- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Shape of a tethered filament in various low-Reynolds-number flows

Christina Kurzthaler1,2,*, Rodolfo Brandão2,3, Ory Schnitzer3,†, and Howard A. Stone2,‡

  • 1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
  • 2Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
  • 3Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

  • *ckurzthaler@pks.mpg.de
  • †o.schnitzer@imperial.ac.uk
  • ‡hastone@princeton.edu
PDF

Phys. Rev. Fluids 8, 014101 – Published 3 January, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.014101

Abstract

We consider the steady-state deformation of an elastic filament in various unidirectional, low-Reynolds-number flows, with the filament either clamped at one end, perpendicular to the flow, or tethered at its center and deforming symmetrically about a plane parallel to the flow. We employ a slender-body model [Pozrikidis, J. Fluids Struct. 26, 393 (2010)] to describe the filament shape as a function of the background flow and a nondimensional compliance η characterizing the ratio of viscous to elastic forces. For η≪1, we describe the small deformation of the filament by means of a regular perturbation expansion. For η≫1, the filament strongly bends such that it is nearly parallel to the flow except close to the tether point; we analyze this singular limit using boundary-layer theory, finding that the radius of curvature near the tether point, as well as the distance of the parallel segment from the tether point, scale like η−1/2 for flow profiles that do not vanish at the tether point, and like η−1/3 for flow profiles that vanish linearly away from the tether point. We also use a Wentzel-Kramers-Brillouin approach to derive a leading-order approximation for the exponentially small slope of the filament away from the tether point. We compare numerical solutions of the model over a wide range of η values with closed-form predictions obtained in both asymptotic limits, focusing on particular uniform, shear and parabolic flow profiles relevant to experiments.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (48)

  1. E. Lauga, The Fluid Dynamics of Cell Motility (Cambridge University Press, Cambridge, UK, 2020), Vol. 62.
  2. S. Gsell, E. Loiseau, U. D'Ortona, A. Viallat, and J. Favier, Hydrodynamic model of directional ciliary-beat organization in human airways, Sci. Rep. 10, 8405 (2020).
  3. O. H. Shapiro, V. I. Fernandez, M. Garren, J. S. Guasto, F. P. Debaillon-Vesque, E. Kramarsky-Winter, A. Vardi, and R. Stocker, Vortical ciliary flows actively enhance mass transport in reef corals, Proc. Natl. Acad. Sci. USA 111, 13391 (2014).
  4. S. Nonaka, Y. Tanaka, Y. Okada, S. Takeda, A. Harada, Y. Kanai, M. Kido, and N. Hirokawa, Randomization of left–right asymmetry due to loss of nodal cilia generating leftward flow of extraembryonic fluid in mice lacking KIF3B motor protein, Cell 95, 829 (1998).
  5. S. Nonaka, H. Shiratori, Y. Saijoh, and H. Hamada, Determination of left–right patterning of the mouse embryo by artificial nodal flow, Nature (London) 418, 96 (2002).
  6. S. Nonaka, S. Yoshiba, D. Watanabe, S. Ikeuchi, T. Goto, W. F. Marshall, and H. Hamada, De novo formation of left–right asymmetry by posterior tilt of nodal cilia, PLoS Biol. 3, e268 (2005).
  7. Y. Okada, S. Takeda, Y. Tanaka, J.-C. I. Belmonte, and N. Hirokawa, Mechanism of nodal flow: A conserved symmetry breaking event in left-right axis determination, Cell 121, 633 (2005).
  8. S. Hanasoge, P. J. Hesketh, and A. Alexeev, Microfluidic pumping using artificial magnetic cilia, Microsyst. Nanoeng. 4, 11 (2018).
  9. H. Gu, Q. Boehler, H. Cui, E. Secchi, G. Savorana, C. De Marco, S. Gervasoni, Q. Peyron, T.-Y. Huang, S. Pane et al., Magnetic cilia carpets with programmable metachronal waves, Nat. Commun. 11, 2637 (2020).
  10. W. Wang, Q. Liu, I. Tanasijevic, M. F. Reynolds, A. J. Cortese, M. Z. Miskin, M. C. Cao, D. A. Muller, A. C. Molnar, E. Lauga et al., Cilia metasurfaces for electronically programmable microfluidic manipulation, Nature (London) 605, 681 (2022).
  11. W. Gilpin, M. S. Bull, and M. Prakash, The multiscale physics of cilia and flagella, Nat. Rev. Phys. 2, 74 (2020).
  12. A. Bausch and K. Kroy, A bottom-up approach to cell mechanics, Nat. Phys. 2, 231 (2006).
  13. A. C. Callan-Jones and R. Voituriez, Actin flows in cell migration: From locomotion and polarity to trajectories, Curr. Opin. Cell Biol. 38, 12 (2016).
  14. J. F. Marko and E. D. Siggia, Stretching DNA, Macromolecules 28, 8759 (1995).
  15. T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, Stretching of a single tethered polymer in a uniform flow, Science 268, 83 (1995).
  16. Y.-N. Young, M. Downs, and C. R. Jacobs, Dynamics of the primary cilium in shear flow, Biophys. J. 103, 629 (2012).
  17. M. D. Koch, C. Fei, N. S. Wingreen, J. W. Shaevitz, and Z. Gitai, Competitive binding of independent extension and retraction motors explains the quantitative dynamics of type IV pili, Proc. Natl. Acad. Sci. USA 118, e2014926118 (2021).
  18. R. G. Cox, The motion of long slender bodies in a viscous fluid Part 1. General theory, J. Fluid Mech. 44, 791 (1970).
  19. G. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. 44, 419 (1970).
  20. J. Lighthill, Flagellar hydrodynamics, SIAM Rev. 18, 161 (1976).
  21. J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705 (1976).
  22. R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech. 99, 411 (1980).
  23. I. Llopis, I. Pagonabarraga, M. Cosentino Lagomarsino, and C. P. Lowe, Sedimentation of pairs of hydrodynamically interacting semiflexible filaments, Phys. Rev. E 76, 061901 (2007).
  24. B. Delmotte, E. Climent, and F. Plouraboué, A general formulation of bead models applied to flexible fibers and active filaments at low Reynolds number, J. Comput. Phys. 286, 14 (2015).
  25. P. J. Żuk, A. M. Słowicka, M. L. Ekiel-Jeżewska, and H. A. Stone, Universal features of the shape of elastic fibres in shear flow, J. Fluid Mech. 914, A31 (2021).
  26. Y. Liu, B. Chakrabarti, D. Saintillan, A. Lindner, and O. du Roure, Morphological transitions of elastic filaments in shear flow, Proc. Natl. Acad. Sci. USA 115, 9438 (2018).
  27. O. Forgacs and S. Mason, Particle motions in sheared suspensions: IX. Spin and deformation of threadlike particles, J. Colloid Sci. 14, 457 (1959).
  28. O. Forgacs and S. Mason, Particle motions in sheared suspensions: X. Orbits of flexible threadlike particles, J. Colloid Sci. 14, 473 (1959).
  29. Y.-N. Young and M. J. Shelley, Stretch-Coil Transition and Transport of Fibers in Cellular Flows, Phys. Rev. Lett. 99, 058303 (2007).
  30. V. Kantsler and R. E. Goldstein, Fluctuations, Dynamics, and the Stretch-Coil Transition of Single Actin Filaments in Extensional Flows, Phys. Rev. Lett. 108, 038103 (2012).
  31. N. Quennouz, M. Shelley, O. du Roure, and A. Lindner, Transport and buckling dynamics of an elastic fibre in a viscous cellular flow, J. Fluid Mech. 769, 387 (2015).
  32. A. M. Słowicka, N. Xue, P. Sznajder, J. K. Nunes, H. A. Stone, and M. L. Ekiel-Jeżewska, Buckling of elastic fibers in a shear flow, New J. Phys. 24, 013013 (2022).
  33. N. Xue, J. K. Nunes, and H. A. Stone, Shear-induced migration of confined flexible fibers, Soft Matter 18, 514 (2022).
  34. X. Xu and A. Nadim, Deformation and orientation of an elastic slender body sedimenting in a viscous liquid, Phys. Fluids 6, 2889 (1994).
  35. L. Li, H. Manikantan, D. Saintillan, and S. E. Spagnolie, The sedimentation of flexible filaments, J. Fluid Mech. 735, 705 (2013).
  36. B. Marchetti, V. Raspa, A. Lindner, O. du Roure, L. Bergougnoux, E. Guazzelli, and C. Duprat, Deformation of a flexible fiber settling in a quiescent viscous fluid, Phys. Rev. Fluids 3, 104102 (2018).
  37. J. S. Wexler, P. H. Trinh, H. Berthet, N. Quennouz, O. du Roure, H. E. Huppert, A. Lindner, and H. A. Stone, Bending of elastic fibres in viscous flows: the influence of confinement, J. Fluid Mech. 720, 517 (2013).
  38. L. Guglielmini, A. Kushwaha, E. S. G. Shaqfeh, and H. A. Stone, Buckling transitions of an elastic filament in a viscous stagnation point flow, Phys. Fluids 24, 123601 (2012).
  39. S. A. Koehler and T. R. Powers, Twirling Elastica: Kinks, Viscous Drag, and Torsional Stress, Phys. Rev. Lett. 85, 4827 (2000).
  40. B. Qian, T. R. Powers, and K. S. Breuer, Shape Transition and Propulsive Force of an Elastic Rod Rotating in a Viscous Fluid, Phys. Rev. Lett. 100, 078101 (2008).
  41. A. Lamura, R. G. Winkler, and G. Gompper, Wall-anchored semiflexible polymer under large amplitude oscillatory shear flow, J. Chem. Phys. 154, 224901 (2021).
  42. P. S. Doyle, B. Ladoux, and J.-L. Viovy, Dynamics of a Tethered Polymer in Shear Flow, Phys. Rev. Lett. 84, 4769 (2000).
  43. T. Y. Lin, A. Saadat, A. Kushwaha, and E. S. G. Shaqfeh, Effect of length on the dynamics of wall tethered polymers in shear flow, Macromolecules 51, 254 (2018).
  44. C. Pozrikidis, Shear flow over cylindrical rods attached to a substrate, J. Fluids Struct. 26, 393 (2010).
  45. E. Wandersman, N. Quennouz, M. Fermigier, A. Lindner, and O. du Roure, Buckled in translation, Soft Matter 6, 5715 (2010).
  46. Wolfram Research Inc., Mathematica, Version 13.1 (Wolfram Research Inc., Champaign, IL, 2022).
  47. S. Alben, M. Shelley, and J. Zhang, Drag reduction through self-similar bending of a flexible body, Nature (London) 420, 479 (2002).
  48. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Mineola, New York, 1964), Vol. 55.

Outline

Information

Phys. Rev. Fluids 8, 014101– Published 3 January, 2023
  • Received 30 August 2022
  • Accepted 15 December 2022
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation