We study helical bodies of arbitrary cross-sectional profile as they swim or transport fluid by the passage of helical waves. Many cases are explored: the external flow problem of swimming in a cylindrical tube or an infinite domain, the internal fluid pumping problem, and confined/unconfined swimming and internal pumping in a viscoelastic (Oldroyd-B) fluid. A helical coordinate system allows for the analytical calculation of swimming and pumping speeds and fluid velocities in the asymptotic regime of nearly cylindrical bodies. In a Newtonian flow, a matched asymptotic analysis results in corrections to the swimming speed accurate to fourth-order in the small wave amplitude, and the results compare favorably with full numerical simulations. We find that the torque-balancing rigid body rotation generally opposes the direction of wave passage, but not always. Confinement can result in local maxima and minima of the swimming speed in the helical pitch, and the effects of confinement decrease exponentially fast with the diameter of the tube. In a viscoelastic fluid, we find that the effects of fluid elasticity on swimming and internal pumping modify the Newtonian results through the mode-dependent complex viscosity, even in a confined domain.
REFERENCES
1.
E.
Lauga
and T. R.
Powers
, “The hydrodynamics of swimming microorganisms
,” Rep. Prog. Phys.
72
, 096601
(2009
).2.
3.
J. R.
Blake
and M. A.
Sleigh
, “Mechanics of ciliary locomotion
,” Biol. Rev.
49
, 85
–125
(1974
).4.
C.
Brennen
and H.
Winet
, “Fluid mechanics of propulsion by cilia and flagella
,” Annu. Rev. Fluid Mech.
9
, 339
–398
(1977
).5.
K.
Drescher
, K. C.
Leptos
, I.
Tuval
, T.
Ishikawa
, T. J.
Pedley
, and R. E.
Goldstein
, “Dancing volvox: Hydrodynamic bound states of swimming algae
,” Phys. Rev. Lett.
102
, 168101
(2009
).6.
7.
I.
Jung
, T. R.
Powers
, and J. M.
Valles
, Jr., “Evidence for two extremes of ciliary motor response in a single swimming microorganism
,” Biophys. J.
106
, 106
–113
(2014
).8.
K.
Fukui
and H.
Asai
, “Spiral motion of paramecium caudatum in a small capillary glass tube
,” J. Eukaryotic Microbiol.
23
, 559
–563
(1976
).9.
J.
Männik
, R.
Driessen
, P.
Galajda
, J. E.
Keymer
, and C.
Dekker
, “Bacterial growth and motility in sub-micron constrictions
,” Proc. Natl. Acad. Sci. U.S.A.
106
, 14861
–14866
(2009
).10.
S.
Jana
, S. H.
Um
, and S.
Jung
, “Paramecium swimming in capillary tube
,” Phys. Fluids
24
, 041901
(2012
).11.
O. S.
Pak
, S. E.
Spagnolie
, and E.
Lauga
, “Hydrodynamics of the double-wave structure of insect spermatozoa flagella
,” J. R. Soc., Interface
9
, 1908
–1924
(2012
).12.
J. B.
Waterbury
, J. M.
Willey
, D. G.
Franks
, F. W.
Valois
, and S. W.
Watson
, “A cyanobacterium capable of swimming motility
,” Science
230
, 74
–76
(1985
).13.
K. M.
Ehlers
, A. D.
Samuel
, H. C.
Berg
, and R.
Montgomery
, “Do cyanobacteria swim using traveling surface waves?
,” PNAS
93
, 8340
–8343
(1996
).14.
H. A.
Stone
and A. D. T.
Samuel
, “Propulsion of microorganisms by surface distortions
,” Phys. Rev. Lett.
77
, 4102
–4104
(1996
).15.
B.
Brahamsha
, “Non-flagellar swimming in marine Synechococcus
,” J. Mol. Microbiol. Biotechnol.
1
, 59
–62
(1999
).16.
K.
Ehlers
and G.
Oster
, “On the mysterious propulsion of Synechococcus
,” PloS One
7
, e36081
(2012
).17.
S. L.
Tamm
, “Ciliary motion in paramecium a scanning electron microscope study
,” J. Cell Biol.
55
, 250
–255
(1972
).18.
H.
Machemer
, “Ciliary activity and the origin of metachrony in Paramecium: Effects of increased viscosity
,” J. Exp. Biol.
57
, 239
–259
(1972
).19.
B.
Nan
, M. J.
McBride
, J.
Chen
, D. R.
Zusman
, and G.
Oster
, “Bacteria that glide with helical tracks
,” Curr. Biol.
24
, R169
–R173
(2014
).20.
G.
Taylor
, “Analysis of the swimming of microscopic organisms
,” Proc. R. Soc. A
209
(1099
), 447
–461
(1951
).21.
M. J.
Lighthill
, “On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers
,” Commun. Pure Appl. Math.
5
, 109
–118
(1952
).22.
J. R.
Blake
, “A spherical envelope approach to ciliary propulsion
,” J. Fluid Mech.
46
, 199
–208
(1971
).23.
J.
Blake
, “Model for micro-structure in ciliated organisms
,” J. Fluid Mech.
55
, 1
–23
(1972
).24.
C.
Brennen
, “An oscillating-boundary-layer theory for ciliary propulsion
,” J. Fluid Mech.
65
, 799
–824
(1974
).25.
S.
Childress
, Mechanics of Swimming and Flying
(Cambridge University Press
, 1981
), Vol. 2
.26.
T.
Ishikawa
, M. P.
Simmonds
, and T. J.
Pedley
, “Hydrodynamic interaction of two swimming model micro-organisms
,” J. Fluid Mech.
568
, 119
–160
(2006
).27.
A.
Kanevsky
, M. J.
Shelley
, and A.-K.
Tornberg
, “Modeling simple locomotors in Stokes flow
,” J. Comput. Phys.
229
, 958
–977
(2010
).28.
Y.
Or
and R. M.
Murray
, “Dynamics and stability of a class of low Reynolds number swimmers near a wall
,” Phys. Rev. E
79
, 045302
(2009
).29.
D.
Crowdy
, “Treadmilling swimmers near a no-slip wall at low Reynolds number
,” Int. J. Nonlinear Mech.
46
, 577
–585
(2011
).30.
S. E.
Spagnolie
and E.
Lauga
, “Hydrodynamics of self-propulsion near a boundary: Predictions and accuracy of far-field approximations
,” J. Fluid Mech.
700
, 105
–147
(2012
).31.
K.
Ishimoto
and E. A.
Gaffney
, “Squirmer dynamics near a boundary
,” Phys. Rev. E
88
(6
), 062702
(2013
).32.
O. S.
Pak
and E.
Lauga
, “Generalized squirming motion of a sphere
,” J. Eng. Math.
88
, 1
–28
(2014
).33.
Z.
Lin
, J.-L.
Thiffeault
, and S.
Childress
, “Stirring by squirmers
,” J. Fluid Mech.
669
, 167
–177
(2011
).34.
S.
Wang
and A. M.
Ardekani
, “Unsteady swimming of small organisms
,” J. Fluid Mech.
702
, 286
–297
(2012
).35.
L.
Zhu
, M.
Do-Quang
, E.
Lauga
, and L.
Brandt
, “Locomotion by tangential deformation in a polymeric fluid
,” Phys. Rev. E
83
, 011901
(2011
).36.
L.
Zhu
, E.
Lauga
, and L.
Brandt
, “Self-propulsion in viscoelastic fluids: Pushers vs. pullers
,” Phys. Fluids
24
, 051902
(2012
).37.
G. J.
Elfring
and E.
Lauga
, “Theory of locomotion through complex fluids
,” in Complex Fluids in Biological Systems
(Springer
, 2015
), pp. 283
–317
.38.
S.
Michelin
and E.
Lauga
, “Efficiency optimization and symmetry-breaking in a model of ciliary locomotion
,” Phys. Fluids
22
, 111901
(2010
).39.
N.
Osterman
and A.
Vilfan
, “Finding the ciliary beating pattern with optimal efficiency
,” Proc. Natl. Acad. Sci. U.S.A.
108
, 15727
–15732
(2011
).40.
K.
Ishimoto
and E. A.
Gaffney
, “Swimming efficiency of spherical squirmers: Beyond the Lighthill theory
,” Phys. Rev. E
90
(1
), 012704
(2014
).41.
J.
Elgeti
and G.
Gompper
, “Emergence of metachronal waves in cilia arrays
,” Proc. Natl. Acad. Sci. U.S.A.
110
, 4470
–4475
(2013
).42.
L.
Zhu
, E.
Lauga
, and L.
Brandt
, “Low-Reynolds-number swimming in a capillary tube
,” J. Fluid Mech.
726
, 285
–311
(2013
).43.
E.
Setter
, I.
Bucher
, and S.
Haber
, “Low-Reynolds-number swimmer utilizing surface traveling waves: Analytical and experimental study
,” Phys. Rev. E
85
, 066304
(2012
).44.
A. T.
Chwang
and T. Y.
Wu
, “Helical movement of micro-organisms
,” Proc. R. Soc. B
178
, 327
–346
(1971
).45.
J. J. L.
Higdon
, “A hydrodynamic analysis of flagellar propulsion
,” J. Fluid Mech.
90
, 685
–711
(1979
).46.
B. U.
Felderhof
, “Swimming at low Reynolds number of a cylindrical body in a circular tube
,” Phys. Fluids
22
, 113604
(2010
).47.
J. J. L.
Higdon
, “The hydrodynamics of flagellar propulsion: Helical waves
,” J. Fluid Mech.
94
, 331
–351
(1979
).48.
N.
Phan-Thien
, T.
Tran-Cong
, and M.
Ramia
, “A boundary-element analysis of flagellar propulsion
,” J. Fluid Mech.
184
, 533
–549
(1987
).49.
B.
Liu
, K. S.
Breuer
, and T. R.
Powers
, “Propulsion by a helical flagellum in a capillary tube
,” Phys. Fluids
26
, 011701
(2014
).50.
H. C.
Fu
, C. W.
Wolgemuth
, and T. R.
Powers
, “Swimming speeds of filaments in nonlinearly viscoelastic fluids
,” Phys. Fluids
21
, 033102
(2009
).51.
S. E.
Spagnolie
, B.
Liu
, and T.
Powers
, “Locomotion of helical bodies in viscoelastic fluids: Enhanced swimming at large helical amplitudes
,” Phys. Rev. Lett.
111
, 068101
(2013
).52.
J.
Blake
, “Mucus flows
,” Math. Biosci.
17
, 301
–313
(1973
).53.
S. M.
Ross
and S.
Corrsin
, “Results of an analytical model of mucociliary pumping
,” J. Appl. Physiol.
37
, 333
–340
(1974
).54.
G. R.
Fulford
and J. R.
Blake
, “Muco-ciliary transport in the lung
,” J. Theor. Biol.
121
, 381
–402
(1986
).55.
M. A.
Sleigh
, J. R.
Blake
, and N.
Liron
, “The propulsion of mucus by cilia
,” Am. Rev. Respir. Dis.
137
, 726
–741
(1988
).56.
D. J.
Smith
, E. A.
Gaffney
, and J. R.
Blake
, “Modelling mucociliary clearance
,” Respir. Physiol. Neurobiol.
163
, 178
–188
(2008
).57.
T. J.
Lardner
and W. J.
Shack
, “Cilia transport
,” Bull. Math. Biophys.
34
, 325
–335
(1972
).58.
J.
Blake
, “Flow in tubules due to ciliary activity
,” Bull. Math. Biol.
35
, 513
–523
(1973
).59.
L.
Li
and S. E.
Spagnolie
, “Swimming and pumping of rigid helical bodies in viscous fluids
,” Phys. Fluids
26
, 041901
(2014
).60.
H.
Power
and G.
Miranda
, “Second kind integral equation formulation of Stokes flows past a particle of arbitrary shape
,” SIAM J. Appl. Math.
47
, 689
–698
(1987
).61.
C.
Pozrikidis
, Boundary Integral and Singularity Methods for Linearized Viscous Flow
(Cambridge University Press
, Cambridge, UK
, 1992
).62.
S.
Kim
and S. J.
Karrila
, Microhydrodynamics: Principles and Selected Applications
(Dover Publications, Inc.
, Mineola, NY
, 1991
).63.
K.
Atkinson
and W.
Han
, Theoretical Numerical Analysis
(Springer
, New York, NY
, 2009
).64.
K. E.
Atkinson
, An Introduction to Numerical Analysis
(John Wiley & Sons
, New York
, 1978
).65.
B.
Liu
, K. S.
Breuer
, and T. R.
Powers
, “Helical swimming in Stokes flow using a novel boundary-element method
,” Phys. Fluids
25
, 061902
(2013
).66.
M.
Sauzade
, G. J.
Elfring
, and E.
Lauga
, “Taylor’s swimming sheet: Analysis and imement of the perturbation series
,” Phys. D
240
, 1567
–1573
(2011
).67.
H.
Aref
, “Stirring by chaotic advection
,” J. Fluid Mech.
143
, 1
–21
(1984
).68.
R. E.
Goldstein
, I.
Tuval
, and J.
van de Meent
, “Microfluidics of cytoplasmic streaming and its implications for intracellular transport
,” Proc. Natl. Acad. Sci. U.S.A.
105
, 3663
–3667
(2008
).69.
R. B.
Bird
, R. C.
Armstrong
, and O.
Hassager
, Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics
(John Wiley and Sons Inc.
, New York, NY
, 1987
).70.
A.
Morozov
and S. E.
Spagnolie
, “Introduction to complex fluids
,” in Complex Fluids in Biological Systems
(Springer
, 2015
), pp. 3
–52
.71.
E.
Lauga
, “Propulsion in a viscoelastic fluid
,” Phys. Fluids
19
, 083104
(2007
).72.
H. C.
Fu
, T. R.
Powers
, and H. C.
Wolgemuth
, “Theory of swimming filaments in viscoelastic media
,” Phys. Rev. Lett.
99
, 258101
–258105
(2007
).73.
B.
Liu
, T. R.
Powers
, and K. S.
Breuer
, “Force-free swimming of a model helical flagellum in viscoelastic fluids
,” Proc. Natl. Acad. Sci. U.S.A.
108
, 19516
–19520
(2011
).74.
E.
Lauga
, “Locomotion in complex fluids: Integral theorems
,” Phys. Fluids
26
, 081902
(2014
).75.
G. J.
Elfring
, O. S.
Pak
, and E.
Lauga
, “Two-dimensional flagellar synchronization in viscoelastic fluids
,” J. Fluid Mech.
646
, 505
(2010
).76.
T.
Honda
, K. I.
Arai
, and K.
Ishiyama
, “Micro swimming mechanisms propelled by external magnetic fields
,” IEEE Trans. Magn.
32
, 5085
–5087
(1996
).77.
A.
Ghosh
and P.
Fischer
, “Controlled propulsion of artificial magnetic nanostructured propellers
,” Nano Lett.
9
, 2243
–2245
(2009
).78.
L.
Zhang
, J. J.
Abbott
, L.
Dong
, K. E.
Peyer
, B. E.
Kratochvil
, H.
Zhang
, C.
Bergeles
, and B. J.
Nelson
, “Characterizing the swimming properties of artificial bacterial flagella
,” Nano Lett.
9
, 3663
–3667
(2009
).79.
S.
Tottori
, L.
Zhang
, F.
Qiu
, K. K.
Krawczyk
, A.
Franco-Obregón
, and B. J.
Nelson
, “Magnetic helical micromachines: Fabrication, controlled swimming, and cargo transport
,” Adv. Mat.
24
, 811
–816
(2012
).80.
S.
Tottori
, L.
Zhang
, K. E.
Peyer
, and B. J.
Nelson
, “Assembly, disassembly, and anomalous propulsion of microscopic helices
,” Nano Lett.
13
, 4263
–4268
(2013
).81.
K. E.
Peyer
, S.
Tottori
, F.
Qiu
, L.
Zhang
, and B. J.
Nelson
, “Magnetic helical micromachines
,” Chem. - Eur. J.
19
, 28
–38
(2013
).82.
Y.
Tsukii
, Protist Information Server, 2005, url: http://protist.i.hosei.ac.jp/.© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.