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Wave‐Speed and Viscous Flow Limitation

Theodore A. Wilson, Joseph R. Rodarte, James P. Butler

10.1002/cphy.cp030305

Source: Supplement 12: Handbook of Physiology, The Respiratory System, Mechanics of Breathing

Originally published: 1986

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Wave‐Speed Flow‐Limiting Mechanism
2 Viscous Flow Limitation
3 Relation Between Wave‐Speed and Viscous Flow‐Limiting Mechanisms
4 Comments on Modeling
Figure 1. Figure 1.

A compliant tube forms the exit nozzle from a large chamber. Area‐pressure behavior of tube is a function of axial position (x). Compressibility of the gas is neglected and absolute pressure is not significant. The pressure in the chamber (P0) and the pressure at the exit of the nozzle, P(x = 2), are pressures relative to the pressure applied to the outside of the tube. The pressure outside the tube is analogous to pleural pressure, chamber pressure is analogous to recoil pressure of the lung, and exit pressure is analogous to pressure at the mouth. Lowering the exit pressure and holding chamber pressure constant in this system is therefore analogous to increasing pleural pressure and holding recoil pressure constant.
Figure 2. Figure 2.

Area‐pressure behavior of compliant tube as a function of axial position.
Figure 3. Figure 3.

Area‐pressure curves of tube with curves showing area‐pressure relation in the flow calculated from Bernoulli equation for 3 different volume flow rates. For each flow rate the area and pressure that occur at each position are given by the intersection of flow and tube curves.
Figure 4. Figure 4.

Pressure (top) and area (bottom) as functions of axial position in compliant tube for each flow rate. For limiting flow () 2 possible branches of the solution are shown downstream of the critical point. Along lower branch a region of supercritical flow ends with a dissipative jump.
Figure 5. Figure 5.

Flow separation occurs downstream of an area minimum. Energy is dissipated in mixing process that reestablishes flow throughout tube cross section. Momentum balance for fluid within the control volume (dashed lines) determines the jump in pressure and area that occurs across mixing region.
Figure 6. Figure 6.

Relation between wave‐speed and viscous flow‐limiting mechanisms. In geometrically similar tubes with the same specific compliance (A), the flow‐pressure behavior in the larger tube (B) is different from the flow‐pressure characteristics of the smaller tube (C). In the larger tube, wave‐speed mechanism limits flow. In the smaller tube, viscous dissipation is relatively more important; for an upstream pressure of 1 cmH2O, flow is limited by viscous dissipation.


Figure 1.

A compliant tube forms the exit nozzle from a large chamber. Area‐pressure behavior of tube is a function of axial position (x). Compressibility of the gas is neglected and absolute pressure is not significant. The pressure in the chamber (P0) and the pressure at the exit of the nozzle, P(x = 2), are pressures relative to the pressure applied to the outside of the tube. The pressure outside the tube is analogous to pleural pressure, chamber pressure is analogous to recoil pressure of the lung, and exit pressure is analogous to pressure at the mouth. Lowering the exit pressure and holding chamber pressure constant in this system is therefore analogous to increasing pleural pressure and holding recoil pressure constant.



Figure 2.

Area‐pressure behavior of compliant tube as a function of axial position.



Figure 3.

Area‐pressure curves of tube with curves showing area‐pressure relation in the flow calculated from Bernoulli equation for 3 different volume flow rates. For each flow rate the area and pressure that occur at each position are given by the intersection of flow and tube curves.



Figure 4.

Pressure (top) and area (bottom) as functions of axial position in compliant tube for each flow rate. For limiting flow () 2 possible branches of the solution are shown downstream of the critical point. Along lower branch a region of supercritical flow ends with a dissipative jump.



Figure 5.

Flow separation occurs downstream of an area minimum. Energy is dissipated in mixing process that reestablishes flow throughout tube cross section. Momentum balance for fluid within the control volume (dashed lines) determines the jump in pressure and area that occurs across mixing region.



Figure 6.

Relation between wave‐speed and viscous flow‐limiting mechanisms. In geometrically similar tubes with the same specific compliance (A), the flow‐pressure behavior in the larger tube (B) is different from the flow‐pressure characteristics of the smaller tube (C). In the larger tube, wave‐speed mechanism limits flow. In the smaller tube, viscous dissipation is relatively more important; for an upstream pressure of 1 cmH2O, flow is limited by viscous dissipation.

References
 1. Dawson, S. V., and E. A. Elliot. Wave‐speed limitation on expiratory flow—a unifying concept. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 43: 498–515, 1977.
 2. Fry, D. L., and R. E. Hyatt. Pulmonary mechanics: a unified analysis of the relationship between pressure, volume, and gas flow in the lungs of normal and diseased human subjects. Am. J. Med. 29: 672–689, 1960.
 3. Griffiths, D. J. Hydrodynamics of male micturition. I. Theory of steady flow through elastic‐walled tubes. Med. Biol. Eng. 9: 581–588, 1971.
 4. Pride, N. B., S. Permutt, R. L. Riley, and B. Bromberger‐Barnea. Determinants of maximal expiratory flow from the lungs. J. Appl. Physiol. 23: 646–662, 1967.
 5. Shapiro, A. H. Steady flow in collapsible tubes. J. Biomech. Eng. 99: 126–147, 1977.

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Article Title: Wave‐Speed and Viscous Flow Limitation
 

How to Cite

Theodore A. Wilson, Joseph R. Rodarte, James P. Butler. Wave‐Speed and Viscous Flow Limitation. Compr Physiol 2011, Supplement 12: Handbook of Physiology, The Respiratory System, Mechanics of Breathing: 55-61. First published in print 1986. doi: 10.1002/cphy.cp030305