Abstract.
Many microorganisms and artificial microswimmers use helical appendages in order to generate locomotion. Though often rotated so as to produce thrust, some species of bacteria such Spiroplasma, Rhodobacter sphaeroides and Spirochetes induce movement by deforming a helical-shaped body. Recently, artificial devices have been created which also generate motion by deforming their helical body in a non-reciprocal way (A. Mourran et al. Adv. Mater. 29, 1604825, 2017). Inspired by these systems, we investigate the transport of a deforming helix within a viscous fluid. Specifically, we consider a swimmer that maintains a helical centreline and a single handedness while changing its helix radius, pitch and wavelength uniformly across the body. We first discuss how a deforming helix can create a non-reciprocal translational and rotational swimming stroke and identify its principle direction of motion. We then determine the leading-order physics for helices with small helix radius before considering the general behaviour for different configuration parameters and how these swimmers can be optimised. Finally, we explore how the presence of walls, gravity, and defects in the centreline allow the helical device to break symmetries, increase its speed, and generate transport in directions not available to helices in bulk fluids.
Graphical abstract
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
G. Taylor, Proc. R. Soc. A: Math. Phys. Eng. Sci. 209, 447 (1951)
E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)
J. Elgeti, R.G. Winkler, G. Gompper, Rep. Prog. Phys. 78, 056601 (2015)
E.A. Gaffney, H. Gadêlha, D.J. Smith, J.R. Blake, J.C. Kirkman-Brown, Annu. Rev. Fluid Mech. 43, 501 (2011)
E. Lauga, Annu. Rev. Fluid Mech. 48, 105 (2016)
L.E. Becker, S.A. Koehler, H.A. Stone, J. Fluid Mech. 490, 15 (2003)
E.M. Purcell, Am. J. Phys 45, 3 (1977)
L. Turner, W. Ryu, H. Berg, J. Bacteriol. 182, 2793 (2000)
S. Chattopadhyay, R. Moldovan, C. Yeung, X. Wu, Proc. Natl. Acad. Sci. U.S.A. 103, 13712 (2006)
R.E. Goldstein, Annu. Rev. Fluid Mech. 47, 343 (2015)
J. Hu, M. Yang, G. Gompper, R.G. Winkler, Soft Matter 11, 7867 (2015)
T.C. Adhyapak, H. Stark, Soft Matter 12, 5621 (2016)
M. Hintsche, V. Waljor, R. Großmann, M.J. Kühn, K.M. Thormann, F. Peruani, C. Beta, Sci. Rep. 7, 16771 (2017)
E.E. Riley, D. Das, E. Lauga, Swimming of peritrichous bacteria is enabled by an elastohydrodynamic instability, arXiv:1806.01902 (2018)
J. Locsei, J. Math. Biol. 55, 41 (2007)
J.T. Locsei, T.J. Pedley, Bull. Math. Biol. 71, 1089 (2009)
K. Drescher, R.E. Goldstein, I. Tuval, Proc. Natl. Acad. Sci. U.S.A. 107, 11171 (2010)
A. Buchmann, L.J. Fauci, K. Leiderman, E. Strawbridge, L. Zhao, Phys. Rev. E 97, 023101 (2018)
S. Spagnolie, E. Lauga, J. Fluid Mech. 700, 105 (2012)
P. Denissenko, V. Kantsler, D.J. Smith, J. Kirkman-Brown, Proc. Natl. Acad. Sci. U.S.A. 109, 8007 (2012)
H. Shum, E.A. Gaffney, Phys. Rev. E 91, 033012 (2015)
S. Bianchi, F. Saglimbeni, R. Di Leonardo, Phys. Rev. X 7, 011010 (2017)
M.J. Kühn, F.K. Schmidt, B. Eckhardt, K.M. Thormann, Proc. Natl. Acad. Sci. U.S.A. 114, 6340 (2017)
F. Ullrich, F. Qiu, J. Pokki, T. Huang, S. Pane, B.J. Nelson, Swimming characteristics of helical microrobots in fibrous environments, in 2016 6th IEEE International Conference on Biomedical Robotics and Biomechatronics, (IEEE, 2016) pp. 470--475
A.K. Balin, A. Zöttl, J.M. Yeomans, T.N. Shendruk, Phys. Rev. Fluids 2, 113102 (2017)
A. Zöttl, J.M. Yeomans, Enhanced bacterial swimming speeds in macromolecular polymer solutions, arXiv:1710.03505 (2017)
N. Ho, K. Leiderman, S. Olson, A 3-dimensional model of flagellar swimming in a Brinkman fluid, arXiv:1804.06271 (2018)
G. Lumay, N. Obara, F. Weyer, N. Vandewalle, Soft Matter 9, 2420 (2013)
G. Grosjean, G. Lagubeau, A. Darras, M. Hubert, G. Lumay, N. Vandewalle, Sci. Rep. 5, 16035 (2015)
A. Walther, A.H.E. Müller, Soft Matter 4, 663 (2008)
R. Mangal, K. Nayani, Y.-K. Kim, E. Bukusoglu, U.M. Córdova-Figueroa, N.L. Abbott, Langmuir 33, 10917 (2017)
L. Zhang, K.E. Peyer, B.J. Nelson, Lab Chip 10, 2203 (2010)
S. Tottori, L. Zhang, F. Qiu, K.K. Krawczyk, A. Franco-Obregón, B.J. Nelson, Adv. Mater. 24, 811 (2012)
S. Tottori, B.J. Nelson, Biomicrofluidics 7, 061101 (2013)
E. Diller, J. Zhuang, G. Zhan Lum, M.R. Edwards, M. Sitti, Appl. Phys. Lett. 104, 174101 (2014)
T. Xu, H. Yu, H. Zhang, C.-I. Vong, and L. Zhang, Morphologies and swimming characteristics of rotating magnetic swimmers with soft tails at low Reynolds numbers, in 2015 IEEE/RSJ International Conference on Intelligent Robots Systems, Hamburg, Germany (IEEE, 2015) pp. 1385--1390
A. Mourran, H. Zhang, R. Vinokur, M. Möller, Adv. Mater. 29, 1604825 (2017)
H. Sayyaadi, A. Motekallem, Int. J. Marit. Technol. 8, 35 (2017)
J. Ali, U.K. Cheang, J.D. Martindale, M. Jabbarzadeh, H.C. Fu, M. Jun Kim, Sci. Rep. 7, 14098 (2017)
J.W. Shaevitz, J.Y. Lee, D.A. Fletcher, Cell 122, 941 (2005)
C.R. Calladine, J. Mol. Biol. 118, 457 (1978)
C.R. Calladine, B.F. Luisi, J.V. Pratap, J. Mol. Biol. 425, 914 (2013)
H. Wada, R.R. Netz, EPL 82, 28001 (2008)
R. Vogel, H. Stark, Eur. Phys. J. E 33, 259 (2010)
W. Ko, S. Lim, W. Lee, Y. Kim, H.C. Berg, C.S. Peskin, Phys. Rev. E 95, 063106 (2017)
H. Berg, Random Walks in Biology (Princeton University Press, Princeton, N.J., 1983)
G. Rosser, R.E. Baker, J.P. Armitage, A.G. Fletcher, J. R. Soc. Interface 11, 20140320 (2014)
S.F. Goldstein, N.W. Charon, Proc. Natl. Acad. Sci. U.S.A. 87, 4895 (1990)
C. Li, Md. A. Motaleb, M. Sal, S.F. Goldstein, N.W. Charon, J. Mol. Microbiol. Biotechnol. 2, 345 (2000)
L. Koens, E. Lauga, Phys. Biol. 11, 066008 (2014)
W. Kan, C.W. Wolgemuth, Biophys. J. 93, 54 (2007)
S. Jung, K. Mareck, L. Fauci, M. Shelley, Phys. Fluids 19, 103105 (2007)
C. Dombrowski, W. Kan, A. Motaleb, N.W. Charon, R.E. Goldstein, C.W. Wolgemuth, Biophys. J. 96, 4409 (2009)
H. Wada, R.R. Netz, Phys. Rev. Lett. 99, 108102 (2007)
J. Yang, C. Wolgemuth, G. Huber, Phys. Rev. Lett. 102, 218102 (2009)
H. Zhang, A. Mourran, M. Möller, Nano Lett. 17, 2010 (2017)
J. Gray, G.J. Hancock, J. Exp. Biol. 32, 802 (1955)
J. Lighthill, SIAM Rev. 18, 161 (1976)
J.B. Keller, S.I. Rubinow, J. Fluid Mech. 75, 705 (1976)
R.E. Johnson, J. Fluid Mech. 99, 411 (1979)
L. Koens, E. Lauga, Phys. Fluids 28, 013101 (2016)
T.D. Montenegro-Johnson, L. Koens, E. Lauga, Soft Matter 13, 546 (2017)
G.I. Taylor, Low Reynolds number Flow, in National Committee for Fluid Mechanics Films (1967) available at https://doi.org/web.mit.edu/hml/ncfmf.html
S. Kim, S.J. Karrila, Microhydrodynamics: Principles and Selected Applications (Courier Corporation, Boston, 2005)
L. Koens, E. Lauga, Phys. Rev. E 93, 043125 (2016)
A. DeSimone, A. Tatone, Eur. Phys. J. E 35, 85 (2012)
R.L. Hatton, H. Choset, Eur. Phys. J. ST 224, 3141 (2015)
G. Cicconofri, A. DeSimone, Eur. Phys. J. E 39, 72 (2016)
R.L. Hatton, T. Dear, H. Choset, IEEE Trans. Robot. 33, 523 (2017)
S. Ramasamy, R.L. Hatton, Geometric gait optimization beyond two dimensions, in 2017 American Control Conference (IEEE, 2017) pp. 642--648
A. Berke, L. Turner, H. Berg, E. Lauga, Phys. Rev. Lett. 101, 038102 (2008)
E. Barta, N. Liron, SIAM J. Appl. Math. 48, 992 (1988)
H. Shum, E. Gaffney, D. Smith, Proc. R. Soc. A: Math. Phys. Eng. Sci. 466, 1725 (2010)
J. Elgeti, U.B. Kaupp, G. Gompper, Biophys. J. 99, 1018 (2010)
H. Shum, E.A. Gaffney, Phys. Rev. E 92, 063016 (2015)
C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977)
R. Cardinaels, H.A. Stone, Phys. Fluids 27, 072001 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://doi.org/creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Koens, L., Zhang, H., Moeller, M. et al. The swimming of a deforming helix. Eur. Phys. J. E 41, 119 (2018). https://doi.org/10.1140/epje/i2018-11728-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2018-11728-2