Purcell’s scallop theorem defines the type of motions of a solid body—reciprocal motions—which cannot propel the body in a viscous fluid with zero Reynolds number. For example, the flapping of a wing is reciprocal and, as was recently shown, can lead to directed motion only if its frequency Reynolds number, , is above a critical value of order one. Using elementary examples, we show the existence of oscillatory reciprocal motions which are effective for all arbitrarily small values of the frequency Reynolds number and induce net velocities scaling as . This demonstrates a continuous breakdown of the scallop theorem with inertia.
REFERENCES
1.
2.
E. M.
Purcell
, “Life at low Reynolds number
,” Am. J. Phys.
45
, 3
(1977
).3.
J.
Lighthill
, “Flagellar hydrodynamics—The John von Neumann lecture, 1975
,” SIAM Rev.
18
, 161
(1976
).4.
C.
Brennen
and H.
Winet
, “Fluid mechanics of propulsion by cilia and flagella
,” Annu. Rev. Fluid Mech.
9
, 339
(1977
).5.
S.
Childress
, Mechanics of Swimming and Flying
(Cambridge University Press
, Cambridge UK
, 1981
).6.
R. M.
Alexander
, Principles of Animal Locomotion
(Princeton University Press
, Princeton NJ
, 2002
).7.
S.
Childress
and R.
Dudley
, “Transition from ciliary to flapping mode in a swimming mollusc: Flapping flight as a bifurcation in
,” J. Fluid Mech.
498
, 257
(2004
).8.
N.
Vandenberghe
, S.
Childress
, and J.
Zhang
, “On unidirectional flight of a free flapping wing
,” Phys. Fluids
18
, 014102
(2006
).9.
N.
Vandenberghe
, J.
Zhang
, and S.
Childress
, “Symmetry breaking leads to forward flapping flight
,” J. Fluid Mech.
506
, 147
(2004
).10.
S.
Alben
and M.
Shelley
, “Coherent locomotion as an attracting state for a free flapping body
,” Proc. Natl. Acad. Sci. U.S.A.
102
, 11163
(2005
).11.
X. Y.
Lu
and Q.
Liao
, “Dynamic responses of a two-dimensional flapping foil motion
,” Phys. Fluids
18
, 098104
(2006
).12.
I.
Proudman
and J. R. A.
Pearson
, “Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder
,” J. Fluid Mech.
2
, 237
(1957
).13.
L. G.
Leal
, “Particle motions in a viscous fluid
,” Annu. Rev. Fluid Mech.
12
, 435
(1980
).14.
G.
Segre
and A.
Silberberg
, “Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams
,” J. Fluid Mech.
14
, 115
(1962
).15.
G.
Segre
and A.
Silberberg
, “Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation
,” J. Fluid Mech.
14
, 136
(1962
).16.
S.
Childress
, “Forward flapping flight as a bifurcation in the frequency Reynolds number
,” in Proceedings of the 2004 International Workshop on Mathematical Fluid Dynamics
, edited by J. R.
Kweon
, S.-C.
Kim
, and S.-I.
Sohn
(Pohang University of Science and Technology
, Korea
), pp. 9
–21
.17.
R. G.
Cox
and S. K.
Hsu
, “The lateral migration of solid particles in a laminar flow near a plane
,” Int. J. Multiphase Flow
3
, 201
(1977
).18.
P.
Vasseur
and R. G.
Cox
, “Lateral migration of spherical particles sedimenting in a stagnant bounded fluid
,” J. Fluid Mech.
80
, 561
(1977
).19.
S. I.
Rubinow
and J. B.
Keller
, “The transverse force on a spinning sphere moving in a viscous fluid
,” J. Fluid Mech.
11
, 447
(1961
).20.
R. G.
Cox
, “The steady motion of a particle of arbitrary shape at small Reynolds numbers
,” J. Fluid Mech.
23
, 625
(1965
).21.
B. P.
Ho
and L. G.
Leal
, “Inertial migration of rigid spheres in two-dimensional unidirectional flows
,” J. Fluid Mech.
65
, 365
(1974
).22.
P.
Vasseur
and R. G.
Cox
, “Lateral migration of a spherical particle in two-dimensional shear flows
,” J. Fluid Mech.
78
, 385
(1976
).23.
P. G.
Saffman
, “The lift on a small sphere in a slow shear flow
,” J. Fluid Mech.
22
, 385
(1965
).24.
H. A.
Stone
, “Philip Saffman and viscous flow theory
,” J. Fluid Mech.
409
, 165
(2000
).25.
H.
Brenner
, “The Oseen resistance of a particle of arbitrary shape
,” J. Fluid Mech.
11
, 604
(1961
).26.
W.
Chester
, “On Oseen’s approximation
,” J. Fluid Mech.
13
, 557
(1962
).27.
H.
Brenner
and R. G.
Cox
, “The resistance to a particle of arbitrary shape in translational motion at small Reynolds number
,” J. Fluid Mech.
17
, 561
(1963
).28.
In the biologically relevant situations where , we have .
29.
Note, however, that the flapping of an asymmetric wing was also considered in Ref. 8 with little influence on the nature of the onset of directed flapping motion.
30.
In a related study,
M.
Roper
and H. A.
Stone
(personal communication) have recently shown that the reciprocal oscillatory translation of a solid body could lead to a net rotation for arbitrarily small values of the frequency Reynolds number.© 2007 American Institute of Physics.
2007
American Institute of Physics
You do not currently have access to this content.
