A numerical investigation was carried out to study the mixing behavior of Stokes flows in a rectangular cavity stirred by three square rods. The square loops of the rods move in such a way that a pseudo-Anosov map can be built in the flow domain in the augmented phase space. The finite volume method was used, and the flow domain was meshed by staggered grids with the periodic boundary conditions of the rod motion being imposed by the mesh supposition technique. Fluid particle tracking was carried out by a fourth-order Runge–Kutta scheme. Tracer stretches from different initial positions were used to evaluate interface prediction by a pseudo-Anosov map. The colored short period Poincaré section was obtained to reveal the size of the domain in which the pseudo-Anosov map was in effect. Dye advection patterns were used to analyze chaotic advection of passive tracer particles using statistical concepts such as “variances” and “complete spatial randomness.” For the fluid in the core region of the cavity, tracer interface stretches experienced exponential increases and had the same power index as that predicted by the pseudo-Anosov map matrix.

References

References
1.
Aref
,
H.
,
1984
, “
Stirring by Chaotic Advection
,”
J. Fluid Mech.
,
143
, pp.
1
21
.10.1017/S0022112084001233
2.
Aref
,
H.
,
2002
, “
The Development of Chaotic Advection
,”
Phys. Fluids
,
14
, pp.
1315
1325
.10.1063/1.1458932
3.
Finn
,
M. D.
,
Cox
,
S. M.
, and
Byrne
,
H. M.
,
2003
, “
Topological Chaos in Inviscid and Viscous Mixers
,”
J. Fluid Mech.
,
493
, pp.
345
361
.10.1017/S0022112003005858
4.
Jana
,
S. C.
,
Metcalfe
,
G.
, and
Ottino
,
J. M.
,
1994
, “
Experimental and Computational Studies of Mixing in Complex Stokes Flows: The Vortex Mixing Flow and Multicellular Cavity Flows
,”
J. Fluid Mech.
,
269
, pp.
199
246
.10.1017/S0022112094001539
5.
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
6.
Meleshko
,
V. V.
,
Galaktionov
,
O. S.
,
Peters
,
G. W. M.
, and
Meijer
,
H. E. H.
,
1999
, “
Three Dimensional Mixing in Stokes Flows: The Partitioned Pipe Mixer Problem Revised
,”
Eur. J. Mech. Fluids
,
18
, pp.
783
792
.10.1016/S0997-7546(99)00120-X
7.
Sivasamy
,
J.
,
Che
,
Z.
,
Wong
,
T. N.
,
Nguyen
,
N.-T.
, and
Yobas
,
L.
,
2010
, “
A Simple Method for Evaluating and Predicting Chaotic Advection in Microfluidic Slugs
,”
Chem. Eng. Sci.
65
, pp.
5382
5391
.10.1016/j.ces.2010.06.017
8.
Ottino
,
J. M.
,
1989
,
The Kinematics of Mixing: Stretching, Chaos, and Transport
,
1st ed.
,
Cambridge University
,
Cambridge, UK
.
9.
Vikhansky
,
A.
,
2002
, “
Enhancement of Laminar Mixing by Optimal Control Methods
,”
Chem. Eng. Sci.
,
57
, pp.
2719
2725
.10.1016/S0009-2509(02)00122-7
10.
Sturman
,
R.
,
Ottino
,
J. M.
, and
Wiggins
,
S.
,
2006
,
The Mathematical Foundations of Mixing
,
1st ed.
,
Cambridge University
,
Cambridge, UK
.
11.
Boyland
,
P. L.
,
Aref
,
H.
, and
Stremler
,
M. A.
,
2000
, “
Topological Fluid Mechanics of Stirring
,”
J. Fluid Mech.
,
403
, pp.
277
304
.10.1017/S0022112099007107
12.
Boyland
,
P. L.
,
Stremler
,
M. A.
, and
Aref
,
H.
,
2003
, “
Topological Fluid Mechanics of Point Vortex Motions
,”
Physica D
,
175
, pp.
69
95
.10.1016/S0167-2789(02)00692-9
13.
Thurston
,
W.
,
1988
, “
On the Geometry and Dynamics of Diffeomorphisms of Surfaces
,”
Bull. Ser., Am. Math. Soc.
,
19
, pp.
417
431
.10.1090/S0273-0979-1988-15685-6
14.
Vikhansky
,
A.
,
2003
, “
Simulation of Topological Chaos in Laminar Flows
,”
Chaos
,
14
(
1
), pp.
14
22
.10.1063/1.1621092
15.
Clifford
,
M. J.
, and
Cox
,
S. M.
,
2006
, “
Smart Baffle Placement for Chaotic Mixing
,”
Nonlinear Dyn.
,
43
, pp.
117
126
.10.1007/s11071-006-0755-9
16.
Finn
,
M. D.
,
Thiffeault
,
J.-L.
, and
Gouillart
,
E.
,
2006
, “
Topological Chaos in Spatially Periodic Mixers
,”
Physica D
,
221
, pp.
92
100
.10.1016/j.physd.2006.07.018
17.
Stremler
,
M. A.
, and
Chen
,
J.
,
2007
, “
Generating Topological Chaos in Lid-Driven Cavity Flow
,”
Phys. Fluids
,
19
, pp.
1
6
.10.1063/1.2772881
18.
Kang
,
T. G.
, and
Kwon
,
T. H.
,
2004
, “
Colored Particle Tracking Method for Mixing Analysis of Chaotic Micromixer
,”
J. Micromec. Microeng.
,
14
, pp.
891
899
.10.1088/0960-1317/14/7/008
19.
Phelps
,
J. H. L.
, and
Tucker
,
C.
, III
,
2006
, “
Lagrangian Particle Calculations of Distributive Mixing: Limitations and Applications
,”
Chem. Eng. Sci.
,
61
, pp.
6826
6836
.10.1016/j.ces.2006.07.008
You do not currently have access to this content.