Rotating helical bodies of arbitrary cross-sectional profile and infinite length are explored as they swim through or transport a viscous fluid. The Stokes equations are studied in a helical coordinate system, and closed form analytical expressions for the force-free swimming speed and torque are derived in the asymptotic regime of nearly cylindrical bodies. High-order accurate expressions for the velocity field and swimming speed are derived for helical bodies of finite pitch angle through a double series expansion. The analytical predictions match well with the results of full numerical simulations, and accurately predict the optimal pitch angle for a given cross-sectional profile. This work may improve the modeling and design of helical structures used in microfluidic manipulation, synthetic microswimmer engineering, and the transport and mixing of viscous fluids.
REFERENCES
1.
T.
Koetsier
and H.
Blauwendraat
, “The Archimedean screw-pump: A note on its invention and the development of the theory
,” in Proceedings of the International Symposium on History of Machines and Mechanisms, 2004
(Springer
, Netherlands
, 2004
), pp. 181
–194
.2.
P. A.
Tanguy
, R.
Lacroix
, F.
Bertrand
, L.
Choplin
, and E. B. De La
Fuente
, “Finite element analysis of viscous mixing with a helical ribbon-screw impeller
,” AIChE J.
38
, 939
–944
(1992
).3.
J. De La
Villeon
, F.
Bertrand
, P. A.
Tanguy
, R.
Labrie
, J.
Bousquet
, and D.
Lebouvier
, “Numerical investigation of mixing efficiency of helical ribbons
,” AIChE J.
44
, 972
–977
(1998
).4.
G.
Delaplace
, J. C.
Leuliet
, and V.
Relandeau
, “Circulation and mixing times for helical ribbon impellers: Review and experiments
,” Exp. Fluids
28
(2
), 170
–182
(2000
).5.
M.
Robinson
and P. W.
Cleary
, “Flow and mixing performance in helical ribbon mixers
,” Chem. Eng. Sci.
84
, 382
–398
(2012
).6.
P. J.
Carreau
, I.
Patterson
, and C. Y.
Yap
, “Mixing of viscoelastic fluids with helical-ribbon agitators. I. Mixing time and flow patterns
,” Can. J. Chem. Eng.
54
, 135
–142
(1976
).7.
F.
Jiang
, K. S.
Drese
, S.
Hardt
, M.
Küpper
, and F.
Schönfeld
, “Helical flows and chaotic mixing in curved micro channels
,” AIChE J.
50
, 2297
–2305
(2004
).8.
T.
Honda
, K. I.
Arai
, and K.
Ishiyama
, “Micro swimming mechanisms propelled by external magnetic fields
,” IEEE Trans. Magn.
32
, 5085
–5087
(1996
).9.
A.
Ghosh
and P.
Fischer
, “Controlled propulsion of artificial magnetic nanostructured propellers
,” Nano Lett.
9
, 2243
–2245
(2009
).10.
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
).11.
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. Mater.
24
, 811
–816
(2012
).12.
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
).13.
K. E.
Peyer
, S.
Tottori
, F.
Qiu
, L.
Zhang
, and B. J.
Nelson
, “Magnetic helical micromachines
,” Chem. Eur. J.
19
, 28
–38
(2013
).14.
B. J.
Nelson
, I. K.
Kaliakatsos
, and J. J.
Abbott
, “Microrobots for minimally invasive medicine
,” Annu. Rev. Biomed. Eng.
12
, 55
–85
(2010
).15.
W.
Gao
, D.
Kagan
, O. S.
Pak
, C.
Clawson
, S.
Campuzano
, E.
Chuluun-Erdene
, E.
Shipton
, E. E.
Fullerton
, L.
Zhang
, E.
Lauga
et al, “Cargo-towing fuel-free magnetic nanoswimmers for targeted drug delivery
,” Small
8
, 460
–467
(2012
).16.
J.
Wang
and W.
Gao
, “Nano/microscale motors: Biomedical opportunities and challenges
,” ACS Nano
6
, 5745
–5751
(2012
).17.
J.
Lighthill
, “Flagellar hydrodynamics - von Neumann lecture, 1975
,” SIAM Rev.
18
(2
), 161
–230
(1976
).18.
E.
Lauga
and T. R.
Powers
, “The hydrodynamics of swimming microorganisms
,” Rep. Prog. Phys.
72
, 096601
(2009
).19.
Z.
Carvalho-Santos
, J.
Azimzadeh
, J. B.
Pereira-Leal
, and M.
Bettencourt-Dias
, “Tracing the origins of centrioles, cilia, and flagella
,” J. Cell Biol.
194
, 165
–175
(2011
).20.
E. M.
Purcell
, “Life at low Reynolds-number
,” Am. J. Phys.
45
(1
), 3
–11
(1977
).21.
J.
Gray
and G. J.
Hancock
, “The propulsion of sea-urchin spermatozoa
,” J. Exp. Biol.
32
(4
), 802
–814
(1955
).22.
S.
Childress
, Mechanics of Swimming and Flying
(Cambridge University Press
, Cambridge, UK
, 1981
), Vol. 2
.23.
B.
Rodenborn
, C.-H.
Chen
, H. L.
Swinney
, B.
Liu
, and H. P.
Zhang
, “Propulsion of microorganisms by a helical flagellum
,” Proc. Natl. Acad. Sci. U.S.A.
110
, E338
–E347
(2013
).24.
M.
Kim
, J. C.
Bird
, A. J.
Van Parys
, K. S.
Breuer
, and T. R.
Powers
, “A macroscopic scale model of bacterial flagellar bundling
,” Proc. Natl. Acad. Sci. U.S.A.
100
, 15481
–15485
(2003
).25.
M.
Kim
and T. R.
Powers
, “Hydrodynamic interactions between rotating helices
,” Phys. Rev. E
69
, 061910
(2004
).26.
M.
Kim
, M.
Kim
, J. C.
Bird
, J.
Park
, T. R.
Powers
, and K. S.
Breuer
, “Particle image velocimetry experiments on a macro-scale model for bacterial flagellar bundling
,” Exp. Fluids
37
, 782
–788
(2004
).27.
M.
Reichert
and H.
Stark
, “Synchronization of rotating helices by hydrodynamic interactions
,” Eur. Phys. J. E
17
, 493
–500
(2005
).28.
H.
Flores
, E.
Lobaton
, S.
Méndez-Diez
, S.
Tlupova
, and R.
Cortez
, “A study of bacterial flagellar bundling
,” Bull. Math. Biol.
67
, 137
–168
(2005
).29.
P. J. A.
Janssen
and M. D.
Graham
, “Coexistence of tight and loose bundled states in a model of bacterial flagellar dynamics
,” Phys. Rev. E
84
, 011910
(2011
).30.
S.
Lim
and C. S.
Peskin
, “Fluid-mechanical interaction of flexible bacterial flagella by the immersed boundary method
,” Phys. Rev. E
85
, 036307
(2012
).31.
S. Y.
Reigh
, R. G.
Winkler
, and G.
Gompper
, “Synchronization and bundling of anchored bacterial flagella
,” Soft Matter
8
, 4363
–4372
(2012
).32.
S. Y.
Reigh
, R. G.
Winkler
, and G.
Gompper
, “Synchronization, slippage, and unbundling of driven helical flagella
,” PloS ONE
8
, e70868
(2013
).33.
H. C.
Fu
, T. R.
Powers
, and H. C.
Wolgemuth
, “Theory of swimming filaments in viscoelastic media
,” Phys. Rev. Lett.
99
, 258101
–258105
(2007
).34.
A. M.
Leshansky
, “Enhanced low-Reynolds-number propulsion in heterogeneous viscous environments
,” Phys. Rev. E
80
, 051911
(2009
).35.
H. C.
Fu
, C. W.
Wolgemuth
, and T. R.
Powers
, “Swimming speeds of filaments in nonlinearly viscoelastic fluids
,” Phys. Fluids
21
, 033102
(2009
).36.
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
).37.
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
).38.
J. J. L.
Higdon
, “The hydrodynamics of flagellar propulsion: Helical waves
,” J. Fluid Mech.
94
, 331
–351
(1979
).39.
E. M.
Purcell
, “The efficiency of propulsion by a rotating flagellum
,” Proc. Natl. Acad. Sci. U.S.A.
94
, 11307
–11311
(1997
).40.
N.
Watari
and R. G.
Larson
, “The hydrodynamics of a run-and-tumble bacterium propelled by polymorphic helical flagella
,” Biophys. J.
98
, 12
–17
(2010
).41.
S. E.
Spagnolie
and E.
Lauga
, “Comparative hydrodynamics of bacterial polymorphism
,” Phys. Rev. Lett.
106
, 058103
(2011
).42.
B.
Liu
, K. S.
Breuer
, and T. R.
Powers
, “Propulsion by a helical flagellum in a capillary tube
,” Phys. Fluids
26
, 011701
(2014
).43.
S.
Childress
, “A thermodynamic efficiency for Stokesian swimming
,” J. Fluid Mech.
705
, 77
–97
(2012
).44.
E. E.
Keaveny
and M. J.
Shelley
, “Hydrodynamic mobility of chiral colloidal aggregates
,” Phys. Rev. E
79
, 051405
(2009
).45.
E. E.
Keaveny
, S.
Walker
, and M. J.
Shelley
, “Optimization of chiral structures for microscale propulsion
,” Nano Lett.
13
, 531
–537
(2013
).46.
K. I.
Morozov
and A. M.
Leshansky
, “The chiral magnetic nanomotors
,” Nanoscale
6
, 1580
–1588
(2014
).47.
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
).48.
C.
Pozrikidis
, Boundary Integral and Singularity Methods for Linearized Viscous Flow
(Cambridge University Press
, Cambridge, UK
, 1992
).49.
K.
Atkinson
and W.
Han
, Theoretical Numerical Analysis
(Springer
, New York, NY
, 2009
).50.
K. E.
Atkinson
, An Introduction to Numerical Analysis
(John Wiley & Sons
, New York
, 1978
).51.
E. E.
Keaveny
and M. J.
Shelley
, “Applying a second-kind boundary integral equation for surface tractions in Stokes flow
,” J. Comput. Phys.
230
, 2141
–2159
(2011
).52.
A. A.
Evans
, S. E.
Spagnolie
, and E.
Lauga
, “Stokesian jellyfish: Viscous locomotion of bilayer vesicles
,” Soft Matter
6
, 1737
–1747
(2010
).53.
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
).54.
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
).55.
S.
Jung
, K.
Mareck
, L.
Fauci
, and M. J.
Shelley
, “Rotational dynamics of a superhelix towed in a Stokes fluid
,” Phys. Fluids
19
, 103105
(2007
).56.
M.
Sauzade
, G. J.
Elfring
, and E.
Lauga
, “Taylor's swimming sheet: Analysis and improvement of the perturbation series
,” Physica D
240
, 1567
–1573
(2011
).57.
C. M.
Bender
and S. A.
Orszag
, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
(Springer
, 1999
), Vol. 1
.58.
S.
Childress
, “Inertial swimmer as a singular perturbation
,” in Proceedings of the ASME 2008 Dynamic Systems and Control Conference
(Ann Arbour
, MI, USA
, 2008
).59.
A. D.
Stroock
, S. K. W.
Dertinger
, A.
Ajdari
, I.
Mezić
, H. A.
Stone
, and G. M.
Whitesides
, “Chaotic mixer for microchannels
,” Science
295
, 647
–651
(2002
).60.
A. D.
Stroock
, S. K.
Dertinger
, G. M.
Whitesides
, and A.
Ajdari
, “Patterning flows using grooved surfaces
,” Anal. Chem.
74
, 5306
–5312
(2002
).61.
S.
Zhou
, A.
Sokolov
, O. D.
Lavrentovich
, and I. S.
Aranson
, “Living liquid crystals
,” Proc. Natl. Acad. Sci. U.S.A.
111
, 1265
–1270
(2014
).62.
S. L.
Tamm
, “Ciliary motion in paramecium a scanning electron microscope study
,” J. Cell Biol.
55
, 250
–255
(1972
).63.
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
).64.
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
).65.
K. M.
Ehlers
, A. D.
Samuel
, H. C.
Berg
, and R.
Montgomery
, “Do cyanobacteria swim using traveling surface waves?
,” Proc. Natl. Acad. Sci. U.S.A.
93
(16
), 8340
–8343
(1996
).66.
H. A.
Stone
and A. D. T.
Samuel
, “Propulsion of microorganisms by surface distortions
,” Phys. Rev. Lett.
77
(19
), 4102
–4104
(1996
).67.
B.
Brahamsha
, “Non-flagellar swimming in marine Synechococcus
,” J. Mol. Microbiol. Biotechnol.
1
, 59
–62
(1999
).68.
K.
Ehlers
and G.
Oster
, “On the mysterious propulsion of Synechococcus
,” PloS ONE
7
, e36081
(2012
).© 2014 AIP Publishing LLC.
2014
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