In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

1.
Tsuda
,
A.
,
Henry
,
F. S.
, and
Butler
,
J. P.
, 1995, “
Chaotic Mixing of Alveolated Duct Flow in Rhythmically Expanding Pulmonary Acinus
,”
J. Appl. Physiol.
8750-7587,
79
(
3
), pp.
1055
1063
.
2.
Tsuda
,
A.
,
Rogers
,
R. A.
,
Hydon
,
P. E.
, and
Butler
,
J. P.
, 2002, “
Chaotic Mixing Deep in the Lung
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
99
, pp.
10173
10178
.
3.
Butler
,
J. P.
, and
Tsuda
,
A.
, 1997, “
Contribution of Convective Stretching and Folding to Aerosol Mixing Deep in the Lung, Assessed by Approximate entropy
,”
J. Appl. Physiol.
8750-7587,
83
(
3
), pp.
800
809
.
4.
Federspiel
,
W. J.
, and
Fredberg
,
J. J.
, 1988, “
Axial Dispersion in Respiratory Bronchioles and Alveolar Ducts
,”
J. Appl. Physiol.
8750-7587,
64
(
6
), pp.
2614
2621
.
5.
Tsuda
,
A.
,
Federspiel
,
W. J.
,
Grant
,
P. A.
, Jr.
, and
Fredberg
,
J. J.
, 1991, “
Axial Dispersion of Inert Species in Alveolated Channels
,”
Chem. Eng. Sci.
0009-2509,
46
(
5/6
),
1419
1426
.
6.
Tsuda
,
A.
,
Butler
,
J. P.
, and
Fredberg
,
J. J.
, 1994, “
Effects of Alveolated Duct Structure on Aerosol Kinetics. Part I. Diffusional Deposition in the Absence of Gravity
,”
J. Appl. Physiol.
8750-7587,
76
(
6
), pp.
2497
2509
.
7.
Darquenne
,
C.
, and
Paiva
,
M.
, 1996, “
Two- and Three-Dimensional Simulations of Aerosol Transport and Deposition in Alveolar Zone of Human Lung
,”
J. Appl. Physiol.
8750-7587,
80
, pp.
1401
1414
.
8.
Tippe
,
A.
, and
Tsuda
,
A.
, 2000, “
Recirculating Flow in an Expanding Alveolar Model: Experimental Evidence of Flow-Induced Mixing of Aerosols in the Pulmonary Acinus
,”
J. Fluid Mech.
0022-1120,
31
(
8
), pp.
979
986
.
9.
Davidson
,
M. R.
, and
Fritz-Gerald
,
J. M.
, 1972, “
Flow patterns in Models of Small Airway Units of the Lung
,”
J. Fluid Mech.
0022-1120,
52
, pp.
161
177
.
10.
Haber
,
S.
,
Butler
,
J. P.
,
Brenner
,
H.
,
Emanuel
,
I.
, and
Tsuda
,
A.
, 2000, “
Flow Field in Self-Similar Expansion on a Pulmonary Alveolus During Rhythmical Breathing
,”
J. Fluid Mech.
0022-1120,
405
, pp.
243
268
.
11.
Lee
,
D. Y.
, and
Lee
,
J. W.
, 2003, “
Characteristics of Particle Transport in an Expanding or Contracting Alveolated Tube
,”
J. Aerosol Sci.
0021-8502,
34
, pp.
1193
1215
.
12.
Haber
,
S.
, and
Tsuda
,
A.
, 2006, “
Cyclic Model for Particle Motion in the Pulmonary Acinus
,”
J. Fluid Mech.
0022-1120,
567
, pp.
157
184
.
13.
Taylor
,
G. I.
, 1960, Low Reynolds Number Flow (16mm film), Newton, MA, Educational Services Inc.
14.
Davies
,
C. N.
, 1972, “
Breathing of Half-Micron Aerosols. II. Interpretation of Experimental Results
,”
J. Appl. Physiol.
0021-8987,
32
, pp.
601
611
.
15.
Tsuda
,
A.
,
Otani
,
Y.
, and
Butler
,
J. P.
, 1999, “
Acinar Flow Irreversibility Caused by Boundary Perturbation of Reversible Alveolar Wall Motion
,”
J. Appl. Physiol.
8750-7587,
86
(
3
), pp.
977
984
.
16.
Heyder
,
J.
,
Blanchard
,
J. D.
,
Feldman
,
H. A.
, and
Brain
,
J. D.
, 1988, “
Convective Mixing in Human Respiratory Tract: Estimates With Aerosol Boli
,”
J. Appl. Physiol.
8750-7587,
64
, pp.
1273
1278
.
17.
Brand
,
P.
,
Rieger
,
C.
,
Shultz
,
H.
,
Beinert
,
T.
, and
Heyder
,
J.
, 1997, “
Aerosol Bolus Dispersion in Healthy Subjects
,”
Eur. Respir. J.
0903-1936,
10
, pp.
460
467
.
18.
Darquenne
,
C.
,
Paiva
,
M.
,
West
,
J. B.
, and
Prisk
,
G. K.
, 1997, “
Effect of Microgravity and Hypergravity on Deposition of 0.5to3-μm Diameter Aerosol in the Human Lung
,”
J. Appl. Physiol.
8750-7587,
83
, pp.
2029
2036
.
19.
Haber
,
S.
,
Yitzhak
,
D.
, and
Tsuda
,
A.
, 2003, “
Gravitational Deposition in a Rhythmically Expanding and Contracting Alveolus
,”
J. Appl. Physiol.
8750-7587,
95
, pp.
657
671
.
20.
Henry
,
F. S.
,
Butler
,
J. P.
, and
Tsuda
,
A.
, 2002, “
Kinematically Irreversible Flow and Aerosol Transport in the Pulmonary Acinus: A Departure From Classical Dispersive Transport
,”
J. Appl. Physiol.
8750-7587,
92
, pp.
835
845
.
21.
Karl
,
A.
,
Henry
,
F. S.
, and
Tsuda
,
A.
, 2004, “
Low Reynolds Number Viscous Flow in an Alveolated Duct
,”
ASME J. Biomech. Eng.
0148-0731,
126
, pp.
420
429
.
22.
Tabor
,
M.
, 1989,
Chaos and Integrability in Nonlinear Dynamics: An Introduction
,
Wiley
,
New York
.
23.
Lichtenberg
,
A. J.
, and
Lieberman
,
M. A.
, 1992,
Regular and Chaotic Dynamics
,
2nd ed.
,
Springer-Verlag
,
Berlin
.
24.
Aref
,
H.
, 1990, “
Chaotic Advection of Fluid Particles
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
333
, pp.
273
288
.
26.
Cartwright
,
J. H. E.
, and
Piro
,
O.
, 1992, “
The Dynamics of Runge–Kutta Methods
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274,
2
, pp.
427
449
.
27.
Moffatt
,
H. K.
, 1964, “
Viscous and Resistive Eddies Near a Sharp Corner
,”
J. Fluid Mech.
0022-1120,
18
, pp.
1
18
.
28.
Ottino
,
J. M.
, 1989,
The Kinematics of Mixing: Stretching, Chaos, and Transport
,
Cambridge University Press
,
Cambridge
.
29.
Laine-Pearson
,
F. E.
, and
Hydon
,
P. E.
, 2006, “
Particle Transport in a Moving Corner
,”
J. Fluid Mech.
0022-1120,
559
, pp.
379
390
.
30.
Zaslavsky
,
G. M.
,
Sagdeev
,
R. Z.
,
Usikov
,
D. A.
, and
Chernikov
,
A. A.
, 1991,
Weak Chaos and Quasi-Regular Patterns
,
Cambridge University Press
,
Cambridge
.
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