A body immersed in a highly viscous fluid can locomote by drawing in and expelling fluid through pores at its surface. We consider this mechanism of jet propulsion without inertia in the case of spheroidal bodies and derive both the swimming velocity and the hydrodynamic efficiency. Elementary examples are presented and exact axisymmetric solutions for spherical, prolate spheroidal, and oblate spheroidal body shapes are provided. In each case, entirely and partially porous (i.e., jetting) surfaces are considered and the optimal jetting flow profiles at the surface for maximizing the hydrodynamic efficiency are determined computationally. The maximal efficiency which may be achieved by a sphere using such jet propulsion is 12.5%, a significant improvement upon traditional flagella-based means of locomotion at zero Reynolds number, which corresponds to the potential flow created by a source dipole at the sphere center. Unlike other swimming mechanisms which rely on the presentation of a small cross section in the direction of motion, the efficiency of a jetting body at low Reynolds number increases as the body becomes more oblate and limits to approximately 162% in the case of a flat plate swimming along its axis of symmetry. Our results are discussed in the light of slime extrusion mechanisms occurring in many cyanobacteria.
REFERENCES
1.
T. Y.
Wu
, C. J.
Brokaw
, and C.
Brennen
, Swimming and Flying in Nature
(Plenum
, New York
, 1975
), Vols. 1
and 2
.2.
E. M.
Purcell
, “Life at low Reynolds-number
,” Am. J. Phys.
45
, 3
(1977
).3.
S.
Childress
, Mechanics of Swimming and Flying
(Cambridge University Press
, Cambridge
, 1981
).4.
5.
J.
Lighthill
, Mathematical Biofluiddynamics
(SIAM
, Philadelphia
, 1975
).6.
C.
Brennen
and H.
Winet
, “Fluid-mechanics of propulsion by cilia and flagella
,” Annu. Rev. Fluid Mech.
9
, 339
(1977
).7.
E.
Lauga
and T. R.
Powers
, “The hydrodynamics of swimming microorganisms
,” Rep. Prog. Phys.
72
, 096601
(2009
).8.
S. E.
Spagnolie
and E.
Lauga
, “The optimal elastic flagellum
,” Phys. Fluids
22
, 031901
(2010
).9.
H. A.
Stone
and A. D. T.
Samuel
, “Propulsion of microorganisms by surface distortions
,” Phys. Rev. Lett.
77
, 4102
(1996
).10.
L. E.
Becker
, S. A.
Koehler
, and H. A.
Stone
, “On self-propulsion of micro-machines at low Reynolds number: Purcell’s three-link swimmer
,” J. Fluid Mech.
490
, 15
(2003
).11.
D.
Tam
and A. E.
Hosoi
, “Optimal stroke patterns for Purcell’s three-link swimmer
,” Phys. Rev. Lett.
98
, 068105
(2007
).12.
J. E.
Avron
and O.
Raz
, “A geometric theory of swimming: Purcell’s swimmer and its symmetrized cousin
,” New J. Phys.
10
, 063016
(2008
).13.
G.
Taylor
, “The action of waving cylindrical tails in propelling microscopic organisms
,” Proc. R. Soc. London, Ser. A
211
, 225
(1952
).14.
I. M.
Kulić
, R.
Thaokar
, and H.
Schiessel
, “Twirling DNA rings—Swimming nanomotors ready for a kickstart
,” Europhys. Lett.
72
, 527
(2005
).15.
R. M.
Thaokar
, H.
Schiessel
, and I. M.
Kulíc
, “Hydrodynamics of a rotating torus
,” Eur. Phys. J. B
60
, 325
(2007
).16.
A. M.
Leshansky
and O.
Kenneth
, “Surface tank treading: Propulsion of Purcell’s toroidal swimmer
,” Phys. Fluids
20
, 063104
(2008
).17.
A. M.
Leshansky
, O.
Kenneth
, O.
Gat
, and J. E.
Avron
, “A frictionless microswimmer
,” New J. Phys.
9
, 145
(2007
).18.
J. E.
Avron
, O.
Gat
, and O.
Kenneth
, “Optimal swimming at low Reynolds numbers
,” Phys. Rev. Lett.
93
, 186001
(2004
).19.
J. E.
Avron
, O.
Kenneth
, and D. H.
Oaknin
, “Pushmepullyou: An efficient micro-swimmer
,” New J. Phys.
7
, 234
(2005
).20.
F. Y.
Ogrin
, P. G.
Petrov
, and C. P.
Winlove
, “Ferromagnetic microswimmers
,” Phys. Rev. Lett.
100
, 218102
(2008
).21.
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
).22.
D. G.
Crowdy
and Y.
Or
, “Two-dimensional point singularity model of a low Reynolds number swimmer near a wall
,” Phys. Rev. E
81
, 036313
(2010
).23.
A.
Najafi
and R.
Golestanian
, “Propulsion at low Reynolds number
,” J. Phys.: Condens. Matter
17
, S1203
(2005
).24.
R.
Zargar
, A.
Najafi
, and M.
Miri
, “Three-sphere low-Reynolds-number swimmer near a wall
,” Phys. Rev. E
80
, 026308
(2009
).25.
B. U.
Felderhof
, “The swimming of animalcules
,” Phys. Fluids
18
, 063101
(2006
).26.
R.
Trouilloud
, T. S.
Yu
, A. E.
Hosoi
, and E.
Lauga
, “Soft swimming: Exploiting deformable interfaces for low Reynolds number locomotion
,” Phys. Rev. Lett.
101
, 048102
(2008
).27.
S. E.
Spagnolie
, “Rehinging biflagellar locomotion in a viscous fluid
,” Phys. Rev. E
80
, 046323
(2009
).28.
A. A.
Evans
, S. E.
Spagnolie
, and E.
Lauga
, “Stokesian jellyfish: Viscous locomotion of bilayer vesicles
,” Soft Matter
6
, 1737
(2010
).29.
A. E.
Walsby
, “Mucilage secretion and the movements of blue-green algae
,” Protoplasma
65
, 223
(1968
).30.
G.
Guglielmi
and G.
Cohen-Bazire
, “Structure et distribution des pores et des perforations de l’enveloppe de peptidoglycane chez quelques cyanobactéries
,” Protistologica
18
, 151
(1982
).31.
E.
Hoiczyk
and W.
Baumeister
, “The junctional pore complex, a prokaryotic secretion organelle, is the molecular motor underlying gliding motility in cyanobacteria
,” Curr. Biol.
8
, 1161
(1998
).32.
E.
Hoiczyk
, “Gliding motility in cyanobacteria: Observations and possible explanations
,” Arch. Microbiol.
174
, 11
(2000
).33.
C.
Wolgemuth
, E.
Hoiczyk
, D.
Kaiser
, and G.
Oster
, “How myxobacteria glide
,” Curr. Biol.
12
, 369
(2002
).34.
C. W.
Wolgemuth
and G.
Oster
, “The junctional pore complex and the propulsion of bacterial cells
,” J. Mol. Microbiol. Biotechnol.
7
, 72
(2004
).35.
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
(1985
).36.
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
, 8340
(1996
).37.
B.
Brahamsha
, “Non-flagellar swimming in marine synechococcus
,” J. Mol. Microbiol. Biotechnol.
1
, 59
(1999
).38.
D.
Weihs
, “Periodic jet propulsion of aquatic creatures
,” Fortschr. Zool.
24
, 171
(1977
).39.
T. L.
Daniel
, “Unsteady aspects of aquatic locomotion
,” Am. Zool.
24
, 121
(1984
).40.
M. E.
Demont
and J. M.
Gosline
, “Mechanics of jet propulsion in the hydromedusan jellyfish, Polyorchis penicillatus: II. Energetics of the jet cycle
,” J. Exp. Biol.
134
, 333
(1988
).41.
P. F.
Linden
and J. S.
Turner
, “‘Optimal’ vortex rings and aquatic propulsion mechanisms
,” Proc. R. Soc. London, Ser. B
271
, 647
(2004
).42.
J. O.
Dabiri
and M.
Gharib
, “The role of optimal vortex formation in biological fluid transport
,” Proc. R. Soc. London, Ser. B
272
, 1557
(2005
).43.
K.
Mohseni
, “Pulsatile vortex generators for low-speed maneuvering of small underwater vehicles
,” Ocean Eng.
33
, 2209
(2006
).44.
J. O.
Dabiri
, “Optimal vortex formation as a unifying principle in biological propulsion
,” Annu. Rev. Fluid Mech.
41
, 17
(2009
).45.
H.
Brenner
, “The Stokes resistance of an arbitrary particle—IV. Arbitrary fields of flow
,” Chem. Eng. Sci.
19
, 703
(1964
).46.
J.
Happel
and H.
Brenner
, Low Reynolds Number Hydrodynamics
(Prentice Hall
, Englewood Cliffs, NJ
, 1965
).47.
S.
Kim
and S. J.
Karrila
, Microhydrodynamics: Principles and Selected Applications
(Dover
, Mineola, NY
, 1991
).48.
A. T.
Chwang
and T. Y. T.
Wu
, “Hydromechanics of low Reynolds number flow. Part 2. Singularity methods for Stokes flows
,” J. Fluid Mech.
67
, 787
(1975
).49.
G. B.
Jeffery
, “The motion of ellipsoidal particles immersed in a viscous fluid
,” Proc. R. Soc. London, Ser. A
102
, 161
(1922
).50.
A. T.
Chwang
and T. Y. T.
Wu
, “Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies
,” J. Fluid Mech.
63
, 607
(1974
).51.
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
(1952
).52.
J. R.
Blake
, “A spherical envelope approach to ciliary propulsion
,” J. Fluid Mech.
46
, 199
(1971
).53.
V.
Magar
and T. J.
Pedley
, “Average nutrient uptake by a self-propelled unsteady squirmer
,” J. Fluid Mech.
539
, 93
(2005
).54.
T.
Ishikawa
, M. P.
Simmonds
, and T. J.
Pedley
, “Hydrodynamic interaction of two swimming model micro-organisms
,” J. Fluid Mech.
568
, 119
(2006
).55.
R. A.
Sampson
, “On Stokes’s current function
,” Philos. Trans. R. Soc. London, Ser. A
182
, 449
(1891
).56.
O.
Pironneau
, “On optimal profiles in Stokes flow
,” J. Fluid Mech.
59
, 117
(1973
).57.
J. -M.
Bourot
, “On the numerical computation of the optimum profile in Stokes flow
,” J. Fluid Mech.
65
, 513
(1974
).58.
A.
Kanevsky
, M. J.
Shelley
, and A. -K.
Tornberg
, “Modeling simple locomotors in Stokes flow
,” J. Comput. Phys.
229
, 958
(2010
).© 2010 American Institute of Physics.
2010
American Institute of Physics
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