The motion of an initially spherical capsule in a wall-bounded oscillating shear flow is investigated via an accelerated boundary integral implementation. The neo-Hookean model is used as the constitutive law of the capsule membrane. The maximum wall-normal migration is observed when the oscillation period of the imposed shear is of the order of the relaxation time of the elastic membrane; hence, the optimal capillary number scales with the inverse of the oscillation frequency and the ratio agrees well with the theoretical prediction in the limit of high-frequency oscillation. The migration velocity decreases monotonically with the frequency of the applied shear and the capsule-wall distance. We report a significant correlation between the capsule lateral migration and the normal stress difference induced in the flow. The periodic variation of the capsule deformation is roughly in phase with that of the migration velocity and normal stress difference, with twice the frequency of the imposed shear. The maximum deformation increases linearly with the membrane elasticity before reaching a plateau at higher capillary numbers when the deformation is limited by the time over which shear is applied in the same direction and not by the membrane deformability. The maximum membrane deformation scales as the distance to the wall to the power 1/3 as observed for capsules and droplets in near-wall steady shear flows.

1.
T.
Fischer
and
H.
Schmid-Schönbein
, “
Tank tread motion of red cell membranes in viscometric flow: Behavior of intracellular and extracellular markers (with film)
,” in
Red Cell Rheology
(
Springer
,
1978
), pp.
347
361
.
2.
T.
Fischer
,
M.
Stohr-Lissen
, and
H.
Schmid-Schonbein
, “
The red cell as a fluid droplet: Tank tread-like motion of the human erythrocyte membrane in shear flow
,”
Science
202
(
4370
),
894
896
(
1978
).
3.
D.
Barthès-Biesel
, “
Motion of a spherical microcapsule freely suspended in a linear shear flow
,”
J. Fluid Mech.
100
(
04
),
831
853
(
1980
).
4.
C. D.
Eggleton
and
A. S.
Popel
, “
Large deformation of red blood cell ghosts in a simple shear flow
,”
Phys. Fluids
10
,
1834
(
1998
).
5.
M.
Kraus
,
W.
Wintz
,
U.
Seifert
, and
R.
Lipowsky
, “
Fluid vesicles in shear flow
,”
Phys. Rev. Lett.
77
(
17
),
3685
(
1996
).
6.
J.
Ha
and
S.
Yang
, “
Electrohydrodynamic effects on the deformation and orientation of a liquid capsule in a linear flow
,”
Phys. Fluids
12
,
1671
(
2000
).
7.
Y.
Sui
,
H. T.
Low
,
Y. T.
Chew
, and
P.
Roy
, “
Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow
,”
Phys. Rev. E
77
(
1
),
016310
(
2008
).
8.
E.
Lac
,
D.
Barthès-Biesel
,
N. A.
Pelekasis
, and
J.
Tsamopoulos
, “
Spherical capsules in three-dimensional unbounded stokes flows: Effect of the membrane constitutive law and onset of buckling
,”
J. Fluid Mech.
516
,
303
334
(
2004
).
9.
D. V.
Le
, “
Effect of bending stiffness on the deformation of liquid capsules enclosed by thin shells in shear flow
,”
Phys. Rev. E
82
(
1
),
016318
(
2010
).
10.
V. V.
Lebedev
,
K. S.
Turitsyn
, and
S. S.
Vergeles
, “
Dynamics of nearly spherical vesicles in an external flow
,”
Phys. Rev. Lett.
99
,
218101
(
2007
).
11.
G.
Ma
,
J.
Hua
, and
H.
Li
, “
Numerical modeling of the behavior of an elastic capsule in a microchannel flow: The initial motion
,”
Phys. Rev. E
79
,
046710
(
2009
).
12.
D.
Cordasco
and
P.
Bagchi
, “
Orbital drift of capsules and red blood cells in shear flow
,”
Phys. Fluids
25
,
091902
(
2013
).
13.
T.
Omori
,
Y.
Imai
,
T.
Yamaguchi
, and
T.
Ishikawa
, “
Reorientation of a nonspherical capsule in creeping shear flow
,”
Phys. Rev. Lett.
108
(
13
),
138102
(
2012
).
14.
H.
Zhao
,
E. S. G.
Shaqfeh
, and
V.
Narsimhan
, “
Shear-induced particle migration and margination in a cellular suspension
,”
Phys. Fluids
24
,
011902
(
2012
).
15.
I.
Koleva
and
H.
Rehage
, “
Deformation and orientation dynamics of polysiloxane microcapsules in linear shear flow
,”
Soft Matter
8
(
13
),
3681
3693
(
2012
).
16.
Z.
Peng
and
Q.
Zhu
, “
Deformation of the erythrocyte cytoskeleton in tank treading motions
,”
Soft Matter
9
,
7617
7627
(
2013
).
17.
D.
Barthès-Biesel
, “
The time-dependent deformation of a capsule freely suspended in a linear shear flow
,”
J. Fluid Mech.
113
,
251
267
(
1981
).
18.
C.
Pozrikidis
, “
Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow
,”
J. Fluid Mech.
297
,
123
152
(
1995
).
19.
R.
Finken
,
S.
Kessler
, and
U.
Seifert
, “
Micro-capsules in shear flow
,”
J. Phys.: Condens. Matter
23
(
18
),
184113
(
2011
).
20.
S. R.
Keller
and
R.
Skalak
, “
Motion of a tank-treading ellipsoidal particle in a shear flow
,”
J. Fluid Mech.
120
,
27
47
(
1982
).
21.
J. M.
Skotheim
and
T. W.
Secomb
, “
Red blood cells and other nonspherical capsules in shear flow: Oscillatory dynamics and the tank-treading-to-tumbling transition
,”
Phys. Rev. Lett.
98
(
7
),
078301
(
2007
).
22.
M.
Abkarian
,
M.
Faivre
, and
A.
Viallat
, “
Swinging of red blood cells under shear flow
,”
Phys. Rev. Lett.
98
(
18
),
188302
(
2007
).
23.
T.
Nakajima
,
K.
Kon
,
N.
Maeda
,
K.
Tsunekawa
, and
T.
Shiga
, “
Deformation response of red blood cells in oscillatory shear flow
,”
Am. J. Physiol.
259
(
4
),
H1071
H1078
(
1990
).
24.
S.
Kessler
,
R.
Finken
, and
U.
Seifert
, “
Elastic capsules in shear flow: Analytical solutions for constant and time-dependent shear rates
,”
Eur. Phys. J. E
29
(
4
),
399
413
(
2009
).
25.
J.
Dupire
,
M.
Abkarian
, and
A.
Viallat
, “
Chaotic dynamics of red blood cells in a sinusoidal flow
,”
Phys. Rev. Lett.
104
(
16
),
168101
(
2010
).
26.
H.
Noguchi
, “
Dynamic modes of red blood cells in oscillatory shear flow
,”
Phys. Rev. E
81
(
6
),
061920
(
2010
).
27.
M.
Zhao
and
P.
Bagchi
, “
Dynamics of microcapsules in oscillating shear flow
,”
Phys. Fluids
23
(
11
),
111901
(
2011
).
28.
R.
Haddock
and
C.
Hill
, “
Rhythmicity in arterial smooth muscle
,”
J. Physiol.
566
(
3
),
645
656
(
2005
).
29.
K.
Shimamura
,
F.
Sekiguchi
, and
S.
Sunano
, “
Tension oscillation in arteries and its abnormality in hypertensive animals
,”
Clin. Exp. Pharmacol. Physiol.
26
(
4
),
275
284
(
1999
).
30.
H.
Nilsson
and
C.
Aalkjær
, “
Vasomotion: Mechanisms and physiological importance
,”
Mol. Interventions
3
(
2
),
79
(
2003
).
31.
C.
Aalkjaer
and
H.
Nilsson
, “
Vasomotion: Cellular background for the oscillator and for the synchronization of smooth muscle cells
,”
Br. J. Pharmacol.
144
(
5
),
605
616
(
2005
).
32.
C.
Meyer
,
G.
De Vries
,
S. T.
Davidge
, and
D. C.
Mayes
, “
Reassessing the mathematical modeling of the contribution of vasomotion to vascular resistance
,”
J. Appl. Physiol.
92
(
2
),
888
889
(
2002
).
33.
T. W.
Secomb
,
M.
Intaglietta
, and
J. F.
Gross
, “
Effects of vasomotion on microcirculatory mass transport
,”
Prog. Appl. Microcirc.
15
,
49
61
(
1989
).
34.
D. D.
Goldman
and
A.
Popel
, “
A computational study of the effect of vasomotion on oxygen transport from capillary networks
,”
J. Theor. Biol.
209
(
2
),
189
199
(
2001
).
35.
R.
Fåhræus
and
T.
Lindqvist
, “
The viscosity of the blood in narrow capillary tubes
,”
Am. J. Physiol.
96
(
3
),
562
568
(
1931
).
36.
S. K.
Doddi
and
P.
Bagchi
, “
Lateral migration of a capsule in a plane Poiseuille flow in a channel
,”
Int. J. Multiphase Flow
34
(
10
),
966
986
(
2008
).
37.
P.
Pranay
,
R. G.
Henríquez-Rivera
, and
M. D.
Graham
, “
Depletion layer formation in suspensions of elastic capsules in newtonian and viscoelastic fluids
,”
Phys. Fluids
24
,
061902
(
2012
).
38.
B.
Kaoui
,
G. H.
Ristow
,
I.
Cantat
,
C.
Misbah
, and
W.
Zimmermann
, “
Lateral migration of a two-dimensional vesicle in unbounded poiseuille flow
,”
Phys. Rev. E
77
(
2
),
021903
(
2008
).
39.
G.
Danker
,
P. M.
Vlahovska
, and
C.
Misbah
, “
Vesicles in poiseuille flow
,”
Phys. Rev. Lett.
102
(
14
),
148102
(
2009
).
40.
H.
Zhao
,
A. P.
Spann
, and
E. S.
Shaqfeh
, “
The dynamics of a vesicle in a wall-bound shear flow
,”
Phys. Fluids
23
(
12
),
121901
(
2011
).
41.
R. K.
Singh
,
X.
Li
, and
K.
Sarkar
, “
Lateral migration of a capsule in plane shear near a wall
,”
J. Fluid Mech.
739
,
421
443
(
2014
).
42.
S.
Nix
,
Y.
Imai
,
D.
Matsunaga
,
T.
Yamaguchi
, and
T.
Ishikawa
, “
Lateral migration of a spherical capsule near a plane wall in stokes flow
,”
Phys. Rev. E
90
(
4
),
043009
(
2014
).
43.
D.
Matsunaga
,
Y.
Imai
,
T.
Yamaguchi
, and
T.
Ishikawa
, “
Deformation of a spherical capsule under oscillating shear flow
,”
J. Fluid Mech.
762
,
288
301
(
2015
).
44.
D.
Barthès-Biesel
,
J.
Walter
, and
A.-V.
Salsac
, “
Flow-induced deformation, of artificial capsules
,” in
Computational Hydrodynamics of Capsules and Biological Cells
, edited by
C.
Pozrikidis
(
CRC Press
,
2010
).
45.
J. P.
Hernández-Ortiz
,
J. J.
de Pablo
, and
M. D.
Graham
, “
Fast computation of many-particle hydrodynamic and electrostatic interactions in a confined geometry
,”
Phys. Rev. Lett.
98
(
14
),
140602
(
2007
).
46.
P. F.
Fischer
,
J. W.
Lottes
, and
S. G.
Kerkemeier
, NEK5000 web page, 2008, http://nek5000.mcs.anl.gov.
47.
H.
Zhao
,
A. H. G.
Isfahani
,
L. N.
Olson
, and
J. B.
Freund
, “
A spectral boundary integral method for flowing blood cells
,”
J. Comput. Phys.
229
,
3726
3744
(
2010
).
48.
Q.
Huang
and
T. A.
Cruse
, “
Some notes on singular integral techniques in boundary element analysis
,”
Int. J. Numer. Methods Eng.
36
,
2643
2659
(
1993
).
49.
L.
Zhu
,
E.
Lauga
, and
L.
Brandt
, “
Low-Reynolds number swimming in a capillary tube
,”
J. Fluid Mech.
726
,
285
311
(
2013
).
50.
L.
Zhu
and
L.
Brandt
, “
The motion of a deforming capsule through a corner
,”
J. Fluid Mech.
770
,
374
397
(
2015
).
51.
L.
Zhu
, “
Simulation of individual cells in flow
,” Ph.D. dissertation (
Royal Institute of Technology
, Stockholm,
2014
).
52.
L.
Zhu
,
C.
Rorai
,
M.
Dhrubaditya
, and
L.
Brandt
, “
A microfluidic device to sort capsules by deformability: A numerical study
,”
Soft Matter
10
,
7705
7711
(
2014
).
53.
C.
Rorai
,
A.
Touchard
,
L.
Zhu
, and
L.
Brandt
, “
Motion of an elastic capsule in a constricted microchannel
,”
Eur. Phys. J. E
38
(
5
),
1
13
(
2015
).
54.
G.
d’Avino
,
P. L.
Maffettone
,
F.
Greco
, and
M. A.
Hulsen
, “
Viscoelasticity-induced migration of a rigid sphere in confined shear flow
,”
J. Non-Newtonian Fluid Mech.
165
(
9
),
466
474
(
2010
).
55.
E.
Lauga
and
T. R.
Powers
, “
The hydrodynamics of swimming microorganisms
,”
Rep. Prog. Phys.
72
,
096601
(
2009
).
56.
O. S.
Pak
,
T.
Normand
, and
E.
Lauga
, “
Pumping by flapping in a viscoelastic fluid
,”
Phys. Rev. E
81
,
036312
(
2010
).
57.
N. C.
Keim
,
M.
Garcia
, and
P. E.
Arratia
, “
Fluid elasticity can enable propulsion at low Reynolds number
,”
Phys. Fluids
24
(
8
),
081703
(
2012
).
58.
T.
Qiu
,
T.-C.
Lee
,
A. G.
Mark
,
K. I.
Morozov
,
R.
Münster
,
O.
Mierka
,
S.
Turek
,
A. M.
Leshansky
, and
P.
Fischer
, “
Swimming by reciprocal motion at low Reynolds number
,”
Nat. Commun.
5
,
11
(
2014
).
59.
E. M.
Purcell
, “
Life at low Reynolds number
,”
Am. J. Phys.
45
,
3
11
(
1977
).
60.
H.
Ma
and
M.D.
Graham
, “
Theory of shear-induced migration in dilute polymer solutions near solid boundaries
,”
Phys. Fluids
17
(
8
),
083103
(
2005
).
61.
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
).
62.
S. E.
Spagnolie
,
B.
Liu
, and
T. R.
Powers
, “
Locomotion of helical bodies in viscoelastic fluids: Enhanced swimming at large helical amplitudes
,”
Phys. Rev. Lett.
111
(
6
),
068101
(
2013
).
63.
C. H.
Wiggins
and
R. E.
Goldstein
, “
Flexive and propulsive dynamics of elastica at low Reynolds number
,”
Phys. Rev. Lett.
80
(
17
),
3879
(
1998
).
64.
S. Y.
Tony
,
E.
Lauga
, and
A. E.
Hosoi
, “
Experimental investigations of elastic tail propulsion at low Reynolds number
,”
Phys. Fluids
18
(
9
),
091701
(
2006
).
65.
R. M.
Arco
,
J. R.
Vélez-Cordero
,
E.
Lauga
, and
R.
Zenit
, “
Viscous pumping inspired by flexible propulsion
,”
Bioinspiration Biomimetics
9
(
3
),
036007
(
2014
).
66.
K. E.
Machin
, “
Wave propagation along flagella
,”
J. Exp. Biol.
35
(
4
),
796
806
(
1958
).
67.
H. M.
Shapiro
,
Practical Flow Cytometry
(
John Wiley & Sons
,
2005
).
68.
S.
Ye
,
X.
Shao
,
Z.
Yu
, and
W.
Yu
, “
Effects of the particle deformability on the critical separation diameter in the deterministic lateral displacement device
,”
J. Fluid Mech.
743
,
60
74
(
2014
).
69.
T.
Krueger
,
D.
Holmes
, and
P. V.
Coveney
, “
Deformability-based red blood cell separation in deterministic lateral displacement devices—A simulation study
,”
Biomicrofluidics
8
(
5
),
054114
(
2014
).
70.
X.
Li
and
K.
Sarkar
, “
Front tracking simulation of deformation and buckling instability of a liquid capsule enclosed by an elastic membrane
,”
J. Comput. Phys.
227
(
10
),
4998
5018
(
2008
).
71.
P. C.-H.
Chan
and
L. G.
Leal
, “
The motion of a deformable drop in a second-order fluid
,”
J. Fluid Mech.
92
(
01
),
131
170
(
1979
).
72.
M.
Shapira
and
S.
Haber
, “
Low Reynolds number motion of a droplet in shear flow including wall effects
,”
Int. J. Multiphase Flow
16
(
2
),
305
321
(
1990
).
73.
G.
Ghigliotti
,
A.
Rahimian
,
G.
Biros
, and
C.
Misbah
, “
Vesicle migration and spatial organization driven by flow line curvature
,”
Phys. Rev. Lett.
106
(
2
),
028101
(
2011
).
You do not currently have access to this content.