Interactions of pulsatile upstream forcing with flow-induced oscillations of a collapsed tube: mode-locking

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Abstract

Interest in flow through collapsible tubes derives from their prevalence in the human body. In the circulation at least, the driving pressure for flow is pulsatile, and it is accordingly appropriate to investigate the response of a collapsible tube to a pulsatile head. A servo-controlled hydraulically powered piston pump was used to add pulsation to the head propelling aqueous flow through a silicone rubber tube. The pump was commanded to execute a sinusoidal variation of piston position at various frequencies on either side of the natural frequency of an operating point where slow (3.75 Hz) repetitive self-excited oscillations in the absence of such forcing could be guaranteed. The resulting time series were characterised by the ratio of the number of collapse cycles to the number of forcing cycles in one overall interaction waveform period. Nonlinear interaction rather than linear superposition was dominant. Many different mode-locked interactions were recorded, and are here systematised in Farey series and plots which show the extent of entrainment. A few of the time series were aperiodic over the time scales investigated, suggesting possible chaos.

Introduction

The dynamics of flow through collapsed tubes assumes importance because of the wide prevalence of elastic conduits which can, under certain conditions, be they physiological or imposed for diagnostic or therapeutic reasons, undergo collapse while continuing to convey their fluid contents. Commonly quoted examples include systemic veins outside the head [1], pulmonary veins [2], the brachial artery during sphygmomanometry [3], the lung airways [4], [5], the trachea [6], and the urethra [7]. A comprehensive review of mammalian examples was given by Shapiro [8]; shorter reviews have been given by Kamm [9] and Bertram [10].

A widespread feature of the occurrences in the human body is the existence of flow through the collapsible conduit impelled by a pressure difference which is pulsatile or fluctuating or transient. Obvious general examples here include the pulsatile impulsion of blood in arteries by the heart ventricles (and retrogradely in large veins by the atria), and the possibilities for rapid variation of driving force under voluntary control in the airways and the urethra. A particular example is the transient production of Korotkov sounds at specific moments during each heartbeat, as the brachial artery pressure passes through the quasi-steady pressure exerted by the sphygmomanometer cuff.

For all that, the investigation of collapsible tubes on the laboratory bench has largely dealt with steady driving pressure head. Transient changes have been examined by Kamm and Shapiro [11] and by Shimizu [12], but only in the context of interest in the spread of an inflation (or deflation) wave along the length of the tube. Upstream pulsatile forcing of a collapsible tube was reported by Conrad [13] as a small part of a more general investigation of collapsible-tube flows, and more recently by Low et al. [14]. The latter work was subsequently described more fully by Low and Chew [15]. Neither Low et al. nor Conrad was concerned with interactions between the forcing and the self-excited oscillations of the tube; indeed, it is not clear from these papers even whether the tube was performing oscillations. The interactions turn out to be interesting and subtle, leading to both complex periodic cycles and aperiodic oscillations.

In the case of steady upstream head alone, behaviour in two-dimensional operating-point space is dominated by broad regions of periodic limit-cycle oscillation. Here we use an operating point which is in the centre of such a well-defined large region of periodic oscillation, and create conditions for the generation of more complex behaviours through the addition of pulsatile forcing, applied at the upstream end of the tube. Thus, for present purposes we treat the whole of the collapsible-tube system as a ‘black box’, to be interrogated with the aid of the added forcing, and look at the interactions which result. We will concentrate on the mode-locked interactions, in which a certain number of ‘response cycles’ from the tube adapt to fit exactly within a number of forcing cycles. Such interactions are periodic, repeating exactly this same adapted fit after a constant time which is a multiple of the forcing period. Aperiodic interactions were infrequent, and are herein mentioned only as needed to give an idea of their distribution and prevalence relative to the mode-locked ones.

Some of the experiments and methodology reported here were first presented at a conference in 1991 [16]. We believe that this is still to date the only report concerned with observed interactions between the forcing and the self-excited oscillations of the tube. However, in other areas of physics, the interaction of such an oscillator with forcing is now recognised as an important way of probing the underlying dynamics of the system. Our methods have accordingly been adapted from investigations of analogous systems, such as (on the theoretical side) the forced van der Pol oscillator [17], [18], and (on the experimental side) externally stimulated embryonic chick hearts [19].

Section snippets

Methods

The collapsible-tube system was, apart from the additional elements to do with forcing, deliberately kept the same as in previously documented series of experiments [20], [21]. The description here will therefore be brief. A silicone–rubber tube of inside diameter 13 mm and wall thickness 2.4 mm, was clamped at each end over a rigid pipe forming part of a steady-flow recirculating system [22]. The unsupported length was 17.4 diameters. The fluid was water at about 25°C with minor admixtures of

Results

Before and after pulsatile forcing, the tube displayed repetitive self-excited oscillations at a frequency of about 3.75 Hz. In a range of forcing frequencies on either side of this natural frequency, the self-excited oscillator locked onto the forcing frequency; one cycle of forcing gave rise to one cycle of response. The adaptation of the response cycle that this entailed was mainly reflected in the variation with frequency of the peak-to-peak excursion of pressure at the tube entrance, as

Discussion

Previous work on collapsible tubes has largely ignored the inherently unsteady nature of physiological flows. We present here a preliminary analysis of the mode-locked interactions in what for many physiological applications is an important system, where one oscillator is acted upon by a non-identical second which is impervious to the first. As alluded to in the Introduction, an example would be the production of the so-called Korotkov sounds when a pressure-inducing cuff is wrapped around the

Acknowledgements

Experiments were funded by the Australian Research Council.

References (27)

  • D.J. Griffiths

    The pressure within a collapsed tube, with special reference to urethral pressure

    Phys. Med. Biol.

    (1985)
  • A.H. Shapiro

    Physiologic and medical aspects of flow in collapsible tubes

  • R.D. Kamm

    Flow through collapsible tubes

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    Subsequently: School of Physics, University of New South Wales, Sydney, NSW 2052, Australia.

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