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Research Papers

Hamiltonian Chaos in a Model Alveolus

[+] Author and Article Information
F. S. Henry

 Molecular and Integrative Physiological Sciences, Harvard School of Public Health, Boston, MA 02115

F. E. Laine-Pearson

Department of Mathematics, University of Surrey, Guildford, UK

A. Tsuda1

 Molecular and Integrative Physiological Sciences, Harvard School of Public Health, Boston, MA 02115atsuda@hsph.harvard.edu

1

Corresponding author.

J Biomech Eng 131(1), 011006 (Nov 19, 2008) (7 pages) doi:10.1115/1.2953559 History: Received June 15, 2007; Revised September 30, 2007; Published November 19, 2008

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.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of model alveolated duct (upper panel) and details of typical solution cell (lower panel). a=duct radius, h=cavity depth, and w=cavity width.

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Figure 2

Streamlines for steady Stokes flow (broken line) and steady flow at Re=1.0 (full line)

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Figure 3

A typical fluid particle path over one cycle for a flow with Re=1.0 and α=0.096(T=3s). (a) Position of the path in the alveolus. (b) An expanded view of the particle path.

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Figure 4

Poincaré sections for a flow with Re=1.0 and α=0.096(T=3s). (a) Full view of all orbits computed. (b) An expanded view of the orbits with islands and their neighbors. Colors have no significance other than to differentiate between individual orbits.

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Figure 5

Poincaré section orbits with islands for three flows with Re=1.0 and three values of α. Red dots: α=0.118(T=2s). Light and dark blue dots: α=0.096(T=3s). Green dots: α=0.083(T=4s). Colors have no significance other than to differentiate between individual orbits.

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Figure 6

Poincaré section orbits with islands for three flows with α=0.096(T=3s) and three values of Re. Yellow dots: Re=0.9. Light and dark blue dots: Re=1.0. Purple dots: Re=1.1. Colors have no significance other than to differentiate between individual orbits.

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Figure 7

Poincaré section for unsteady Stokes flow with α=0.096(T=3s). Colors have no significance other than to differentiate between individual orbits.

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Figure 8

Time history of the exponent σn for two fluid particle pairs. The initial separation, d0=10−12; see text for equation and further details. One pair are located in the lower lobe (specifically, the fifth orbit from top right in Fig. 4) and the other pair in the upper lobe (specifically, the second orbit from top right in Fig. 4) of the Poincaré section given in Fig. 4. Upper panel: Breathing Cycles 0–10. Lower panel: Breathing Cycles 90–100.

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