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The instability of a sedimenting suspension of weakly flexible fibres

  • Harishankar Manikantan (a1), Lei Li (a2), Saverio E. Spagnolie (a2) and David Saintillan (a1)

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.

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Corresponding author

Email address for correspondence: dstn@ucsd.edu

References

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Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245268.
Bergougnoux, L., Ghicini, S., Guazzelli, E. & Hinch, J. 2001 Spreading fronts and fluctuations in sedimentation. Phys. Fluids 15, 18751887.
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, vol. I, Fluid Mechanics. Wiley Interscience.
Brenner, M. P. 1999 Screening mechanisms in sedimentation. Phys. Fluids 11, 754772.
Butler, J. E. & Shaqfeh, E. S. G. 2002 Dynamic simulations of the inhomogeneous sedimentation of rigid fibres. J. Fluid Mech. 468, 205237.
Caflisch, R. E. & Luke, J. H. C. 1985 Variance in the sedimentation speed of a suspension. Phys. Fluids 28, 759760.
Cosentino Lagomarsino, M., Pagonabarraga, I. & Lowe, C. P. 2005 Hydrodynamic induced deformation and orientation of a microscopic elastic filament. Phys. Rev. Lett. 94, 148104.
Dahlkild, A. 2011 Finite wavelength selection for the linear instability of a suspension of settling spheroids. J. Fluid Mech. 689, 183202.
Doi, M. & Edwards, S. F. 1986 The Theory of Polymer Dynamics. Oxford University Press.
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371394.
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.
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, 17621767.
Goubault, C., Jop, P., Fermigier, M., Baudry, J., Bertrand, E. & Bibette, J. 2003 Flexible magnetic filaments as micromechanical sensors. Phys. Rev. Lett. 91, 260802.
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 5355.
Guazzelli, É. 2001 Evolution of particle-velocity correlations in sedimentation. Phys. Fluids 13, 15371540.
Guazzelli, É. & Hinch, J. 2011 Fluctuations and instability in sedimentation. Annu. Rev. Fluid Mech. 43, 97116.
Gustavsson, K. & Tornberg, A.-K. 2009 Gravity induced sedimentation of slender fibres. Phys. Fluids 21, 123301.
Ham, J. M. & Homsy, G. M. 1988 Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions. Intl J. Multiphase Flow 14, 533546.
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, 317328.
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, 290293.
Hinch, E. J. 1987 Sedimentation of small particles. In Disorder and Mixing (ed. Guyon, E., Nadal, J.-P. & Pomeau, Y.), chap. 9, pp. 153161. Kluwer.
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, 361388.
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161179.
Keshtkar, M., Heuzey, M. C. & Carreau, P. J. 2009 Rheological behaviour of fibre-filled model suspensions: effect of fibre flexibility. J. Rheol. 53, 631650.
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.
Koch, D. L. & Shaqfeh, E. S. G. 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521542.
Koch, D. L. & Shaqfeh, E. S. G. 1991 Screening in sedimenting suspensions. J. Fluid Mech. 224, 275303.
Kuusela, E., Lahtinen, J. M. & Ala-Nissila, T. 2003 Collective effects in settling of spheroids under steady-state sedimentation. Phys. Rev. Lett. 90, 094502.
Ladd, A. J. C. 2002 Effects of container walls on the velocity fluctuations of sedimenting spheres. Phys. Rev. Lett. 88, 048301.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.
Li, L., Manikantan, H., Saintillan, D. & Spagnolie, S. 2013 The sedimentation of flexible filaments. J. Fluid Mech. 735, 705736.
Luke, J. H. C. 2000 Decay of velocity fluctuations in a stably stratified suspension. Phys. Fluids 12, 16191621.
Mackaplow, M. B. & Shaqfeh, E. S. G. 1998 A numerical study of the sedimentation of fibre suspensions. J. Fluid Mech. 376, 149182.
Manga, M. & Stone, H. A. 1995 Collective hydrodynamics of deformable drops and bubbles in dilute low Reynolds number suspensions. J. Fluid Mech. 300, 231263.
Metzger, B., Butler, J. E. & Guazzelli, E. 2007 Experimental investigation of the instability of a sedimenting suspension of fibres. J. Fluid Mech. 575, 307332.
Metzger, B., Guazzelli, E. & Butler, J. E. 2005 Large-scale streamers in the sedimentation of a dilute fibre suspension. Phys. Rev. Lett. 95, 164506.
Mucha, P. J. & Brenner, M. P. 2003 Diffusivities and front propagation in sedimentation. Phys. Fluids 15, 13051313.
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, 71104.
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.
Ramaswamy, S. 2001 Issues in the statistical mechanics of steady sedimentation. Adv. Phys. 50, 297341.
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.
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.
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, 347388.
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006c Stabilization of a suspension of sedimenting rods by induced-charge electrophoresis. Phys. Fluids 18, 121701.
Saintillan, D. & Shelley, M. J. 2012 Emergence of coherent structures and large-scale flows in motile suspensions. J. R. Soc. Interface 9, 571585.
Segrè, P. N., Herbolzheimer, E. & Chaikin, P. M. 1997 Long-range correlations in sedimentation. Phys. Rev. Lett. 79, 25742577.
Switzer, L. H. III & Klingenberg, D. J. 2003 Rheology of sheared flexible fibre suspensions via fibre-level simulations. J. Rheol. 47, 759778.
Tornberg, A.-K. & Gustavsson, K. 2006 A numerical method for simulations of rigid fibre suspensions. J. Comput. Phys. 215, 172196.
Tornberg, A.-K. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibres in Stokes flows. J. Comput. Phys. 196, 840.
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.
Van Der Schoot, P. 1996 The nematic–smectic transition in suspensions of slightly flexible hard rods. J. Physique II 6, 15571569.
Vishnampet, R. & Saintillan, D. 2012 Concentration instability of sedimenting spheres in a second-order fluid. Phys. Fluids 24, 073302.
Xu, X. & Nadim, A. 1994 Deformation and orientation of an elastic slender body sedimenting in a viscous liquid. Phys. Fluids 6, 28892893.
Zhang, F., Dahlkild, A. & Lundell, F. 2013 Nonlinear disturbance growth during sedimentation of dilute fibre suspensions. J. Fluid Mech. 719, 268294.
Zinchenko, A. Z. & Davis, R. H. 2003 Large-scale simulations of concentrated emulsion flows. Phil. Trans. R. Soc. Lond. A 361, 813845.
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The instability of a sedimenting suspension of weakly flexible fibres

  • Harishankar Manikantan (a1), Lei Li (a2), Saverio E. Spagnolie (a2) and David Saintillan (a1)

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