibility in sedimenting suspensions in the asymptotic regime of weakly flexible bodies (large elasto-gravitation number). Unlike the behaviour of straight fibres, individual flexible filaments rotate as they sediment, leading to an anisotropic base state of fibre orientations in an otherwise homogeneous suspension. A mean-field theory is derived to describe the evolution of fibre concentration and orientation fields, and we explore the stability of the base state to perturbations of fibre concentration. We show that fibre flexibility affects suspension stability in two distinct and competing ways: the anisotropy of the base state renders the suspension more unstable to perturbations, while individual particle self-rotation acts to prevent clustering and stabilizes the suspension. In the presence of thermal noise, the dominant effect depends critically upon the relative scales of flexible fibre self-rotation compared to rotational Brownian motion.

JFM classification

Instability: Instability Low-Reynolds-number flows: Low-Reynolds-number flows Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 756 , 10 October 2014 , pp. 935 - 964
Copyright
© 2014 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

Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.Google Scholar
Bergougnoux, L., Ghicini, S., Guazzelli, E. & Hinch, J. 2001 Spreading fronts and fluctuations in sedimentation. Phys. Fluids 15, 1875–1887.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, vol. I, Fluid Mechanics. Wiley Interscience.Google Scholar
Brenner, M. P. 1999 Screening mechanisms in sedimentation. Phys. Fluids 11, 754–772.Google Scholar
Butler, J. E. & Shaqfeh, E. S. G. 2002 Dynamic simulations of the inhomogeneous sedimentation of rigid fibres. J. Fluid Mech. 468, 205–237.Google Scholar
Caflisch, R. E. & Luke, J. H. C. 1985 Variance in the sedimentation speed of a suspension. Phys. Fluids 28, 759–760.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
Dahlkild, A. 2011 Finite wavelength selection for the linear instability of a suspension of settling spheroids. J. Fluid Mech. 689, 183–202.CrossRefGoogle Scholar
Doi, M. & Edwards, S. F. 1986 The Theory of Polymer Dynamics. Oxford University Press.Google Scholar
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371–394.CrossRefGoogle Scholar
Gao, T., Blackwell, R., Glaser, M. A., Betterton, M. D. & Shelley, M. J.2014 A multiscale active nematic theory of microtubule/motor-protein assemblies. ArXiv Preprint arXiv:1401.8059.Google Scholar
Gardel, M. L., Nakamura, F., Hartwig, J. H., Crocker, J. C., Stossel, T. P. & Weitz, D. A. 2006 Prestressed F-actin networks cross-linked by hinged filamins replicate mechanical properties of cells. Proc. Natl Acad. Sci. USA 103, 1762–1767.CrossRefGoogle ScholarPubMed
Goubault, C., Jop, P., Fermigier, M., Baudry, J., Bertrand, E. & Bibette, J. 2003 Flexible magnetic filaments as micromechanical sensors. Phys. Rev. Lett. 91, 260802.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.CrossRefGoogle Scholar
Guazzelli, É. 2001 Evolution of particle-velocity correlations in sedimentation. Phys. Fluids 13, 1537–1540.CrossRefGoogle Scholar
Guazzelli, É. & Hinch, J. 2011 Fluctuations and instability in sedimentation. Annu. Rev. Fluid Mech. 43, 97–116.Google Scholar
Gustavsson, K. & Tornberg, A.-K. 2009 Gravity induced sedimentation of slender fibres. Phys. Fluids 21, 123301.Google Scholar
Ham, J. M. & Homsy, G. M. 1988 Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions. Intl J. Multiphase Flow 14, 533–546.Google Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions to the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5, 317–328.CrossRefGoogle Scholar
Herzhaft, B., Guazzelli, É., Mackaplow, M. B. & Shaqfeh, E. S. G. 1996 Experimental investigation of the sedimentation of a dilute fibre suspension. Phys. Rev. Lett. 77, 290–293.Google Scholar
Hinch, E. J. 1987 Sedimentation of small particles. In Disorder and Mixing (ed. Guyon, E., Nadal, J.-P. & Pomeau, Y.), chap. 9, pp. 153–161. Kluwer.Google Scholar
Hoffman, B. D. & Shaqfeh, E. S. G. 2009 The effect of Brownian motion on the stability of sedimenting suspensions of polarizable rods in an electric field. J. Fluid Mech. 624, 361–388.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Keshtkar, M., Heuzey, M. C. & Carreau, P. J. 2009 Rheological behaviour of fibre-filled model suspensions: effect of fibre flexibility. J. Rheol. 53, 631–650.Google Scholar
Kim, S. & Karrila, S. J. 2005 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
Koch, D. L. & Shaqfeh, E. S. G. 1991 Screening in sedimenting suspensions. J. Fluid Mech. 224, 275–303.Google Scholar
Kuusela, E., Lahtinen, J. M. & Ala-Nissila, T. 2003 Collective effects in settling of spheroids under steady-state sedimentation. Phys. Rev. Lett. 90, 094502.Google Scholar
Ladd, A. J. C. 2002 Effects of container walls on the velocity fluctuations of sedimenting spheres. Phys. Rev. Lett. 88, 048301.CrossRefGoogle ScholarPubMed
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.CrossRefGoogle Scholar
Li, L., Manikantan, H., Saintillan, D. & Spagnolie, S. 2013 The sedimentation of flexible filaments. J. Fluid Mech. 735, 705–736.CrossRefGoogle Scholar
Luke, J. H. C. 2000 Decay of velocity fluctuations in a stably stratified suspension. Phys. Fluids 12, 1619–1621.Google Scholar
Mackaplow, M. B. & Shaqfeh, E. S. G. 1998 A numerical study of the sedimentation of fibre suspensions. J. Fluid Mech. 376, 149–182.CrossRefGoogle Scholar
Manga, M. & Stone, H. A. 1995 Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions. J. Fluid Mech. 300, 231–263.Google Scholar
Metzger, B., Butler, J. E. & Guazzelli, E. 2007 Experimental investigation of the instability of a sedimenting suspension of fibres. J. Fluid Mech. 575, 307–332.Google 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.CrossRefGoogle ScholarPubMed
Mucha, P. J. & Brenner, M. P. 2003 Diffusivities and front propagation in sedimentation. Phys. Fluids 15, 1305–1313.Google Scholar
Mucha, P. J., Tee, S.-Y., Weitz, D. A., Shraiman, B. I. & Brenner, M. P. 2004 A model for velocity fluctuations in sedimentation. J. Fluid Mech. 501, 71–104.Google Scholar
Narsimhan, V. & Shaqfeh, E. S. G. 2010 Lateral drift and concentration instability in a suspension of bubbles induced by Marangoni stresses at zero Reynolds number. Phys. Fluids 22, 101702.CrossRefGoogle Scholar
Ramaswamy, S. 2001 Issues in the statistical mechanics of steady sedimentation. Adv. Phys. 50, 297–341.Google Scholar
Saintillan, D., Darve, E. & Shaqfeh, E. S. G. 2005 A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: the sedimentation of fibres. Phys. Fluids 17, 033301.Google Scholar
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006a The effect of stratification on the wavenumber selection in the instability of sedimenting spheroids. Phys. Fluids 18, 121503.Google Scholar
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006b The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation. J. Fluid Mech. 553, 347–388.Google Scholar
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006c Stabilization of a suspension of sedimenting rods by induced-charge electrophoresis. Phys. Fluids 18, 121701.Google Scholar
Saintillan, D. & Shelley, M. J. 2012 Emergence of coherent structures and large-scale flows in motile suspensions. J. R. Soc. Interface 9, 571–585.Google Scholar
Segrè, P. N., Herbolzheimer, E. & Chaikin, P. M. 1997 Long-range correlations in sedimentation. Phys. Rev. Lett. 79, 2574–2577.Google Scholar
Switzer, L. H. III & Klingenberg, D. J. 2003 Rheology of sheared flexible fibre suspensions via fibre-level simulations. J. Rheol. 47, 759–778.Google Scholar
Tornberg, A.-K. & Gustavsson, K. 2006 A numerical method for simulations of rigid fibre suspensions. J. Comput. Phys. 215, 172–196.Google Scholar
Tornberg, A.-K. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibres in Stokes flows. J. Comput. Phys. 196, 8–40.Google Scholar
Van den Heuvel, M., Bondesan, R., Lagomarsino, M. C. & Dekker, C. 2008 Single-molecule observation of anomalous electrohydrodynamic orientation of microtubules. Phys. Rev. Lett. 101, 118301.Google Scholar
Van Der Schoot, P. 1996 The nematic–smectic transition in suspensions of slightly flexible hard rods. J. Physique II 6, 1557–1569.Google Scholar
Vishnampet, R. & Saintillan, D. 2012 Concentration instability of sedimenting spheres in a second-order fluid. Phys. Fluids 24, 073302.Google Scholar
Xu, X. & Nadim, A. 1994 Deformation and orientation of an elastic slender body sedimenting in a viscous liquid. Phys. Fluids 6, 2889–2893.CrossRefGoogle Scholar
Zhang, F., Dahlkild, A. & Lundell, F. 2013 Nonlinear disturbance growth during sedimentation of dilute fibre suspensions. J. Fluid Mech. 719, 268–294.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2003 Large-scale simulations of concentrated emulsion flows. Phil. Trans. R. Soc. Lond. A 361, 813–845.Google Scholar