The slow motion of a circular cylinder in a plane Poiseuille flow in a microchannel is analyzed for a wide range of cylinder radii and positions across the channel. The cylinder translates parallel to the channel walls and rotates about its axis. The Stokes approximation is used and the problem is solved analytically using the Papkovich-Fadle eigenfunction expansion and the least-squares method. The stream function and the pressure distribution of the flow field are obtained as results. The force and moment exerted on the cylinder, and the pressure change far from the cylinder, are calculated and shown as functions of the size and location of the cylinder. The results confirm some reciprocal relations exactly. In particular, the translational and rotational velocities of the drifting cylinder in the existing Poiseuille flow are determined. The induced pressure change, when the cylinder drifts in the Poiseuille flow, is also calculated. Some typical streamline patterns, depending on the size and location of the cylinder, are shown and discussed. When the cylinder translates and/or rotates in the channel blocked at infinity, a series of Moffatt eddies appears far from the cylinder in the channel, as expected.

1.
N. T.
Nguyen
and
S. T.
Wereley
,
Fundamentals and Applications of Microfluidics
, 2nd ed. (
Artech House Inc.
,
2006
).
2.
M.
Gad-el-Hak
,
The MEMS Handbook MEMS: Introduction and Fundamentals
, 2nd ed. (
CRC Press Taylor & Francis Group
,
2006
).
3.
M.
Bahramia
,
M. M.
Yovanovich
, and
J. R.
Culham
, “
Pressure drop of fully developed laminar flow in microchannels of arbitrary cross-section
,”
J. Fluids Eng.-Trans. ASME
128
,
1036
1044
(
2006
).
4.
A.
Petropoulos
,
G.
Kaltsas
,
D.
Randjelovic
, and
E.
Gogolides
, “
Study of flow and pressure field in micro-channels with various cross-section areas
,”
Microelectron. Eng.
87
,
827
829
(
2010
).
5.
E.
Buyruk
,
M. W.
Johnson
, and
I.
Owen
, “
Numerical and experimental study of flow and heat transfer around a tube in cross flow at low Reynolds number
,”
Int. J. Heat Fluid Flow
19
,
223
232
(
1998
).
6.
A.
Golpaygan
and
N.
Ashgriz
, “
Multiphase flow model to study channel flow dynamics of PEM fuel cells: Deformation and detachment of water droplets
,”
Int. J. Comput. Fluid Dyn.
22
,
85
95
(
2008
).
7.
A.
Armellini
,
L.
Casarsa
, and
P.
Giannattasio
, “
Separated flow structures around a cylindrical obstacle in a narrow channel
,”
Exp. Therm. Fluid Sci.
33
,
604
619
(
2009
).
8.
A. S.
Dvinsky
and
A. S.
Popel
, “
Motion of a rigid cylinder between parallel plates in Stokes flow, Part 1: Motion in a quiescent fluid and sedimentation
,”
Comput. Fluids
15
,
391
404
(
1987
).
9.
A. S.
Dvinsky
and
A. S.
Popel
, “
Motion of a rigid cylinder between parallel plates in Stokes flow, Part 2: Poiseuille and Couette flow
,”
Comput. Fluids
15
,
405
419
(
1987
).
10.
R. F.
Day
and
H. A.
Stone
, “
Lubrication analysis and boundary integral simulations of a viscous micropump
,”
J. Fluid Mech.
416
,
197
216
(
2000
).
11.
S.
Champmartin
and
A.
Ambari
, “
Kinematics of symmetrically confined cylindrical particle in a ‘Stokes-type’ regime
,”
Phys. Fluids
19
,
073303
(
2007
).
12.
S.
Champmartin
,
A.
Ambari
, and
N.
Roussel
, “
Flow around confined rotating cylinder at small Reynolds number
,”
Phys. Fluids
19
,
103101
(
2007
).
13.
J.
Yang
,
G.
Huber
, and
C. W.
Wolgemuth
, “
Forces and torques on rotating spirochete flagella
,”
Phys. Rev. Lett.
107
,
268101
(
2011
).
14.
J.
Yang
,
C. W.
Wolgemuth
, and
G.
Huber
, “
Forces and torques on a cylinder rotating in a narrow gap at low Reynolds number: Scaling and lubrication analyses
,”
Phys. Fluids
25
,
051901
(
2013
).
15.
I. O.
Götze
and
G.
Gompper
, “
Dynamic self-assembly and directed flow of rotating colloids in microchannels
,”
Phys. Rev. E
84
,
031404
(
2011
).
16.
M. E.
Staben
,
A. Z.
Zinchenko
, and
R. H.
Davis
, “
Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow
,”
Phys. Fluids
15
,
1711
1733
(
2003
).
17.
O. H.
Faxén
, “
Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes
,”
Proc. R. Swed. Inst. Eng. Res. (Stockholm)
187
,
1
13
(
1946
).
18.
Y.
Takaisi
, “
The drag on a circular cylinder moving with low speeds in a viscous liquid between two parallel walls
,”
J. Phys. Soc. Jpn.
10
,
685
693
(
1955
).
19.
Y.
Takaisi
, “
The drag on a circular cylinder placed in a stream of viscous liquid midway between two parallel planes
,”
J. Phys. Soc. Jpn.
11
,
1092
1095
(
1956
).
20.
C. Y.
Wang
, “
Stokes flow through a channel obstructed by horizontal cylinders
,”
Acta Mech.
157
,
213
221
(
2002
).
21.
S.-H.
Yoon
and
J.-T.
Jeong
, “
Stokes flow through a microchannel obstructed by a vertical plate
,”
Eur. J. Mech. B Fluids
34
,
64
69
(
2012
).
22.
J.-T.
Jeong
and
S.-H.
Yoon
, “
Two-dimensional Stokes flow around a circular cylinder in a microchannel
,”
J. Mech. Sci. Technol.
28
,
573
579
(
2014
).
23.
J.
Happel
and
H.
Brenner
,
Low Reynolds Number Hydrodynamics
(
Prentice-Hall, Inc.
,
Englewood Cliffs, NJ
,
1965
).
24.
A. M. J.
Davis
, “
Periodic blocking in parallel shear or channel flow at low Reynolds number
,”
Phys. Fluids A
5
,
800
809
(
1993
).
25.
C. Y.
Wang
, “
Stokes flow through a transversely finned channel
,”
J. Fluids Eng.-Trans. ASME
119
,
110
114
(
1997
).
26.
J.-T.
Jeong
, “
Two-dimensional Stokes flow through a slit in a microchannel with slip
,”
J. Phys. Soc. Jpn.
75
,
094401
(
2006
).
27.
P. N.
Shankar
,
Slow Viscous Flows
(
Imperial College Press
,
London
,
2007
).
28.
S. D.
Conte
and
C.
de Boor
,
Elementary Numerical Analysis
(
McGraw-Hill Kogakusa Ltd.
,
Tokyo
,
1980
).
29.
E. Y.
Harper
and
I.-D.
Chang
, “
Drag on a cylinder between parallel walls in Stokes’ flow
,”
Phys. Fluids
10
,
83
88
(
1967
).
30.
H. K.
Moffatt
, “
Viscous and resistive eddies near a sharp corner
,”
J. Fluid Mech.
18
,
1
18
(
1964
).
You do not currently have access to this content.