The Stokes flow in a cylindrical quadrant duct with a rotating wall was analytically and numerically studied. Based on mathematics and fluid dynamics theory, the analytical expressions of three velocity components were achieved by solving a Poisson's equation and a biharmonic equation. Especially, a closed-form analytical expression of axial velocity was obtained, which can greatly improve the calculating accuracy and speed in analyzing Stokes flow. The velocity distributions for different Reynolds numbers were investigated numerically to insure the accuracy of the analytical results at low Reynolds numbers and to confirm the error range of the analytic results at higher Reynolds numbers. The conclusion indicates that there exists an infinite sequence of eddies that decrease exponentially in size towards the sectorial vertex. The width of the first eddy region reached 99.4% of the sector radius; the sum of the width of other eddies is only 0.6% of the sector radius, which cannot be easily displayed graphically, while the sequence of eddies contributes to form the chaotic flow. The maximum deviations of the velocity components between the analytical results and simulated ones are all less than 1% when Re < 0.1, which verifies the validity and accuracy of the analytical expressions in the creeping flow regime. The analytical expressions are not only suitable for creeping flow but also for laminar flow with smaller Reynolds number (Re < 50).

References

1.
Galaktionov
,
O. S.
,
Meleshko
,
V. V.
,
Peters
,
G. W. M.
, and
Meijer
,
H. E. H.
,
1999
, “
Stokes Flow in a Rectangular Cavity With a Cylinder
,”
Fluid Dyn. Res.
,
24
(
2
), pp.
81
102
.10.1016/S0169-5983(98)00013-6
2.
Shankar
,
P. N.
,
Meleshko
,
V. V.
, and
Nikiforovich
,
E. I.
,
2002
, “
Slow Mixed Convection in Rectangular Containers
,”
J. Fluid Mech.
,
471
, pp.
203
217
.10.1017/S0022112002002069
3.
Ng
,
C. N.
, and
Wang
,
C. Y.
,
2010
, “
Stokes Flow Through a Periodically Grooved Tube
,”
ASME J. Fluids. Eng.
,
132
(
10
), p.
101204
.10.1115/1.4002654
4.
Gürcan
,
F
.,
2005
, “
Streamline Topologies Near a Stationary Wall of Stokes Flow in a Cavity
,”
Appl. Math. Comput.
,
165
(
2
), pp.
329
345
.10.1016/j.amc.2004.06.017
5.
Gürcan
,
F.
, and
Bilgil
,
H.
,
2013
, “
Bifurcations and Eddy Genesis of Stokes Flow Within a Sectorial Cavity
,”
Eur. J. Mech. B/Fluids
,
39
, pp.
42
51
.10.1016/j.euromechflu.2012.11.002
6.
Aref
,
H
.,
2002
, “
The Development of Chaotic Advection
,”
Phys. Fluids
,
14
(
4
), pp.
1315
1325
.10.1063/1.1458932
7.
Rodrigo
,
A. J. S.
,
Mota
,
J. P. B.
,
Lefèvre
,
A.
, and
Saatdjian
,
E.
,
2003
, “
On the Optimization of Mixing Protocol in a Certain Class of Three-Dimensional Stokes Flows
,”
Phys. Fluids
,
15
(
6
), pp.
1505
1516
.10.1063/1.1572492
8.
Rodrigo
,
A. J. S.
,
Mota
,
J. P. B.
,
Lefevre
,
A.
,
Leprevost
,
J. C.
, and
Saatdjian
,
E.
,
2003
, “
Chaotic Advection in a Three-Dimensional Stokes Flow
,”
AIChE J.
,
49
(
11
), pp.
2749
2758
.10.1002/aic.690491107
9.
Lefevre
,
A.
,
Mota
,
J. P. B.
,
Rodrigo
,
A. J. S.
, and
Saatdjian
,
E.
,
2003
, “
Chaotic Advection and Heat Transfer Enhancement in Stokes Flows
,”
Int. J. Heat Fluid Flow
,
24
(
3
), pp.
310
321
.10.1016/S0142-727X(03)00022-5
10.
Saatdjian
,
E.
,
Rodrigo
,
A. J. S.
, and
Mota
,
J. P. B.
,
2011
, “
Stokes Flow Heat Transfer in an Annular, Rotating Heat Exchanger
,”
Appl. Therm. Eng.
,
31
(
8
), pp.
1499
1507
.10.1016/j.applthermaleng.2011.01.037
11.
Khakhar
,
D. V.
,
Franjione
,
J. G.
, and
Ottino
,
J. M.
,
1987
, “
A Case Study of Chaotic Mixing in Deterministic Flows: The Partitioned-Pipe Mixer
,”
Chem. Eng. Sci.,
42
(12)
, pp. 2909–2926.
12.
Mizuno
,
Y.
, and
Funakoshi
,
M.
,
2005
, “
Reynolds Number Dependences of Velocity Field and Fluid Mixing in Partitioned-Pipe Mixer
,”
J. Phys. Soc. Jpn.
,
74
(
5
), pp.
1479
1489
.10.1143/JPSJ.74.1479
13.
Ling
,
F. H.
,
1993
, “
Chaotic Mixing in a Spatially Periodic Continuous Mixer
,”
Phys. Fluids A: Fluid Dyn.
,
5
(
9
), pp.
2147
2160
.10.1063/1.858554
14.
Mizuno
,
Y.
, and
Funakoshi
,
M.
,
2002
, “
Chaotic Mixing Due to a Spatially Periodic Three-Dimensional Flow
,”
Fluid Dyn. Res.
,
31
(
2
), pp.
129
149
.10.1016/0009-2509(87)87056-2
15.
Meleshko
,
V. V.
,
Galaktionov
,
O. S.
,
Peters
,
G. W. M.
, and
Meijer
,
H. E. H.
,
1999
, “
Three-Dimensional Mixing in Stokes Flow: The Partitioned Pipe Mixer Problem Revisited
,”
Eur. J. Mech. B/Fluids
,
18
(
5
), pp.
783
792
.10.1016/S0997-7546(99)00120-X
16.
Kusch
,
H. A.
, and
Ottino
,
J. M.
,
1992
, “
Experiments on Mixing in Continuous Chaotic Flows
,”
J. Fluid Mech.
,
236
, pp.
319
348
.10.1017/S0022112092001435
17.
Mizuno
,
Y.
, and
Funakoshi
,
M.
,
2004
, “
Chaotic Mixing Caused by an Axially Periodic Steady Flow in a Partitioned-Pipe Mixer
,”
Fluid Dyn. Res.
,
35
(
3
), pp.
205
227
.10.1016/j.fluiddyn.2004.05.003
18.
Russell
,
R. D.
, and
Shampine
,
L. F.
,
1972
, “
A Collocation Method for Boundary Value Problems
,”
Numer. Math.
,
19
(
1
), pp.
1
28
.10.1007/BF01395926
19.
Prudnikov
,
A. P.
,
Brychkov
,
Y. A.
, and
Marichev
,
O. I.
,
1986
,
Integrals and Series
, Vol.
1
,
Gordon and Breach
,
London
.
20.
Moffatt
,
H. K.
and
Duffy
,
B. R.
,
1980
, “
Local Similarity Solutions and Their Limitations
,”
J. Fluid Mech.
,
96
, pp.
299
313
.10.1017/S0022112080002133
21.
Hancock
,
C
.,
1984
, “
The Angle of Separation in Stokes Flow Near a Sharp Corner
,”
Q. J. Mech. Appl. Math.
,
37
(
1
), pp.
113
119
.10.1093/qjmam/37.1.113
You do not currently have access to this content.