## Abstract

The motivation for this subject comes from physiology: Air-flow in the lungs, where flow limitation during forced expiration is a consequence of large-airway collapse, and wheezing, which is a manifestation of self-excited mechanical oscillations; Blood flow in veins, such as those of giraffes, in which the return of blood to the heart from the head must be accompanied by partial venous collapse, and in arteries, which exhibit self-excited oscillations (Korotkov sounds) when compressed by a blood-pressure cuff. Laboratory experiments are frequently conducted in a Starling resistor, a finite length of flexible tube, mounted between two rigid tubes and contained in a pressurised chamber. Steady conditions upstream and downstream give rise not only to steady flows, but also to a rich variety of self-excited oscillations, which theoreticians have been seeking to understand for at least five decades. Some of the observations have been reproduced in full Navier–Stokes computations for a two-dimensional model, but these do not provide physical understanding. We seek a self-consistent mathematical model for the oscillations. We concentrate first on 1D models, in which the key dependent variables are the cross-sectional area *A* and the cross-sectionally averaged velocity *u* and pressure *p*, all taken to be functions of longitudinal coordinate *x* and time *t*. The governing equations are those of conservation of mass and momentum and a tube law representing the elastic properties of the vessel. In the momentum equation, the viscous resistance term is conventionally modelled either as a linear function of fluid velocity, accurate at low Reynolds number, or with an ad hoc representation of the energy loss at flow separation. Even with such crude approximations, the predictions of 1D models agree quite well both with observations in the giraffe and with some of the 2D computations and 3D experiments. For a more rational model, we examine a 2D model problem, in which part of one wall of a parallel sided channel is replaced by a membrane under tension. One approach, for large Reynolds-number flow, and a long membrane, is to consider small deflections of the membrane and use interactive boundary-layer theory. This leads to interesting predictions, such as the impossibility of simultaneously prescribing the flow rate and the upstream pressure, but not to oscillations, except in cases where wall inertia is important (flutter). Another approach is to assume a parabolic velocity profile everywhere, leading to a rational choice for the inertia and viscous terms in the 1D momentum equation. If, further, the undisturbed membrane is taken to be flat, by a suitable choice of external pressure distribution, the system leads to an oscillatory instability even without wall inertia. Whether these oscillations have the same physics as those computed numerically at lower Reynolds number remains to be seen.

This is a preview of subscription content, access via your institution.

## References

Bertram C D 1986 Unstable equilibrium behaviour in collapsible tubes.

*J. Biomech.*19: 61–69Bertram C D, Raymond C J and Pedley T J 1991 Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a flow.

*J. Fluids Struct.*5: 391–426Brook B S, Falle S A E G and Pedley T J 1999 Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state.

*J. Fluid Mech.*396: 223–256Cancelli C and Pedley T J 1985 A separated flow model for collapsible tube oscillations.

*J. Fluid Mech.*157: 375–404Gavriely N, Shee T R, Cugall D W and Grotberg J B 1989 Flutter in flow-limited collapsible tubes: A mechanism for generation of wheezes.

*J. Appl. Physiol.*66: 2251–2261Guneratne J and Pedley T J 2006 High-Reynolds-number steady flow in a collapsible channel.

*J. Fluid Mech.*569: 151–184Hargens A R, Millard R W, Pettersson K and Johansen K 1987 Gravitational haemodynamics and oedema prevention in the giraffe.

*Nature*329: 59–60Hicks J W and Badeer H S 1989 Siphon mechanism in collapsible tubes: application to circulation in the giraffe head.

*Am. J. Physiol.*256: R567–R571Hyatt R E, Schilder D P and Fry D L 1958 Relationship between maximum expiratory flow and degree of lung inflation.

*J. Appl. Physiol.*13: 331–336Jensen O E 1990 Instabilities of flow in a collapsed tube.

*J. Fluid Mech.*220: 623–659Jensen O E 1992 Chaotic oscillations in a simple collapsible-tube model.

*ASME J. Biomech. Eng.*144: 55–59Jensen O E and Heil M 2003 High-frequency self-excited oscillations in a collapsible-channel flow.

*J. Fluid Mech.*481: 235–268Jensen O E and Pedley T J 1989 The existence of steady flow in a collapsed tube.

*J. Fluid Mech.*206: 339–374Kudenatti R B, Bujurke N M and Pedley T J 2012 Stability of two-dimensional collapsible-channel flow at high Reynolds number.

*J. Fluid Mech.*705: 371–386Kudenatti R B, Bujurke N M and Pedley T J 2015 Chapter in this volume

Luo X Y and Pedley T J 1996 A numerical simulation of unsteady flow in a 2-D collapsible channel.

*J. Fluid Mech.*314: 191–225Luo X Y and Pedley T J 1998 The effects of wall inertia on flow in a 2-D collapsible channel.

*J. Fluid Mech.*363: 253–280Pedley T J 1980

*The fluid mechanics of large blood vessels.*Cambridge University PressPedley T J 2000 Blood flow in arteries and veins. In: Batchelor G K, Moffatt HK, Worster MG (eds)

*Perspectives in fluid dynamics*. Cambridge University Press, pp 105–158Pedley T J, Brook B S and Seymour R S 1996 Blood pressure and flow rate in the giraffe jugular vein.

*Phil. Trans. R Soc Lond. B*351: 855–866Pedley T J and Luo X Y 1998 Models of flow and self-excited oscillations in collapsible tubes.

*Theor. Comp. Fluid Dyn.*10: 277–294Pedley T J and Stephanoff K D 1985 Flow along a channel with a time-dependent indentation in one wall: The generation of vorticity waves.

*J. Fluid Mech.*160: 337–367Pihler-Puzović D and Pedley T J 2013 Stability of high-Reynolds-number flow in a collapsible channel.

*J. Fluid Mech.*714: 536–561Pihler-Puzović D and Pedley T J 2014 Flutter in a quasi-one-dimensional model of a collapsible channel.

*Proc. R. Soc. Lond. A*470: 20140,015Shapiro A H 1977 Steady flow in collapsible tubes.

*ASME J. Biomech. Eng.*99: 126–147Smith F T 1976a Flow through constricted or dilated pipes and channels: Part i.

*Q. J. Mech. Appl. Math.*29: 343–364Smith F T 1976b Flow through constricted or dilated pipes and channels: Part ii.

*Q. J. Mech. Appl. Math.*29: 365–376Stewart P S, Heil M, Waters S L and Jensen O E 2010 Sloshing and slamming oscillations in collapsible-channel flow.

*J. Fluid Mech.*662: 288–319Stewart P S, Waters S L and Jensen O E 2009 Local and global instabilities of flow in a flexible-walled channel.

*Eur. J. Mech. B/Fluids*28: 541–557Xu F, Billingham J and Jensen O E 2013 Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux.

*J. Fluid Mech.*723: 706–733Xu F, Billingham J and Jensen O E 2014 Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment.

*J. Fluid Mech.*746: 368–404

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

PEDLEY, T.J., PIHLER-PUZOVIĆ, D. Flow and oscillations in collapsible tubes: Physiological applications and low-dimensional models.
*Sadhana* **40, **891–909 (2015). https://doi.org/10.1007/s12046-015-0363-9

Received:

Revised:

Accepted:

Published:

Issue Date:

### Keywords

- Collapsible tubes
- flow-structure interactions
- 1D models
- 2D models.