Oscillatory flow in thin-walled curved elastic tubes

A numerical analysis is carried out for oscillatory flow of a viscous incompressible fluid through thin-walled elastic tubes with small curvature. A wave propagation analysis for sinusoidal input pressure disturbance yields a resulting asymmetrical flow pattern and the secondary flow pattern induced by centrifugal effects due to curvature. Essential differences between the effects of curvature on steady and oscillatory flows are obtained. A complicated quadrihelical secondary flow pattern is obtained. The flow pattern and computed wall shear in curved tubes are determined and compared with those of oscillatory flow through straight elastic tubes. Parameters in the computation are chosen to correspond to physiological flows at arterial sites of small curvature.     

Role of viscosity in frequency-dependent reflection from discontinuities in liquid-filled tubes

The paper investigates the effect of fluid viscosity on wave reflection at sites of geometric and/or mechanical discontinuity in a system of liquid-filled tubes, when the transmission of pressure pulses along these tubes is governed by a linear medium wavelength (LMW) theory. This theory is capable of describing the propagation of pressure impulses that contain frequencies in the range 0–500 Hz. We derive formulae for the reflection coefficients at a junction between two dissimilar tubes and examine the influence of fluid viscosity on these coefficients. Over much of the frequency spectrum under consideration, viscosity changes the reflection coefficients very little. The influence of viscosity is significant, however, at frequencies below 1 Hz.     

Clinical implications of pressure deformation analysis of curved elastic tubes

The clinical implications of the pressure-deformation characteristics of a curved tapered elastic tube segment simulating the aortic arch without branches, subjected to an internal pressure, are presented. A finite-element technique was employed to analyse the asymmetrical deformation of the tube. Large deformation characteristics and the non-linear stress-dependent elastic stiffness property of the vessel wall were incorporated by a step-wise incremental analysis method, with a linear small deformation finite-element formulation being employed for the analysis at each time step increment. For a healthy vessel, simulated by uniform Young's modulus against stress characteristics, and a simulated arteriosclerotic vessel (having some wall segments with elevated Young's modulus against stress characteristics), the distributions of the deformations, wall stress and compliance were determined during the cardiac cycle, and graphically illustrated. The results enabled us to appreciate the sensitivity and utility of pressure-deformation characteristics to distinguish between healthy and arteriosclerotic vessels. The methodology can be employed to generate pulse pressure-deformation patterns which, by comparison with fluoroscopically or echocardiographically monitored deformation patterns, may help determine the distribution of the stress-dependent Young's modulus of the vessel wall. This technique may provide a tool for the detection of arteriosclerosis in the human circulatory system especially with the recent developments in the image-processing techniques, such as videodensitometric analysis.     

Negative-resistance effects in flow through collapsible tubes: 3 Two-dimensional treatment of the elastic properties of elastic constriction

The elastic behaviour of a 2-dimensional elastic constriction is treated by two different methods: in the first, allowance is made for axial tension in the thin wall of the channel; in the second, a stress analysis of the block of elastic material which, by compression, forms the elastic constriction is carried out. In both cases the result is to reduce the flow resistance in the sonic configuration, leaving the transonic resistance unchanged from the 1-dimensional treatment. In practice, the 2-dimensional elastic correction exceeds the opposite hydrodynamic correction found previously, and so the predicted flow resistance is greater in the transonic than in the sonic configuration, as observed. Therefore it is the elastic deviation from the one-dimentional theory which accounts for the experimental difference between the two flow resistances, and hence for the negative-resistance behaviour. In the appropriate limit, all the 2-dimensional corrections vanish and the results of the 1-dimensional theory are recovered.     

Pulse wave attenuation measurement by linear and nonlinear methods in nonlinearly elastic tubes.

Reasons for the continuing difficulty in making definitive measurements of pulse wave attenuation in elastic tubes and arteries in the presence of reflections are sought. The measurement techniques available were re-examined in elastic tubes mimicking the arterial compliance nonlinearity, under conditions of strong reflection. The pulse was of physiological shape, and two different pulse amplitudes in the physiological range were used. Measurements of pressure, flow-rate and diameter pulsation allowed the deployment of four of the classical linear methods of analysis. In addition, a method of separating the forward- and backward-travelling waves that does not require linearising assumptions was used, and the attenuation in the forward and reverse directions was calculated from the resulting waveforms. Overall, the results obtained here suggest that a fully satisfactory way of measuring arterial attenuation has yet to be devised. The classical linear methods all provided comparable attenuation estimates in terms of average value and degree of scatter across frequency. Increased scatter was generally found at the higher pulse amplitude. When the forward waveforms from the separation were similarly compared in terms of frequency components, the average value at energetic harmonics was similar to both the value indicated by the linear methods and the values predicted from linear theory on the basis of estimated viscous and viscoelastic parameter data. The backward waveforms indicated a physically unreasonable result, attributed as the expression for this technique of the same difficulties that normally manifest in scatter. Data in the literature suggesting that one of the classical methods, the three-point, systematically over-estimates attenuation were not supported, but it was confirmed that this method becomes prone to negative attenuation estimates at low harmonics as pulse amplitude increases. Although the goal of definitive attenuation measurement remains elusive, the task provides a sensitive tool for the examination of the effect of nonlinearities in the arterial system.     

Fluid velocity greater than wavespeed and the transition from supercritical to subcritical flow in elastic tubes

The behaviour of the flow regime downstream of the choke point in a flow-limited water filled penrose tube was examined. The transmural pressure along the length of the tube was measured with a moveable side-tap catheter and tube area and stiffness were derived from the tube's static pressure/area curve. Stable supercritical flow, in which the local fluid velocity is greater than the local speed of wave propagation, was demonstrated to extend downstream from the choke point. Speed ratios (fluid velocity divided by tube wavespeed) as large as ten were measured in tube segments in which the area changed so gradually with length as to rule out significant longitudinal tension effects on the tube pressure/area curve. The predicted transition from supercritical to subcritical velocity, or elastic jump, was also studied. Sidewall friction along the jump and longitudinal tension effects due to longitudinal wall curvature were found to be significant factors governing the variation of pressure within the jump. Taking friction into account, the flow momentum equation was found to describe the overall size of the elastic jump adequately if its upstream and downstream limits were taken at points where wall curvature effects were negligible.     

Negative-resistance effects in flow through collapsible tubes: 2 Two-dimensional theory of flow near an elastic constriction

In an attempt to account for the experimental difference between the flow resistances of an elastic constriction in the sonic and transonic configurations (which leads to the observed negative-resistance behaviour), a 2-dimensional hydrodynamic treatment of the steady flow of an ideal fluid near an elastic constriction is carried out. It is found that the resistance in the sonic configuration is the higher, in direct disagreement with experiment. In the appropriate limit, this 2-dimensional treatment gives the same results as the older 1-dimensional theory, despite the fact that, starting from a parabolic instead of hyperbolic equation of motion, it appears quite different. In an appendix some new results of the 1-dimensional theory, needed in the main paper, are given.     

Pressure-radius relationships for elastic tubes and their application to arteries: Part 1-Theoretical relationships

The displacement relationships describing the deformation of an elastic vessel under excess internal pressure which are derived from different theories of elasticity are compared. The main result of the comparison is that theories which take account of the thickness of the wall of the vessel produce a significantly better representation than those theories which treat the wall as a membrane. The classical and statistical theories of thick-walled tubes result in complicated pressure-radius relationships. It is shown that there is little difference between the results of the more exact theories and those for a thin membrane corrected by means of a simple thickness factor. A review of the different theories is necessary to decide which pressure-displacement relationship to apply as an approximation for the elastic properties of arteries. An indication is given of the manner in which the relationship is used in numerical computations. In Part 2 the experimental determination of the pressure-radius relationship for a rubber tube is described. The results are in agreement with the conclusions of the comparison of theoretical treatments in Part 1.     

Self-excited oscillations in thin-walled collapsible tubes

A simple flow system incorporating a collapsible tube of silicon rubber is tested to determine the conditions under which the tube will experience self-excited oscillation of its walls. The results are presented as curves for the dimensionless frequency and time-averaged flow rate for a range of tube length-to-diameter ratios, Reynolds numbers and the ratios of transmural pressure to flow pressure drop. It is significant that no oscillations were detected for Reynolds numbers below about 1500, which is considerably higher than is found in the veins of a resting patient. The flow rate in general varies as should be expected, except, perhaps, at the onset of oscillation at low transmural pressures when it suffers a fairly abrupt decrease from its steady full flow value.