Flow limitation in collapsible tubes
The velocity of gas molecules (v, cm/s) and the cross-sectional area (A, cm²) determine the volume flow passing through a tube per unit of time (mL/s):
flow = velocity·area = v·A
Area: In a collapsible tube the walls approach each other as outside pressure increases relative to inside pressure; the cross-sectional area is therefore a function of the pressure difference across the wall and the elastic properties of that wall. On that account resistance to flow increases, and pressure increases upstream from the narrowed airway segment. In spite of increasing driving pressure the speed of gas flowing through a narrow tube never exceeds the speed (see footnote) with which a pressure wave propagates through the wall of the tube (wave-speed limitation). This is akin to the fact that a loud sound does not travel faster than a soft sound. Similarly, if a stone is thrown into the water with great force, the resulting pressure wave does not travel any faster than when little force is applied. It can also be likened to a waterfall: whatever the height of the waterfall and whatever the flow in the upstream river, the speed of the free falling water is the same; however, the wider the waterfall, the more water is displaced.
Propagation of a pressure wave in collapsible tubes
- Griffiths DJ. Hydrodynamics of male micturition. I. Theory of steady flow through elastic-walled tubes. Med Biol Eng 1971; 9: 581-588.
- Pedersen OF, Nielsen TM. The critical transmural pressure of the airway. Acta Physiol Scand 1976; 97: 426-446.
- Dawson SV, Elliott EA. Wave-speed limitation on expiratory flow – a unifying concept. J Appl Physiol 1977; 43: 498-515.
Wave speed in collapsible tubes
In a collapsible tube, the cross-sectional
area of the tube (A) determines wave speed, how this area changes
with the pressure difference across its wall (dA/dP), and by gas
density (r):
v = square root [A·(dA/dP)/r)].
In this equation dA/dP is a measure of the elastic properties of the tube.