Haigh.Alexander

University of Southampton
Faculty of Engineering and the Environment
Aerodynamics and Flight Mechanics Research Group
Flow through Collapsible Tubes
IP026
Alexander Haigh ID: 2550457
Supervisor: Professor Owen Tutty
This report is submitted in partial fulfilment of the requirements for the degree
of Master of Engineering (Mechanical Engineering, Aerospace), Faculty of
Engineering and the Environment, University of Southampton
MEng Mechanical Engineering (Aerospace)
May 2015
Word Count: 9529

Alexander Haigh Flow through Collapsible Tubes
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Declaration:
I, Alexander Haigh declare that this thesis and the work presented in it are my own and has
been generated by me as the result of my own original research.
I confirm that:
 This work was done wholly or mainly while in candidature for a degree at this University;
 Where any part of this thesis has previously been submitted for any other qualification at
this University or any other institution, this has been clearly stated;
 Where I have consulted the published work of others, this is always clearly
attributed;
 Where I have quoted from the work of others, the source is always given. With the
exception of such quotations, this thesis is entirely my own work;
 I have acknowledged all main sources of help;
 Where the thesis is based on work done by myself jointly with others, I have made clear
exactly what was done by others and what I have contributed myself;
 None of this work has been published before submission.

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Acknowledgments
I would like to firstly thank Professor Owen Tutty for his support and guidance throughout the
project and I hope that this work will be useful for his teaching activities in the future.
I would also like thank the staff of the electronics workshop and particularly Mr Rob Stansbridge and
Mr Anthony Wood for their help on electronic matters.
Finally I thank Mr Maurice Jones for all of his support in the Building 25 design studio.

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Table of Contents
Abstract..............................................................................................................................................5
1.Introduction ...................................................................................................................................6
1.1. Aim..........................................................................................................................................7
1.2.Important Equations and Concepts.........................................................................................7
2. Literature Review..........................................................................................................................9
2.1. History.....................................................................................................................................9
2.2 Starling Resistor .................................................................................................................... 12
2.3. Basic Mechanisms................................................................................................................ 13
2.3.1. Degrees of Collapse........................................................................................................ 13
2.3.2. Flow Limitation............................................................................................................... 14
2.3.3. Self-Induced Oscillations ................................................................................................ 16
3. Apparatus Modifications.............................................................................................................. 18
3.1. Modification Requirements.................................................................................................... 18
3.1.1. Pressure Control............................................................................................................. 18
3.1.2. Pressure Measurement.................................................................................................. 19
3.1.3. Flow Meter..................................................................................................................... 21
3.1.4. Electronics and User Interface ....................................................................................... 22
3.1.5. Test Tubes ...................................................................................................................... 25
3.1.6. Airtightness .................................................................................................................... 25
3.1.7. Air Pump......................................................................................................................... 26
3.1.8. Sensor Calibration .......................................................................................................... 27
3.1.9. Initial Testing.................................................................................................................. 29
4. Experiments .................................................................................................................................. 30
4.1. Effects of Flow Rate on Pressure Gradient ............................................................................. 30
4.1.1. Aim ................................................................................................................................. 30
4.1.2. Method........................................................................................................................... 30
4.1.3. Results ............................................................................................................................ 31
4.1.4. Discussion....................................................................................................................... 37
4.1.5. Wear on Tube................................................................................................................. 38
4.1.6. Reynolds Number........................................................................................................... 40
4.2. Reduction in Downstream Pressure........................................................................................ 41
4.2.1. Method........................................................................................................................... 41
4.2.2. Results ............................................................................................................................ 41
4.2.3. Discussion....................................................................................................................... 42
4.2.4. Reynolds Number........................................................................................................... 43
4.3. Moens-Korteweg Wave Speed ................................................................................................ 48

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4.3.1. Calculation...................................................................................................................... 48
4.3.2. Method........................................................................................................................... 49
4.3.3. Results ............................................................................................................................ 49
4.3.4. Discussion ...................................................................................................................... 50
5. Conclusion..................................................................................................................................... 57
6. Recommendations........................................................................................................................ 59
7. References..................................................................................................................................... 60
APPENDIX A ....................................................................................................................................... 62

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Abstract
The study of flow through collapsible tubes is important because of its relevance to human
physiology and application to medicine. The study of collapsible tubes phenomena is the subject of
this individual project. This final reports contains a literature review of the theory and mathematical
models that exist for flow through collapsible tubes. It also describes the experimental campaign
detailing results and analysis. The experimental campaign was carried out using existing equipment
which required modification in order to start the project.
The study of flow through collapsible tubes has interested researchers because of its complexity and
potential application to medicine. The interactions between the Fluid Mechanics of the flow and the
Solid Mechanics of the tube mean that even a simple configuration displays unusual behaviours. One
such configuration is flow through an elastic tube contained within a pressurised box known as a
Starling Resistor.
The University’s Starling Resistor required modifications to improve flow control and airtightness. In
this area significant progress was made. Electronic pressure sensors to record the downstream and
upstream flow pressures of the tube were installed as well as a sensor to record the air pressure of
the chamber. A flow meter was also installed to record the volumetric flow rate through the device.
The sensors were connected to an Arduino Uno microprocessor that allowed the sensor readings to
be displayed graphically on a laptop in real time. This was particularly useful for experiments in which
reading could be observed as they happened and did not require further processing.
Degrees of collapse were demonstrated by slowly increasing the flow rate through the tube. Self-
induced oscillations were well demonstrated by experiments. Studies into self-induced oscillations
included the parameters at which they occurred as well as a validation of theoretical predictions of
the Moens-Korteweg equation. The recording of the self-induced oscillations that propagated
upstream did not match with the calculated Moens-Korteweg speed but did match with theory
suggesting the oscillations occur once the flow speed u matches the wave speed c.

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1. Introduction
The study of flow through collapsible tubes differs from that of rigid tubes in that the collapsible tube
cannot support a circular cross section at standard conditions (20°C and 101 KPa) when there is no
flow passing through it. The tubes thin wall make it highly compliant to changes between the
pressure outside the tube and the pressure inside (the transmural pressure 𝑃𝑡𝑚). This transmural
pressure along with the pressure gradient (the variation of pressure along the tube) are the main
parameters of the system.
The unusual coupling between the flow pressure forces and the material properties of the tube (such
as elasticity) give rise to several unusual behaviours. As the pressure external to the tube exceeds the
pressure inside of the tube the cross sectional area becomes highly compliant and exhibits degrees of
collapse. With decreasing transmural pressure the tube cross sectional area will partially collapse,
from a circular to a fully collapsed state where the cross section becomes flat resembling a dumbbell
shape.
Additionally whereas reduction in downstream pressures typically causes higher flow rates a partially
collapsed tube exhibits “flow limitation”. During which reductions in downstream pressure have no
effect on the flow rate.
A significant enough transmural pressure will provoke pressure pulses to propagate up the tube in
the opposite direction to the flow. These “self-induced oscillations” are a result of instabilities in the
flow and are a significant area of research in collapsible tubes.
Experiments into flow through collapsible tubes are mostly conducted with a pressurised box
through which a flow in an elastic tube flows. This apparatus is known as a “Starling Resistor” (figure
1). For meaningful experiments to take place the resistance at each end must be variable along with
the pressure inside the box
Figure 1. Starling Resistor apparatus used for experiments involving collapsible tubes.

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Early experimental work into collapsible tubes managed to demonstrate the unusual behaviours such
as flow limitation and self-induced oscillations. This work served as the foundation for theoreticians
attempting to construct models of the flows during these behaviours.
The study of flow through collapsible tubes is of particular interest in physiology where numerous
behaviours collapsible tubes are comparable to various phenomena in the human body. For example
onset of “flow limitation” during wheezing due to forced expiration in the respiratory system. Self-
induced oscillations that occur in collapsible tubes also manifest themselves as snoring in the human
airways and as Korotkoff sounds in the vascular system (Ur and Gordon 1970).
1.1. Aim
The aim of the project was to record the parameter setups at which the main phenomena of
collapsible tubes occurred. Through quantitatively recorded experiments the aim was to allow
users to be able to recreate the phenomena, for educational purposes within the faculty. In
doing this a thorough review of the literature looking in to the experimental methods, theory and
mechanisms related to research into collapsible tubes was been undertaken.
Work was also carried out in modifying the University’s Starling Resistor (a pressurised chamber
through which fluid flows via a collapsible tube). The apparatus previously had no way of
measuring or varying the required parameters for experiments and was unable to demonstrate
the main phenomena such as self-induced oscillations.
1.2. Important Equations and Concepts
The volumetric flow rate is defined as the flow velocity multiplied by the area of the conduit, in
this case the area if the circular tube (equation 1).
𝑄 = 𝑉𝐴 (𝐸𝑞. 1)
Where Q corresponds to flow rate, V is flow velocity and A is the cross sectional area of the tube.
The transmural pressure is the Pressure outside the tube minus the pressure inside the tube
(equation 2).
𝑃𝑡𝑟𝑎𝑛𝑠 = 𝑃𝑜𝑢𝑡 − 𝑃𝑖𝑛 (𝐸𝑞. 2)
Where Ptrans is the transmural pressure, Pout is the pressure outside the tube and Pin is the
pressure inside of the tube.

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Furthermore the flow speed (and hence Q) through the tube is not only affected by the pressure
gradient along the tube (Pup – Pdown) but also, because of Bernoulli’s equation (equation 3) the
cross sectional area of the tube.
𝑝 +
1
2
𝜌𝑢2
+ 𝜌𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐸𝑞. 3)
A decrease in the cross sectional area will increase the flow rate causing the local pressure inside
the tube to drop.
The fundamental equation for describing fluid flow is the Navier-Stokes equation (equation 4). By
assuming properties of the flow (such as steadiness and viscosity) simple solutions of the
equation can be obtained to approximately describe the flow.
𝜌
𝜕𝑢𝑖
𝜕𝑡
+ 𝜌𝑢𝑗
𝜕𝑢𝑖
𝜕𝑥𝑗
= −
𝜕𝑝
𝜕𝑥𝑗
𝛿𝑖𝑗 −
𝜕
𝜕𝑥𝑗
(
2
3
𝜇
𝜕𝑢 𝑘
𝜕𝑥 𝑘
) 𝛿𝑖𝑗 +
𝜕
𝜕𝑥𝑗
(𝜇
𝜕𝑢𝑖
𝜕𝑥𝑗
+ 𝜇
𝜕𝑢𝑗
𝜕𝑥𝑖
) + 𝜌𝑔𝑖 (𝐸𝑞. 4)
Where u is the fluid velocity in the i, j or k direction, x is the position in either the i,j or k
direction, ρ is the density of the fluid, µ is the dynamic viscosity of the fluid, g is the acceleration
due to gravity, is is the fluid pressure and 𝛿𝑖𝑗 is the Kronecker delta.

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2. Literature Review
2.1. History
The idea of engineering analysis to biological problems is not a new one. Indeed as far back as
1808 Thomas Young at a lecture to the Royal Society said that the laws of mechanics of
inanimate objects are no different to the laws of mechanics that govern the human body (Caro et
Al 1978). He went on further to say that if the mechanical properties of the heart, vessels and
muscles were known then the problem of blood flow around the body simply became a question
of hydraulics.
Thomas Young (1773 - 1829) was a professor of physics at Cambridge University and was also a
practising physician. It is worth noting that it this multidisciplinary approach (physiology with
engineering analysis) is what makes the study of collapsible tubes one of particular interest. Not
least because of the complex concepts of fluid mechanics it presents but also their relevance to
human physiology.
Early studies into flow through collapsible tubes were done by physiologists wishing to gain
further understanding into the flow of fluid through various conduits in the human body (namely
blood flow through veins and arteries). Because of the limited information available in physics
and engineering physiologists often resorted to analogies based on other types of flow occurring
in nature such as flow in open channels and waterfalls (Shapiro 1977).
These studies could only go so far, during the 1950’s and 1960’s early engineering analyses into
collapsible flow were done. Early experimental work (Holt 1959, Katz 1969) investigated the main
phenomena of collapsible tubes using a Starling Resistor configuration and attempted to explain
them with 1 dimensional lumped parameter models. A lumped parameter model is a simple
model that reduces a spatially varying system to just a few parameters (namely time and
position). Early experimental work demonstrated flow limitation, degrees of collapse and self-
induced oscillations.
During the 1970’s attention focused on the study of flow through collapsible tubes and bio fluid
dynamics in general. Books published such as Mathematical Biofluiddynamics (Lighthill 1974) and
Mechanics of the Circulation (Caro et al 1978) provided rigorous analyses to many physiological
examples of flow through collapsible tubes at an appropriate level of detail that both engineers
and physiologists could appreciate.

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This early work provided good experimental results by measuring flow rates against pressure
differences and vice versa. Theoretically they were explained by 1 dimensional lumped
parameter models (Shapiro 1977). The geometry of the tube was modelled with simple time
dependent variables like cross sectional area. This lead to a form of the continuity equation that
applies for collapsible tubes (equation 5);
𝜕𝐴
𝜕𝑡
+
𝜕(𝑢𝐴)
𝜕𝑥
= 0 (𝐸𝑞. 5)
Where u is fluid velocity, A is the cross sectional area of the tube, t is time and x is position along
the length of the tube.
The 1D conservation of momentum equation can be written as (Equation 7);
𝑢
𝑑𝑢
𝑑𝑥
= −
1
𝜌
𝑑𝑝
𝑑𝑥
− 𝑅(𝐴, 𝑢)𝑢 (𝐸𝑞. 6)
Where p is pressure, 𝜌 is density and R is the viscous resistance that is a function of A and u.
For a full mathematical model there must also be an equation of state that relates the main
parameters as well as a wave equation. This is elegantly described in (Shapiro 1977) in which a
model for collapsible tubes is compared to the model of 1D gas dynamics that requires an
equation for continuity (conservation of mass), a wave speed equation and finally a state relation
(ideal gas law).
For flow through collapsible tubes the equation of state is known as the “Tube Law”. It relates
the transmural pressure to the cross sectional area of the tube (equation 7).
𝑃(𝐴) = 𝑝𝑡𝑟𝑎𝑛𝑠 = 𝑝𝑖𝑛 − 𝑝 𝑜𝑢𝑡 (𝐸𝑞. 7)
Where P(A) is a function of the cross sectional area, 𝑝 𝑜𝑢𝑡 is the external pressure, pin is the
pressure inside the tube and 𝑝𝑡 is the transmural pressure.
Although there is no exact equation that fully describes the tube it can be calculated for certain
ranges where the flow can be modelled. Studies by (Katz 1969, Shapiro 1977, Koslovsky et al
2014) have derived their own tube laws from experimental results.
For small amplitude pressure waves propagated at wave speed (𝑐) (equation 8);
𝑐 = (
𝐸ℎ
𝜌𝑑
)
1
2
(𝐸𝑞. 8)
Where E is the Young’s Modulus, h is the wall thickness, ρ is the density of the fluid and d is the
diameter of the tube.

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These early studies provided a theoretical basis for the development of research into collapsible
tubes. As numerical simulation capabilities developed there was somewhat of a split with
researchers either studying the theoretical numerical side or further experimenting with
different configurations and producing more data.
In later years many experiments were repeated particularly by Bertram in the 1980’s (Bertram
2002) due to measurement and apparatus configurations not having been adequately recorded
during the early years. Further criticism of early experimental work (Shapiro 1977, Barclay 1986)
was made of the approach in early experiments. The parameters varied were not systematically
recorded and failed to appreciate the fact that the Starling Resistor system is driven by two
pressure differences rather than three individual pressures. Also the effects of tension in the
tube between the rigid connection and collapsed part were not considered.
A lot of this scrutiny of early experimental work came as theoreticians tried to explain the
behaviour (particularly self-induced oscillations). At the time of these experiments it was not
known what parameters may have had subtle effects on the behaviour (such as geometry, tube
longitudinal tension) and hence were not described in detail.
As observed in (Heil & Hazel 2011) 1D lumped parameters manage to describe the overall
characteristics of the flow such as collapse and flow limitation however they did not provide
much of an insight into the mechanisms of self-induced oscillations as they failed to describe the
interactions between the solid and fluid mechanics.
Theoreticians also continued developing models and as computational capabilities increased in
the 80’s and 90’s more sophisticated models were developed. One such 2D model was the
subject of much attention during the 1990’s (Luo and Pedley 1997).
The model consists of two rigid plates parallel to each other that are separated by a distance ℎ 𝑜,
opposite and parallel to these two plates is a continuous rigid plate. Between the separated plates
is an elastic membrane of length L with longitudinal tension T (figure 2).
Figure 2. Diagram of 2D model.

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The simplified mechanics of the elastic membrane allowed for it to be more easily considered
when analysing the system. Incorporating equations for the tension of the membrane into the
relevant Navier Stokes equations allows for full numerical simulation at steady flows.
These models adequately described the phenomenon for steady flow however validation
required a separate experimental set up. Experiments involving these two dimensional models
are limited. The experimental work for collapsible tubes has largely been done through
apparatus’ of a Starling Resistor type configuration, hence the data is highly 3 dimensional.
Towards the late 1990’s and early 2000’s 3D models were investigated notably by (Hazel & Heil
2003). With numerical methods the 3D Navier Stokes equations together with geometric
equations for the tube deformations were solved for steady flow of a viscous fluid. From these
models the extent to which the Bernoulli Effect causes further collapse of tubes was thoroughly
analysed. Also the behaviour of the fluid as the tube forms a dumbbell shape at negative 𝑃𝑡𝑚. As
the contact between the middle parts of the tube increases the flow is restricted to two lobes at
each side. The further constriction causes increased velocity forming two jets on either side.
These computational results allowed for the velocity profiles and geometries to be visualised in 3
dimensions (figure 3).
Figure 3. 3D simulation of the geometry of a collapsed tube during steady flow.[8]
2.2. Starling Resistor
The main apparatus for studying flow through collapsible tubes is the Starling Resistor (figure 4).
The system consists of a pressured chamber with two rigid tubes going through opposite walls.
These are the inlet and outlet. Between these rigid tubes a collapsible tube is connected. The
pressure in the chamber is typically varied but maintained above ambient pressure. The pressure

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at both the inlet and outlet can be varied individually. These three pressures (𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟,
𝑃𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚,𝑃𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚) define the two pressure differences that control the system. The two
pressure differences being the 𝑃𝑢𝑝 –𝑃𝑑𝑜𝑤𝑛 and the transmural pressure. The transmural pressure
is the difference between the external and internal pressure of the tube. The pressure that is
subtracted from 𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟 is the one that is varied.
Figure 4. Starling Resistor apparatus
2.3. Basic Mechanisms
There are three mechanisms of collapsible tubes that are of interest in this project. Degrees of
collapse, flow limitation and self-induced oscillations.
2.3.1. Degrees of collapse
Consider a Starling Resistor setup where the external pressure along all points on the tube is
set to constant. The pressure gradient along the tube is fixed.
At positive transmural pressures the tube is circular and behaves like a circular, rigid tube.
The flow through the tube corresponds to Poiseuille flow, with the pressure gradient being
proportional to the flow speed. Taking the full Navier-Stokes equation from section 1.2 and
assuming a steady, laminar, incompressible flow yields the following relation for a circular
tube (equation 9);
𝑢 ∝
𝑑𝑃
𝑑𝑥
(𝐸𝑞. 9)
Where u is the fluid velocity in the in the longitudinal and
𝑑𝑃
𝑑𝑥
is the pressure gradient along
the tube.

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As the external pressure increases transmural pressure 𝑃𝑖𝑛 –𝑃𝑜𝑢𝑡 will drop, while the
transmural pressure 𝑃𝑡𝑟𝑎𝑛𝑠 is positive the tube will remain fully open and distended. At this
point the tube is circular.
As transmural pressure approaches 0 the thin wall’s small resistance to bending will mean
that small changes in the transmural pressure will strongly affect the tube cross section.
Once the 𝑃𝑡𝑟𝑎𝑛𝑠 falls below 0 the tube will collapse. In this region the changes in cross
sectional area are large compared to small changes in the transmural pressure.
Figure 5. Cross sectional area variation with transmural pressure.[9]
For regions (ii) and (iii) of figure 5 the change in cross sectional area is large for small
variations in 𝑃𝑡𝑟𝑎𝑛𝑠. As 𝑃𝑡𝑟𝑎𝑛𝑠 continues to drop the tube once again undergoes only small
variations in cross sectional area. Note also that at this point the top and bottom part of the
tube are in contact. This contact increases as 𝑃𝑡𝑟𝑎𝑛𝑠 is decreased until the cross sectional
area forms a dumbbell shape.
2.3.2. Flow limitation
Consider a Starling Resistor setup where the chamber pressure 𝑃𝑐 is kept constant and the
flow rate (Q) is increased by reducing the downstream pressure. As the downstream
pressure is decreased the pressure gradient driving the flow is increased. At some point after
the downstream pressure has fallen below 𝑃𝑐 (region 1 in figure 6) there will be a small drop
in the flow rate as Pdown is further decreased and there will come a point (region 2 figure 6) in
which the flow rate stagnates with further reductions in 𝑃𝑜𝑢𝑡 . At this point any increases in
the pressure gradient have no effect on the flow rate through the tube. This behaviour that
the tube exhibits is known as “flow limitation”. It is comparable to “choking” that occurs in
nozzles in 1D gas dynamics.

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Figure 6. Onset of flow limitation when reducing downstream pressure.[5]
The mechanism for this is interesting as is comparable to the onset of “choking” that occurs
in 1D gas dynamics (Shapiro 1977). By combining the 1D steady conservation of momentum
equation with a resistance term and the continuity equation (equation 10).
𝑑𝐴
𝑑𝑥
=
−𝑅𝑢𝐴
𝑐2 − 𝑢2
(𝐸𝑞. 10)
The small amplitude pulse propagation wave speed is given by (equation 11).
𝑐2
=
𝐴
𝜌
𝑑𝑃
𝑑𝐴
(𝐸𝑞. 11)
It is clear the fluid velocity u should be less than c. Considering that R is larger than 0 this
implies that the change in area along the x axis should be negative (
𝑑𝐴
𝑑𝑥
> 0). Also for continuity
it is required that the velocity increases in the x direction.
Now, for a flow where c is only slightly larger than u to begin with (i.e. u/c is close to 1) then
as the flow velocity increases along the tube length the corresponding dA/dx according to
relations previously established will approach -∞. As this is impossible the flow is more likely

16
to become unsteady. As with acoustic waves in 1D gas dynamics the small amplitude waves
travel at speeds u±c, hence when u/c is above 1 information cannot propagate upstream
(Anderson 2011). The fact that information can no longer propagate upstream result in flow
limitation. Increases in downstream pressure have no effect on the overall flow rate.
2.3.3. Self-Incduced Oscillations
Between the regions were the tube is fully opened and partially collapsed as well as the
region where the tube is partially and fully collapsed the tube exhibits self-induced
oscillations. Small pressure waves appear to pulsate along the tube in the upstream
direction. (Bertram 1982) describes the steady oscillations as “milking”, with the tubs
remaining open for the majority if the cycle followed by a rapid collapse at the end.
Despite this behaviour receiving much attention from researchers the full mechanics of these
oscillations are still not fully understood. The problem is highly three dimensional and
involves complex interactions between the fluid and structural mechanics of the tube.
Several models for these oscillations have been proposed and while managing to encapsulate
many of the behaviours fail to fully explain all of the different mechanisms.
One such model was described by (Barclay et al 1986) in which the collapsible tube is
modelled as a RLC circuit. With the pressure difference over flow rate as resistance
(analogous to R = V/I), capacitance as the change in volume with transmural pressure and
inductance as the fluid mass inertia. This simple lumped parameter model is able to predict
qualitative changes in the frequency of the oscillations but however does not address more
complex occurrences such as wave propagation and turbulence.
Experimental studies (Bertram 1982, Bertram an Raymond 1989) have found that collapsible
tubes of different length can exhibit different modes of oscillations. Furthermore these
experiments demonstrated that it is possible for these modes to interact and cause highly
erratic responses. (Jensen 1989) suggested that longer tubes can exhibit several steady
modes of oscillations. For these different modes it is uncertain to what extent the influence
of flutter (the dynamic instability of the tube wall from fluid flow) and flow separation have
on these self-induced oscillations, particularly for shorter tubes.
An important example of self-induced oscillations is their use in medicine when recording the
blood pressure of patients. When a cuff is inflated the arm of a patient a sharp snapping
sound can be heard which corresponds to the relaxation part of the self-induced oscillation

17
occurring in the artery. Although this was an accepted medical practice for many years it was
not until (Ur & Gordon 1970) when it was conclusively found that these sounds
corresponded to self-induced oscillations in the artery. In addition the intensity of the sounds
was found to correspond to the flow rate passing through the artery, something that had
previously not been considered in medical practice.

18
3. Apparatus Modifications
3.1. Modification Requirements
After initial testing of the Starling Resistor during week 2 various areas of improvement were
identified.
 Purchase suitable tubes for experiments.
 Install devices for measuring the flow rate through the system.
 Install devices to measure and vary the flow pressure at the inlet/outlet.
 Ensure that the chamber is airtight.
Essentially the Starling Resistor worked well to a degree, however for the purpose of
experimentation it had no way of measuring any of the parameters meaning that no results
could meaningfully be recreated. Previous experiments had also failed to demonstrate self-
induced oscillations due to the tube being too thick.
3.1.1. Pressure Control
A standard pressure varying valve was purchased so that the pressure at the downstream
end of the tube could be varied. This model was also chosen because it had a mechanical
pressure gauge installed in it (figure 7). This solved the both the problem of pressure
measurement and variation at the downstream end. A separate pressure gauge was
purchased and fitted at the upstream end (figure 8).
Figure 7. Pressure varying valve with inbuilt pressure gauge

19
Figure 8. Upstream pressure gauge
Upon testing the pressure taps were found to provide inadequate control of the flow
pressure, the valves were adjusted via screwdriver. This was inconvenient for experiments.
A set of standard garden taps were purchased from a local hardware store. These provided
more convenient control of the pressure at each end (figure 9).
Figure 9. Standard Garden Taps installed for pressure control at each end of the Starling Resistor
3.1.2. Pressure Measurement
The pressure varying valves were of a 22 mm diameter which required the use of copper
22mm to 25mm reducers (figure 10). The 22 mm end was simply attached to the valve by
compression joint while the 15mm end was solder to the copper pipe. This was done by
solder in the Engineering Design and Manufacture Centre (EDMC).

20
Figure 10. 22mm reducers soldered to segments 15mm copper pipe
Upon testing the pressure gauges were not sensitive enough to give adequate readings for
the pressure.
A new system for pressure measurement was installed. The aim was obtain measurements
for the pressures (upstream, downstream and within the chamber) electronically and display
them on an interface with the measurements from the flow sensor.
Three pressure sensors (Model MPXV5004GVP, RS components) were purchased via the
electronics workshop (figure 11). For both the upstream and downstream pressures a 1.7mm
diameter hole was drilled between the tap and the pressure chamber. Two tappings in the
form of a 1.3mm stainless steel pipes were inserted into the holes and bonded with hot glue
(figure 12). The tappings were connected to the pressure sensors via rubber tubing.
Figure 11. Pressure Sensor that were used to measure upstream, downstream and chamber pressure

21
Figure 12. Tappings installed to measure static pressure in tube.
For the chamber pressure a hole was drilled into the side of the pressure chamber and a
rubber tube bonding with hot glue. The pressure sensor was attached via a rubber tube
(Figure 13).
Figure 13. Tube attached to pressure sensor to measure chamber pressure.
3.1.3. Flow Meter
Previously the flow rate had been calculated using a measuring a cylinder and by dividing the
cylinder volume by the time it took to fill the cylinder. To overcome this rather ad hoc
method an electronic flow meter was purchased and installed (figure 14).
The flow meter worked by having a small jagged wheel exposed to the flow, the angular
velocity of this wheel in turn increase linearly with the flow. On either side of the wheel was
LED and photo-sensitive diode. The photo diode only passed current when light is shone onto

22
it, this light source comes from the LED. As the wheel rotates it blocks the light coming from
the LED to the photo-diode. The output of this signal a series of peaks in the output voltage
that is proportional to the rpm of the wheel which is proportional to the flow rate.
The advantage of this type of sensor is that the output is inherently digital and hence does
not require any conversion and is simple to process. The sensor was run through an Arduino
Uno microprocessor. A simple code was written to count the number of peaks for a time
period and divide it by this period of time, giving the flow rate.
Figure 14. Flow sensor attached to Starling Resistor
3.1.4. Electronics and User Interface
The three pressure sensors and flow meter were connected to an Arduino Uno
Microprocessor (figure 16). Each signal from the three pressure sensors ran through the
Arduino analogue to digital converter (figure 15). The signal was then multiplied by a suitable
coefficient in order to obtain a value for the pressure.
For the flow meter the signal ran through the Arduino digital input. The number of “drops” in
the signal per second was obtained through the microprocessor. This signal was then
multiplied by the appropriate coefficient in order to obtain the flow rate (figure 17).

23
Figure 15. Wiring for sensors through Arduino Uno. 1) +5V power supply. 2) +3.3V power supply. 3)
Ground 4) Analogue outputs from pressure sensors into A/D converter. 5 – 7) Pressure sensors. 8)
Flow Sensor. 9) Flow Sensor output into Digital Input
Figure 16. Arduino Uno with wiring

24
Figure 17. Block diagram of sensor electronics
The signals were multiplied by the appropriate factors in order to obtain accurate
quantitative measurements. These readings were then displayed graphically in real time on a
laptop screen via an open source “Arduino Monitor” program (figure 18).
Figure 18. Arduino Monitor screenshot
The Arduino Monitor Connects to the Arduino Uno and displays the signals of each sensor on
a graph that updates in real time. The colour that each parameter was allocated on the
interface changed each time the system was turned on and off. Each figure has explanation
as to what colours correspond to what parameters.

25
3.1.5. Test Tubes
The tube purchased was a Penrose surgical drainage tube that was made of natural latex
rubber (figure 19). The tube was 320 mm in length and 12.7 mm in diameter. This tube was
selected as it was used in several experiments from the literature (Katz 1969). The tube was
suitably thin and was a suitable size for the Starling Resistor.
Figure 19. Silicone tube attached to Starling Resistor
3.1.6. Airtightness
Following initial tests it was found that one of the seals between a wall and the base had
split. The joint was resealed using silicone adhesive. To ensure airtightness all other joins
were resealed by applying the adhesive externally around the join (figure 20).
Figure 20. Sealant applied for improved airtightness

26
It proved difficult to make the Starling Resistor fully airtight. Fortunately once the finished
set-up was tested with the air pump the pressure in the chamber was found to increase to a
steady value of 200KPa. The air entered the chamber stabilized with the air leaking the
chamber resulting in a steady, elevated pressure in the chamber.
The area most vulnerable to air leaks was the region mid-way along the long seal between
the longitudinal wall and the base. In order to provide further strength and relive pressure
from the adhesive sealant, duct tape was applied to the seal in this area (figure 21).
Figure 21. Duct tape applied to seal along the length of the Starling Resistor
3.1.7. Air Pump
The pressure pump inherited from the previous project did not work. Upon inquiring the
retailer (Halfords) a replacement was provided at no further cost. (Figure 22).
Figure 22. Air Pump used to elevate chamber pressure

27
3.1.8. Sensor Calibration
The sensors were purchased new and required calibration. The sensors had in-built signal
conditioning hence no other circuitry was required.
Flow Sensor
The flow sensor required calibration in order to validate its linearity and obtain a coefficient
by which the signal could be multiplied by in order to obtain an actual measurement of the
volumetric flow rate. Although the linearity of the sensor was guaranteed by the
manufacturer testing was done as to ensure accuracy and the functionality of the code that
was written.
Method
Six separate flow rates were run through the flow sensor. For each flow the volumetric flow
rate was measured experimentally by timing how long it took for the flow to fill a 500ml
flask. In addition to this flow rate the signal measured from the flow sensor was recorded.
Plotting the signal reading and measured flow rate the linearity of the sensor could be
validated. Dividing the flow rate by the signal the coefficient for obtaining the flow rate could
be calculated. The time for the flows to fill the 500mL container were recorded and the flow
rates were calculated (table 1).
Recorded
Signal
Time (s) Amount of
Water (mL)
Flow Rate
(mL/s)
Flow Rate
(m3
/s)
15.5 51.98 500 9.61 0.00000961
31.5 26.06 500 19.19 0.00001919
46 17 500 29.4 0.0000294
53.5 14.48 500 34.53 0.00003453
62.5 12.5 500 41.67 0.00004167
74 10 500 50 0.00005
Table 1. Flow Rate measurements taken during sensor calibration

28
Figure 23. Graph of Arduino output signal against measured flow rate.
(Figure 23) shows that the signal outputted by the flow sensor through the Arduino Uno
varies linearly with the flow rate.
By dividing each measured flow rate (in m3
/s) by the corresponding recorded output signal
and averaging the results a coefficient for obtaining the flow rate can be obtained. The
resulting coefficient is 0.00000066.
Multiplying the signal by this coefficient will result in very small results the will not show up
on the Arduino Monitor. In order for the signal to be of a significant enough magnitude, the
signal was multiplied by 6.6. Hence the flow rate shown on the graphs are m3
/s × 10−7
or 10
mL/s.
Pressure Sensor
The pressure sensors required a coefficient by which to be multiplied by in order to give a
reading of the pressure in KPa. The linearity of the sensors was assumed as all required signal
conditioning was included in the sensor.
At normal atmospheric conditions (101 KPa) all three sensors were found to give a reading of
256. A simple division resulted in a coefficient by which the signal could be multiplied in
order to obtain accurate measurements of pressure.
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60
Signal
Flow Rate (mL/s)
Signal vs Flow Rate

29
101
256
= 0.3937
3.1.9. Initial Testing
An early experiment was undertaken to see if self-induced oscillation could take place within
the constraints of the experimental setup (flow rate from a normal household tap). The early
results were encouraging with self-induced oscillations occurring (figure 24). This was a
qualitative experiment and no parameters were taken.
Figure 24. Self-induced oscillations exhibited during early experiments

30
4. Experiments
Having read the relevant literature concerning experimental work particularly (Bertram 2006,
Barclay et al 1986) the main parameter varied was the downstream pressure while the upstream
and external pressures were kept constant. Decreasing the downstream pressure will increase
both the pressure gradient and transmural pressure with one parameter.
For experiments with self-induced oscillations a more methodical approach was used in
experimenting with different initial flow rates and then decreasing the downstream pressure.
Another objective of experiments was to look into the relations between the pulse propagations
and the physical properties of the tube and fluid. Validating (equation 12) was an area to explore.
𝑐0 = (
𝐸ℎ
𝜌𝑑
)
1
2
(𝐸𝑞. 12)
As the apparatus may be used for educational purposes the aim was to get a quantitative set of
measurements so that all of the phenomena can be reproduced. For experiments involving the
wave speed videos of the oscillations were observed frame-by-frame in order to determine the
speed at which the wave propagated.
4.1. Effect of Flow Rate on Pressure Gradient
4.1.1. Aim
The aim of this experiment was to investigate the effect that increasing the flow rate would
have on the pressure gradient along the tube. In addition to the pressure measurements,
pictures of the tube were taken to demonstrate the degrees of collapse in the tube as the
flow rate was increased.
4.1.2. Method
For this experiment the upstream and downstream resistances were kept constant while the
flow rate Q (controlled by the main tap) was gradually increased from an initially low flow
rate. The pressure of the air chamber was kept at atmospheric pressure (101 KPa) so that the
transmural pressure remained mostly positive to minimize the effect of oscillations. A second
test experiment was then run at chamber pressure 200 KPa for comparison. The flow rate

31
and the pressure gradient (Pup – Pdown) were measured using the Arduino Uno and Arduino
monitor. Images of the tube were also taken to illustrate the different degrees of collapse.
4.1.3. Results
The chamber pressure was kept constant at atmospheric pressure (101 KPa) along with the
upstream and downstream resistances. Initially as the flow rate was increased the pressure
gradient was observed to drop.
As the flow rate reached about 10 mL/s (100 on figure 25) oscillations occured as the
downstream pressure fell below the chamber pressure (101 KPa) causing oscillations to occur
in the tube.
The oscillations consisted of a rapid collapse followed a more gradual relaxation and filling of
the tube. (Figures 29 and 30) show the tube at different stages of oscillation. Although (figure
30) is at a higher flow rate than 29 the downstream end of the tube is more collapsed. This is
because the tube has just compressed in its oscillation.
When the flow rate was about 30 mL/s (labelled 6 on figure 25) the increases in flow rate
were matched with an increase in the pressure gradient. From figure 32 the tube was full and
behaved as a circular tube.
There came a point at which the increases in the flow rate were no longer matched by an
increase in pressure gradient. At this point (labelled 7 on figure 25) pressure gradient
dropped with increasing flow rate. From (figure 33) the downstream end was fully distended.
If the flow rate was increased beyond this point there was a “dip” in both the pressure
difference and flow rate, this occured at (region 8 on figure 25). From (figure 35) it was
observed that the downstream end of the tube had a build-up of fluid causing the wall of the
tube to stretch.
As the tube was filled with water the flow rate was quickly dropped and the experiment was
concluded.
The experiment was repeated at a higher chamber pressure (200KPa) with similar trends
were demonstrated with sharper oscillations (figure 26).

32
Figure 25. Pressure gradient (brown) and flow rate (Yellow) plotted against time. With regions labelled 1-9.
Figure 26. Experiment done at a higher chamber pressure. Pressure gradient (yellow), Flow rate (pink) and
Chamber pressure (green).

33
Figure 27. Collapsible tube during flow rate variation
Figure 28. Collapsible tube during flow rate variation, corresponding to point 1 on figure 25

34

35

36

37
4.1.4. Discussion
The initial increases in the flow rate (up to region 6 on figure 25) caused the pressure
gradient along the tube to decrease. At this point there was very little flow going through the
tube and from (figure 27) the tube was observed to be fully collapsed. Once a sufficient flow
was supplied this increased the pressure at the downstream end. From the early tests the
increases in the flow rate had little effect on the upstream pressure, hence the pressure
gradient fell initially.
At higher flow rates (up to region 6) when the resistance of the tap was negligible, increases
in flow (and hence fluid velocity) corresponded to an increase in the pressure gradient. This
conformed to Poisseuille flow (equation 9) which is assumed for a fully open tube with a
positive transmural pressure (Heil and Jensen 2002).
At point 7 on (figure 25) (flow rate = 55 mL/s) the increases in the flow was matched by a
significant drop in the pressure gradient. From (figure 34) it can be seen that the
downstream end of the tube started to accumulate water. This indicated that the material
properties of the downstream end of the tube have allowed it to accumulate water.

38
As the flow rate supplying the system was increased, fluid started to build up in the
downstream end of the tube giving the appearance of water filling a balloon. The fluid build-
up in the tube resulted in less fluid flowing downstream to the flow sensor and in turn
relieved the pressure downstream of the tube. This accounts for the “dip” in signal that was
observed (figure 25 region 8).
This “water ballooning” effect is worth noting. At this point the resistance at the downstream
tap became so significant (due to a higher flow rate) that it overcame the stiffness of the
collapsible tube wall and in turn the flow (taking the path of least resistance) filled the tube.
Interesting also that the fluid built up at one end of the tube and did not distribute through
the length of the tube (as if one was inflating a long balloon). Although these observations
indicated that that choking may have occurred (as in the system could not accept any higher
flow rates) the drop in pressure gradient and subsequent ballooning was more likely down to
the material properties being compromised.
4.1.5. Wear on Tube
The surgical drainage tube was purchased in early November and was used for experiments
continuously through to March. During this period the downstream end was subjected to
several experiments involving self-induced oscillations and excessive flows that caused a
build-up of fluid at the downstream end. The stretching of the walls that were observed from
this fluid build-up may have changed the mechanical properties of the wall at this region.
Stretching the walls may have plastically deformed the tube making it more susceptible to
“ballooning” and fluid build-up at higher flow rates.
During one experiment done at higher flow rates the tube ruptured at the downstream end.
The tube was removed and the downstream end was observed to have become thinner and
stretched. This implied that the downstream end of the tube had become worn and had
deformed plastically. (Figures 36 and 37) show the downstream end after rupture and how
the this part of the tube had become worn and damaged, in contrast to the upstream end
(figure 38).

39
Figure 36. Downstream end of tube after rupture during experiments
Figure 37. Downstream end of tube after rupture during experiments
Figure 38. Upstream end of tube after rupture during experiments

40
4.1.6. Reynolds Number
Another explanation for these unexpected results is found by looking at the Reynolds
number. When the flow rate was below 50 mL/s the Reynolds number was below 4000. At
flow rates higher than this the flow rate is in the transitional laminar to turbulent region. The
Reynolds number peaked at 4675 which is in the turbulent region.
Once the flow is turbulent this breaks down one of the fundamental assumptions of
Poisseuille’s law (that the flow must be laminar). Once the flow is turbulent the flow rate is
proportional to the square root of the pressure gradient.
Is it clear to see that the complex behaviour of the oscillations, changing the material of the
tube and the transition of the flow from laminar to turbulent all interact to give the unusual
behaviour observed in experiments.

41
4.2. Reduction in Downstream Pressure
4.2.1. Method
This experiment looked into the effect of reducing the downstream pressure. Reducing the
downstream pressure reduces the transmural pressure (Ptrans = Pout – Pin) at the downstream
end so that eventually Ptrans is negative. Once the transmural pressure becomes negative self-
induced oscillations occur. Before the experiment was run the downstream resistance was
set to as high as possible without “ballooning” occurring.
4.2.2. Results
The downstream resistance was gradually reduced decreasing the downstream pressure. The
upstream resistance was kept constant while the chamber pressure was kept at its elevated
state of 200 KPa. Initial reductions in Pdown had little effect on the other parameters of the
system except for a steady decline in the upstream pressure.
As soon as Pdown fell below the chamber pressure (and the transmural pressure at this point
became negative) there was jump in the flow rate from 10 mL/s and 15 mL/s (figure 39).
Once the flow rate had increased the tube started to oscillate. These initial oscillations
indicated a simultaneous drop in both the flow rate and the downstream pressure.
Around the 170 KPa region the tube was very sensitive to changes in Pdown. At first the
oscillations were infrequent. Once Pdown was decreased slightly further the oscillations
become more frequent (although interestingly of a similar magnitude).
Once Pdown had been further reduced to 150 KPa (so Ptrans = -50KPa) the oscillations increased
in frequency and the behaviours of the flow rate and downstream pressure became more
erratic with larger fluctuations (Figure 39).
The experiment was repeated for starting flow rates of 10 mL/s, 20 mL/s and 30 mL/s. All
experiments demonstrated similar trends with oscillations becoming increasingly erratic as
Ptrans becomes more negative (figures 40 to 42).

42
Figure 39. Effect of reducing downstream pressure. Downstream pressure (Yellow), Flow Rate
(Green), Chamber Pressure (Brown), Test Signal (Pink).
4.2.3. Discussion
As Pdown is initially reduced there was very little effect on the flow rate. This is due to the fact
that the pressure difference was not the main driving force of the flow (it is in fact the
household tap that supplies the water).
The “jump” in flow rate occured when the transmural pressure was negative but before
oscillations had occurred. This indicated that the fluid was exiting the tube at a quicker rate
because the path had been constricted, although by the time the fluid reached the pressure
sensor the conduit had returned to the 15 mm diameter.
The spikes in the downstream pressure in (figure 39) illustrate mechanics of behaviour.
During oscillations the downstream end of the tube started from collapsed and gradually
filled, as it was filling less fluid iwas travelling to the flow sensor. This would account for the
temporary drop in flow rate and pressure. Once the tube was full the tube then contracted
rapidly.
This 170 KPa point seems to be a critical point for the system. There is fine line between the
gradual, low frequency oscillations and the erratic, high frequency oscillations beyond the
third vertical gridline. Over the course of the experiment Pdown is decreased further. For the
initial set of regular oscillations the drops in both the flow rate and the downstream pressure
appear to be in phase with each other.

43
Explanations for this could be; a decrease in the flow rate means that less flow getting to
sensor meaning fluid is building up in the tube, this momentary stagnation in the flow as the
tube refills results in a higher pressure at the downstream end. Almost immediately
afterwards the systems stabilises.
Once Pdown has been reduced around the 150 KPa mark the oscillations become highly erratic,
the fluctuations in Pdown and flow rate no longer appear to be in phase with each other.
Furthermore although the amplitude of the fluctuations in the flow rate increase the flow
rate appears to actually increase as well.
The different frequencies of oscillations indicate different modes of oscillation occurring and
possibly interacting. (Bertram 1982, Bertram an Raymond 1989) have demonstrated similar
behaviours involving low frequency oscillations at low flow rates transitioning to higher
frequency, “chaotic” oscillations at lower transmural pressures. (Jensen 1989, and Bertram
and Raymond 1990) have suggested that for short tubes the modes of oscillations are more
likely to interact with other instabilities such as flow separation and flutter. (Walsh 1995) also
suggests that flutter can onset pulses at a lower flow velocity than the critical flow velocity
for self-induced oscillations.
From the results presented in (figure 39) it would seem that once the downstream pressure
reached 170 KPa the onset of self-induced oscillations caused the steady dips in pressure and
flow rate. Afterwards, as the downstream pressure was further decreased the influence of
flutter became more prominent. This complex interaction between flutter and self-induced
oscillations caused the chaotic unsteady response on the graph.
4.2.4. Reynolds Number
The Reynolds number in these experiments did not exceed 4000 hence the flow was not
turbulent. In the regions of interest the Re was below 2100 (27 mL/s) and hence the flow was
laminar. The behaviours observed however arise from the elastic properties of the tube
rather than the transition of the flow.

44
Figure 40. Reduction in Downstream Pressure at Flow Rate 30 mL/s. Downstream pressure
(Blue), Flow Rate (Pink), Chamber Pressure (Brown), Test Signal (Purple).
(Green), Flow Rate (Pink), , Test Signal (Yellow).

45
(Pink), Flow Rate (Purple), Chamber Pressure (Blue), Test Signal (Brown).
Figure 43. Collapsible tube during downstream pressure reduction experiment, corresponding to a positive
transmural pressure

46
Figure 44. Collapsible tube during downstream pressure reduction experiment, corresponding to a positive
transmural pressure
Figure 45. Collapsible tube during downstream pressure reduction experiment, corresponding to where the
downstream pressure is at the critical 170 KPa region, just before collapse.

47
Figure 46. Collapsible tube during downstream pressure reduction experiment, corresponding to a negative
transmural pressure with self-induced oscillations.
Figure 47. Collapsible tube during downstream pressure reduction experiment, corresponding to a negative
transmural pressure with self-induced oscillations.

48
4.3. Moens-Korteweg Wavespeed
This experiment was to investigate the Moens-Korteweg wave speed equation. This included
comparing the value of pulse propagations obtained experimentally with the wave speed
predicted by the Moens-Koretweg equation.
The Starling Resistor was set-up so that it would start from a high flow rate. The flow rate would
be decreased gradually. The point at which oscillations just started to occur (low frequency
oscillations) were taken to be the point at which the flow speed u matched the wave speed c of
small pressure pulses up the tube, this was predicted to be the point at which oscillations occur
(Shapiro 1977). Hence the flow speed at which oscillations started to occur gave an indication of
the pressure pulse wave speed and could be compared to the prediction from Moens-Korteweg
equation.
4.3.1. Calculation
The predicted wave speed is given by the Moens-Korteweg equation;
𝑐 = √
𝐸ℎ
𝜌𝑑
Were c is the wave speed, E is young’s modulus, h is the tube thickness, ρ is the density of
the fluid and d is the diameter of the tube.
The following properties of the tube are shown in table 2.
Property Value
Young’s Modulus 1.6×105
Pa
Tube Thickness 0.0005 m
Density of Fluid (Water) 999.97 kg/m3
Diameter of Tube 0.0127 m
Table 2. Properties of tube used to calculate Moens Korteweg Speed
The values yielded the following result;
𝑐 = √
𝐸ℎ
𝜌𝑑
= √
1.6×105×0.0005
999.97×0.0127
= 2.5 𝑚/𝑠

49
4.3.2. Method
The Arduino was programmed so that it would display the upstream pressure, downstream
pressure, the flow rate and the flow speed on the monitor. The pressure values were used as
they would indicate when the tube oscillated through pressure drops. As the signals from the
pressure sensors were analogue there were fewer fluctuations in the signal compared to the
flow meter resulting in a clearer indication of oscillations. The signal for velocity was
multiplied by 1000 so that it was more clearly visible on the monitor.
The pulse velocity was also recorded by timing how long the pulse took to travel down the
length of the tube. This could then be used to calculate the wave speed. This was done by
observing videos of the pulse propagations frame by frame.
The flow rate was lowered from an initially high value until the first evidence of oscillations
were observed. This was velocity was recorded and compared to the predicted wave speed.
The point at which oscillations occurred was around 25 mL/s, the Reynolds number at this
point is 2125 which is just above the laminar limit into the transitional region. The flow rate
was slightly varied about this point and the flow rate was assumed to be laminar.
4.3.3. Results
Figure 48. Investigations in Moens Korteweg Speed.
Flow Rate (Pink), Flow Velocity (Yellow), Upstream Pressure (Blue), Downstream Pressure (Brown).

50
The flow rate was dropped from an initial value of 30 mL/s and as expected the flow velocity
dropped proportionally. The pressure values (both upstream and downstream) fell in
accordance with Q as well, in line with Poisseuilles’ law. (Figures 49 – 51) show that the tube
was full during these flow rate indicating a positive transmural pressure.
When the flow velocity was around the 0.22 m/s region the first signs of instability in the
tube were observed. At this point the tube was collapsing with the tube area closing. At the
0.2 m/s region the tube started to exhibit low frequency oscillations. This was characterised
by a period were the tube would appear to collapse at the downstream end (figure 56)
followed by a pulse of fluid with starting at the downstream end and propagating upstream.
The pressure pulse starting at the downstream end can be seen in (figure 57).
The flow rate was then increased in order to see if any oscillations occurred beyond this
point. Once the flow rate was decreased again oscillations were again observed to occur.
Figure (56) clearly demonstrates the propagation of the pressure pulse up the tube.
The critical value of flow velocity was observed to be around 0.22 m/s
Frame by frame Pressure Pulse observation
The video taken was viewed on Windows Movie Maker which allowed the video frame to be
observed individually by a hundredth of a second. (Figures 56 -57) show in detail the
pressure pulse propagation corresponding to region 4 on (figure 48).
The pulse was observed to take 1.23 seconds to travel the length of the tube. The length of
the tube was 32 cm giving a wave speed of 0.26 m/s
4.3.4. Discussion
The flow exhibited instability and started to oscillation once the flow speed fell below 0.22
m/s. Collapsible tube theory implies that once the flow speed u and wave speed c are equal
the flow becomes unstable and starts to oscillate. From this it can be deduced that the
pressure waves are approximately 0.22m/s.
From the video recording of the pressure pulses the wave speed does appear to be 0.26 m/s.
Which is an error of about 15%. This is nonetheless encouraging that the observed wave
speed is the same as that predicted by the theory analogous to 1D gas dynamics (Shapiro
1977).

51
There is however quite a large discrepancy between these wave speeds and that predicted
by the Moens Korteweg equation. In fact the predicted wave speed is about 10 times larger
than the ones observed from the experiments. The Moens-Korteweg speed assumes small
amplitude waves and a positive transmural pressure (hence circular cross section), both of
which do not apply to this case. Given these assumptions it is little surprise that the
discrepancy between the predicted and recorded wave speed was so large.
Figure 49. Tube during Moens Korteweg experiment corresponding to region 1 on graph

52

53

54

55
05.04
05.05
05.05
05.06
05.07
Figure 56. Screenshots of a pressure wave propagating upstream

56
1:22.24
1:22.77
1:23.14
1:23.47
Figure 57. Screenshots of pressure wave propagating upstream the tube over a period of 1.23 seconds

57
5. Conclusion
The project overall has been a success. The three main objectives of the project have been met.
These objectives were;
 Conduct a literature review into the theory of flow through collapsible tubes. Determine
the behaviours to investigate.
 Modify the Starling Resistor so that it can support quantitative experiments, this included
obtaining suitable tubes and installing suitable sensors and electronics.
 Conduct experiments that demonstrate the behaviours of collapsible tubes.
Modifications were particularly successful with the Starling Resistor being fitted with electronic
sensors. The measurements can be displayed through a standard laptop on an interface that updates
in real time. This makes future experiments either in a research or teaching capacity much more
useful. Suitable tubes and garden taps were purchased which also enhanced experimental work.
With regards to degrees of collapse and self-induced oscillations the experiments that were
conducted demonstrated these behaviours well with good images and measurements being taken
for the parameters at which these phenomena occurred. Flow limitation and choking were not
exhibited because the flow was driven by the sink tap rather than the pressure difference upstream
and downstream.
The results for the Moens Korteweg wave speed were mixed. The experimental observations both
found the pressure wave speed to be between 0.22 and 0.26 m/s. This in turn can be interpreted as a
validation of the theory that oscillations occur when the flow speed equals the wave speed. Both of
these values however, were out by a factor of ten to the wave-speed predicted by the Moens-
Korteweg equation.
In terms of organisation the first semester plan was well adhered to. The literature review was
completed by week 6 and the first set of modifications were completed by week 10 ready for the
interim report.
The second semester activities were delayed by the need for electrical sensors as oppose to
mechanical gauges. This was due to the lack of sensitivity of the pressure gauges. This setback
however had little impact on the overall plan. Although modifications took longer than expected,
once the apparatus was functioning the experiments took quicker than originally planned. The first
draft of the final report was finished by the second week of the Easter break, only a week behind
schedule just under a month before the final report deadline.

58
Overall this project has resulted in a significantly enhanced Starling Resistor that has demonstrated
some of the interesting behaviour that occurs in a collapsible tube. The current Starling Resistor
provides a good basis for future work in area for either research or teaching.

59
6. Recommendations
The main modification that could be made to the Starling Resistor is a tank that could supply a more
compliant flow rate. Currently the flow is dictated solely by a household tap. This means that
although that upstream and downstream taps have some influence it means that the range at which
the pressure can be varied is rather limited.
If the water was supplied to the resistor from the tap via an intermediate tank then the flow rate
would be more compliant and ultimately depend on the upstream and downstream resistances. This
would improve the likelihood of observing flow limitation.
For validating the Moens-Kortweg wave speed an alternative approach is required. This would
involve maintaining a positive transmural pressure and increasing the flow speed until it matched the
wave speed predicted by the Moens-Korteweg equation.

60
7. References
1. Anderson, J.D. 2011. Fundamentals of Aerodynamics. Singapore. McGraw-Hill
2. Barclay, W.H., Thalayasingam, S., 1986 Self-excited oscillations in thin-walled collapsible
tubes. Medical and Biological Engineering and Computing. 24, 482-487
3. BERTRAM, C. D. 1982 Two modes of instability in a thick-walled collapsible tube conveying a
flow. Journal of Biomechanics 15, 223-224.
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Appendix A – Risk Assessment

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