PSZ 19: 16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
BORANG PENGESAHAN STATUS TESIS♦
JUDUL :
MATHEMATICAL MODELLING OF NON-NEWTONIAN BLOOD
FLOW THROUGH A TAPERED STENOTIC ARTERY
SESI PENGAJIAN :
2005/ 2006
ZUHAILA BINTI ISMAIL
Saya
(HURUF BESAR)
mengaku membenarkan tesis (PSM/Sarjana/Doktor Falsafah)* ini disimpan di Perpustakaan
Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut :
1. Tesis adalah hak milik Universiti Teknologi Malaysia.
2. Perpustakaan Universiti Teknologi Malaysia dibenarkan membuat salinan untuk tujuan
pengajian sahaja.
3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan penukaran antara
institusi pengajian tinggi.
4. ** Sila tandakan (√ )
√
SULIT
(Mengandungi maklumat yang berdarjah keselamatan atau
kepentingan Malaysia seperti yang termaktub di dalam AKTA
RAHSIA RASMI 1972)
TERHAD
(Mengandungi maklumat TERHAD yang telah ditentukan oleh
organisasi/badan di mana penyelidikan dijalankan)
Disahkan oleh
TIDAK TERHAD
(TANDATANGAN PENULIS)
(TANDATANGAN PENYELIA)
Alamat Tetap :
NO 22, TINGKAT DAMAI 1
TAMAN PERMATA,
14000 BKT MERTAJAM, P. PINANG
Tarikh :
CATATAN :
9 MEI 2006
PROF DR NORSARAHAIDA S.AMIN
Nama Penyelia
Tarikh:
9 MEI 2006
* Potong yang tidak berkenaan.
** Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak
berkuasa/organisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis ini perlu
dikelaskan sebagai SULIT atau TERHAD.
♦ Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara
penyelidikan, atau disertai bagi pengajian secara kerja kursus dan penyelidikan, atau
Laporan Projek Sarjana Muda (PSM).
“I declare that I have read through this dissertation and in my opinion it has fulfilled
the requirements in terms of the scope and quality for the purpose of awarding the
Master of Science (Mathematics) degree.”
Signature
:
Name of Supervisor : PROF DR NORSARAHAIDA S. AMIN
Date
:
9 MAY 2006
MATHEMATICAL MODELLING OF NON-NEWTONIAN BLOOD FLOW
THROUGH A TAPERED STENOTIC ARTERY
ZUHAILA BINTI ISMAIL
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
MAY 2006
ii
“I declare that this dissertation entitled “Mathematical Modelling of Non-Newtonian
Blood Flow through a Tapered Stenotic Artery” is the result of my own research
except as cited in references. The dissertation has not been accepted for any degree
and is not concurrently
submitted in candidature of any degree”.
Signature
:
Name
:
Date
:
ZUHAILA BINTI ISMAIL
9 MAY 2006
iii
For My Dear Family
iv
ACKNOWLEDGEMENT
First of all, thanks to Almighty Allah s.w.t. for graciously bestowing me the
perseverance to undertake this research. I sincerely have to thank all those who have
supported and helped me writing this report. A special thanks and a deepest
appreciation to my supervisor, Prof Dr Norsarahaida S. Amin who provided guidance
and advice. Dr P.K. Mandal of the Department of Mathematics, Visva-Bharati, West
Bengal, India who assisted me with some materials for this dissertation.
A warmest gratitude to my family especially my parents, Pn. Hajjah Zaiton
Mohd Yusof and Tuan Haji Ismail Mat Hanif for their constant encouragement and
advice. Without their support, it would not have been possible for me to complete
this project.
I also would like to thank my special friend, Nurul Anwar Abdul Aziz for his
willingness to help me when I really need it. Finally, I wish to thank my friends
especially Ilyani Abdullah, Norzieha Mustapha, Wan Rukaida Wan Abdullah, Siti
Aisyah Zulkifli, Nur Ilyana Anwar Apandi, Norhafizah Md Sarif, Noor Zakiah
Yahya for all the wonderful times we have had. I would not have lasted through the
research without your encouragement and support.
v
ABSTRACT
A mathematical model of non-Newtonian blood flow through a tapered
stenotic artery is considered. It has been established that the regional blood rheology
is altered once a stenosis develops. A stenosis is defined as the partial occlusion of
the blood vessels due to the accumulation of cholesterols and fats and the abnormal
growth of tissue. The non-Newtonian model chosen is characterized by the
generalized Power-Law model and the effect of tapering on the arterial segment is
incorporated in the analysis due to the pulsatile nature of blood flow. The flow is
assumed to be unsteady, laminar, two-dimensional and axisymmetric. The equations
of motion in terms of the viscous shear stress in the cylindrical coordinate system are
first derived and then transformed using the radial coordinate transformation before
they are solved numerically using a finite difference scheme. Numerical results
obtained show that the blood flow characteristics such as the velocity profiles, flow
rate, and wall shear stress have lower values while the resistive impedances have
higher values compared to the values obtained from the Newtonian model.
vi
ABSTRAK
Model matematik bagi aliran darah tak Newtonan melalui arteri berstenosis
yang menirus dipertimbangkan. Sifat semulajadi darah akan berubah di sekitar
kawasan di mana terdapatnya stenosis, iaitu himpunan kolesterol dan lemak-lemak
serta pertumbuhan tisu yang luar biasa pada dinding arteri. Model tak Newtonan
yang dipilih bercirikan model ‘Power-Law’, sementara kesan arteri yang menirus dan
mengembang diambil kira di dalam analisis berdasarkan sifat semulajadi aliran darah
yang bergantung kepada denyutan jantung. Aliran darah yang dipertimbangkan
adalah aliran tak mantap dalam dua matra, lamina dan berpaksi simetri. Persamaan
gerakan dalam sebutan tegasan ricih dalam koordinat silinder diterbit dan
dijelmakan menggunakan jelmaan jejari koordinat Persamaan ini diselesaikan secara
berangka menggunakan kaedah beza terhingga. Keputusan berangka yang diperoleh
menunjukkan ciri-ciri aliran darah seperti halaju, kadar aliran dan tegasan ricih
mempunyai nilai yang lebih rendah manakala jumlah rintangan adalah lebih tinggi
berbanding nilai yang diperoleh daripada model Newtonan.
vii
CONTENTS
CHAPTER
I
II
SUBJECT
PAGE
TITLE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABTRACTS
v
ABSTRAK
vi
CONTENTS
vii
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiii
LIST OF TERMINOLOGY
xiv
INTRODUCTION
1
1.1
Introduction
1
1.2
Research Background
3
1.3
Objectives of Research
6
1.4
Scope of Research
7
1.5
Significance of Study
7
1.6
Outline of Dissertation
8
DERIVATION OF THE GOVERNING EQUATIONS
10
2.1
10
Introduction
viii
III
2.2
The Equation of Continuity
10
2.3
The Equations of Motion
15
2.3.1
x - Component of Momentum
15
2.3.2
y - Component of Momentum
19
2.3.3
z - Component of Momentum
22
2.3.4
The Power-law Model
26
2.4
The Governing Equations in Cylindrical Coordinates
29
2.5
Derivation of the Mathematical Model
33
2.6
The Boundary Conditions
34
2.6.1
The Pressure Gradient
35
THE GEOMETRY OF STENOSIS
36
3.1
Introduction
36
3.2
The Geometry of Stenosis
36
3.3
Formulation of the Geometry of Mild Stenosis in a
41
non Tapered Artery
3.4
Formulation of the Geometry of Mild Stenosis in
44
a Tapered Artery
IV
SOLUTION PROCEDURE
49
4.1
Introduction
49
4.2
Transformation of the Governing Equations using
49
Radial Coordinate Transformation
4.3
4.2.1
Transformation of the z Momentum
50
4.2.2
Transformation of the Continuity Equation
52
4.2.3
Transformation of the Normal Stress (τ zz )
52
4.2.4
Transformation of the Shear Stress (τ xz )
53
4.2.5
Transformation of the Boundary Conditions
54
Derivation of the Radial Velocity Component, vr ( x, z , t )
54
ix
V
VI
4.4
Discretization of the Axial Velocity component, vz ( x, z , t )
58
4.5
Discritized Forms of Blood Flow Characteristics
61
4.6
The Numerical Procedure
64
4.7
Some Comments
65
NUMERICAL RESULTS AND DISCUSSION
69
5.1
Introduction
69
5.2
Effect of Tapering on Axial and Radial velocity
69
5.2.1
Different Taper Angle under Stenotic Conditions
69
5.2.2
Effect of Stenosis on Axial and Radial Velocity
71
5.3
Axial and Radial Velocity at Different Times
73
5.4
Axial and Radial Velocity at Different Axial Positions
75
5.5
Variation of Blood Flow Characteristics
77
5.5.1
Variation of the Rate of Flow with Time
77
5.5.2
Variation of the Resistance of Flow with Time
78
5.5.3
Variation of the Wall Shear Stress with Time
79
CONCLUSION
81
6.1
Summary of Research
81
6.2
Suggestions for Future Research
83
REFERENCES
84
Appendix A
89
Appendix B
95
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1.1
Atherosclerosis
2
1.1.2
Stenosis that Exists in Coronary Artery Restricts Blood Flow
2
to the Heart.
2.2.1 Region of Volume ∆x ∆y ∆z Fixed in Space through which
11
a Fluid is Flowing.
2.3.1
Volume Element ∆x ∆y ∆z with Arrows Indicating the
15
Direction in which the x-component of Momentum is
Transported through the Surfaces.
2.4.1
Cylindrical Coordinates.
29
3.2.1
The Geometry of Mild Stenosis.
37
3.2.2
The Geometry of a Cosine-Shaped Stenosis.
38
3.2.3
The Geometry of Bell-Shaped Stenosis.
38
3.2.4
The Geometry of Two Overlapping Stenosis.
39
3.2.5
The Geometry of Three Overlapping Stenosis.
39
3.2.6
The Irregular Stenosis.
40
3.2.7
The Geometry of Multi-Mild Stenosis.
40
3.2.8
The Geometry of Multi-Irregular Stenosis.
41
3.2.9
The Smooth Profile used to Approximate Multi Irregular
41
Stenosis.
3.3.1
The Geometry of Mild Stenosis in non-Tapered (φ = 0 )
42
Artery.
3.4.1
The Geometry of Mild Stenosis in Tapered (φ > 0 ) Artery.
44
xii
3.4.2
The Geometry of Mild Stenosis for Different Angles of
48
Tapering.
4.7.1
Radial Velocity Profile for φ = 0 D
65
4.7.2
Axial Velocity Profile for φ = 0 D
66
4.7.3
Variation of the Rate of Flow for φ = 0 D
66
4.7.4
Variation of the Resistance for φ = 0 D
67
4.7.5
Variation of the Wall Shear Stress for φ = 0 D
67
5.2.1
Axial Velocity Profiles for Different Taper Angles at
70
t = 0.45s.
5.2.2
Radial Velocity Profiles for Different Taper Angles at
71
t = 0.45s.
5.2.3
Axial Velocity Profiles at z = 28mm for t = 0.45s.
72
5.2.4
Radial Velocity Profiles at z = 28mm for t = 0.45s.
73
5.3.1
Axial Velocity Profiles for Different Times at z = 28mm.
74
5.3.2
Radial Velocity Profiles for Different Times at z = 28mm.
75
5.4.1
Axial Velocity Profiles for Different Axial Positions at
76
t = 0.45s.
5.4.2
Radial Velocity Profiles for Different Axial Positions at
77
t = 0.45.
5.5.1
Variation of the Rate of Flow with Time at z = 28mm.
78
5.5.2
Variation of the Resistance of Flow with Time at z = 28mm.
79
5.5.3
Variation of the Wall Shear Stress with Time at z = 28mm.
80
xiii
LIST OF SYMBOLS
R(z, t )
-
the radius of the tapered arterial segment in the stenotic region
a
-
the constant radius of the non-tapered artery in the
non-stenotic region
φ
-
the angle of tapering
lD
-
the length of the stenosis
d
-
the location of the stenosis
-
the critical height of the stenosis for the tapered artery
τ m sec φ
appearing at z = d +
lD
.
2
m
-
the slope of the tapered vessel
ω
-
the angular frequency
fp
-
the pulse frequency
b
-
constant variable
L
-
the finite difference arterial segment
φ<0
-
the converging tapering
φ =0
-
the non-tapered artery
φ >0
-
the diverging tapering
τ rz
-
shear stress
vz ( r , z , t )
-
the axial velocity component
vr ( r , z , t )
-
the radial velocity component
p
-
pressure
ρ
-
density of blood
λ
-
wavelength
xiv
∂p
∂z
-
pressure gradient
AD
-
constant amplitude of the pressure gradient
A1
-
amplitude of the pulsatile component
∆x
-
increment in the radial directions
∆z
-
increment in the axial directions
∆t
-
small time increment
Q
-
volumetric flow rate
∧
-
resistance
τw
-
wall shear stress
V
-
volume
xiv
LIST OF TERMINOLOGY
Atherosclerosis
-
arteriosclerosis characterized by irregularly distributed
lipid deposits in the intima of large and medium-sized
arteries; such deposits provoke fibrosis and
calcification. Atherosclerosis is set in motion when
cells lining the arteries are damaged as a result of high
blood pressure, smoking, toxic substances in the
environment, and other agents. Plaques develop when
high density lipoproteins accumulate at the site of
arterial damage and platelets act to form a fibrous cap
over this fatty core. Deposits impede or eventually shut
off blood flow.
Blood
-
the fluid and its suspended formed elements that are
circulated through the heart, arteries, capillaries, and
veins; blood is the means by which 1) oxygen and
nutritive materials are transported to the tissues, and 2)
carbon dioxide and various metabolic products are
removed for excretion. The blood consists of a pale
yellow or gray-yellow fluid, plasma, in which are
suspended red blood cells (erythrocytes), white blood
cells (leukocytes), and platelets.
Cardiac arrest
-
complete cessation of cardiac activity either electric,
mechanical, or both; may be purposely induced for
therapeutic reasons.
xv
Cardiac cycle
-
the complete round of cardiac systole and diastole with
the intervals between, commencing with any event in
the heart's action and ending when same event is
repeated.
Cardiovascular
-
relating to the heart and the blood vessels or the
circulation.
Cerebrovascular
-
relating to the blood supply to the brain, particularly
with reference to pathologic changes.
Diastole
-
normal postsystolic dilation of the heart cavities,
during which they fill with blood; diastole of the atria
precedes that of the ventricles; diastole of either
chamber alternates rhythmically with systole or
contraction of that chamber.
Diastolic
-
relating to diastole.
Disease
-
an interruption, cessation, or disorder of body
functions, systems, or organs.
Elastic
-
having the property of returning to the original shape
after being compressed, bent, or otherwise distorted or
a rubber or plastic band used in orthodontics as either a
primary or adjunctive source of force to move teeth.
The term is generally modified by an adjective to
describe the direction of the force or the location of the
terminal connecting points.
Erythrocyte
-
a mature red blood cell. Synonym: red blood cell,
haemacyte, red corpuscle, hemacyte.
Hemoglobin
-
the red respiratory protein of erythrocytes, consisting
xvi
of approximately 3.8% heme and 96.2% globin, with a
molecular weight of 64,450, which as oxyhemoglobin
(HbO2) transports oxygen from the lungs to the tissues
where the oxygen is readily released and HbO2
becomes Hb. When Hb is exposed to certain
chemicals, its normal respiratory function is blocked;
e.g., the oxygen in HbO2 is easily displaced by carbon
monoxide, thereby resulting in the formation of fairly
stable carboxyhemoglobin (HbCO), as in asphyxiation
resulting from inhalation of exhaust fumes from
gasoline engines. When the iron in Hb is oxidized from
the ferrous to ferric state, as in poisoning with nitrates
and certain other chemicals, a nonrespiratory
compound, methemoglobin (MetHb), is formed.
Hypertension
-
high blood pressure; generally established guidelines
are values of more than 140 mmHg systolic, or more
than 90 mmHg diastolic blood pressure. Despite many
discrete and inherited but rare forms that have been
identified, the evidence is that for the most part blood
pressure is a multifactorial, perhaps galtonian trait.
Laminar
-
arranged in plates or laminae.
Laminar flow
-
the relative motion of elements of a fluid along smooth
parallel paths, which occurs at lower values of
Reynolds number.
Stenosis
-
a stricture of any canal; especially, a narrowing of one
of the cardiac valves.
Stenotic
-
narrowed; affected with stenosis.
xvii
Stress
-
reactions of the body to forces of a deleterious nature,
infections, and various abnormal states that tend to
disturb its normal physiologic equilibrium
(homeostasis) or psychological stimulus such as very
high heat, public criticism, or another noxious agent or
experience which, when impinging upon an individual,
produces psychological strain or disequilibrium.
Systole
-
contraction of the heart, especially of the ventricles, by
which the blood is driven through the aorta and
pulmonary artery to traverse the systemic and
pulmonary circulations, respectively; its occurrence is
indicated physically by the first sound of the heart
heard on auscultation, by the palpable apex beat, and
by the arterial pulse.
Systolic
-
relating to, or occurring during cardiac systole.
Vessel
-
a structure conveying or containing a fluid, especially a
liquid.
Viscosity
-
in general, the resistance to flow or alteration of shape
by any substance as a result of molecular cohesion;
most frequently applied to liquids as the resistance of a
fluid to flow because of a shearing force.
Viscous
-
sticky; marked by high viscosity.
CHAPTER I
INTRODUCTION
1.1
Introduction
Heart problem is one of the most common causes of death. Angina pectoris
and myocardial infarction are two examples of heart diseases. Angina pectoris is the
term used to describe the pain cause when the vessel is not carrying enough blood to
the heart muscle. The pain occurs especially when the heart muscle needs more
blood. In the case of myocardial infarction or heart attack, part of the heart muscle is
destroyed. This causes severe pain in the chest that can lead to death. A healthy
person is not aware of having a heart, or the important work it does in making the
body works properly. The heart is the strongest organ in the body, and it works like
a pump. The heart is like an engine, which can wear out, and breaks down
completely, this is called cardiac arrest. Actually how likely someone is to have a
heart attack depends on a number of things. The main cause that leads to a heart
attack is atherosclerosis (see Figure 1.1.1).
2
Figure 1.1.1 Atherosclerosis
Chakravarty (1987) mentioned that atherosclerosis occurs when the nature of
blood flow changes from its usual state to a disturbed flow condition due to the
presence of a stenosis in an artery. Stenosis is defined as a partial occlusion of the
vessels caused by abnormal growth of tissues or the deposition of cholesterol as
substances on the arterial wall. This research considers the study of blood flow
through arteries in the presence of stenosis because it can cause the development of
cardiovascular diseases such as stroke and heart attack (see Figure 1.1.2).
Figure 1.1.2 Stenosis that Exists in Coronary Artery
Restricts Blood Flow to the Heart.
The fact that blood exhibits non – Newtonian behaviour was actually first
recognised around the turn of the century (Enderle et al. (2000)). From a biofluid
mechanics point of view, blood would not be expected to obey the very simple, one
parameter, and linearized law of viscosity as developed by Newton. Blood is
3
nonhomongeneous, anisotropic ionic, composite fluid composed of a suspension of
many asymmetric, relatively large, viscoelastic particles carried in a liquid that
contains high molecular weight, asymmetric, ionic that behaves in a complicated way
under shear – type loading. Therefore, blood exhibits non-Newtonian (nonlinear),
time dependent (viscoelastic) deformation (flow) characteristics that can only be
modelled by higher order constitutive equations, such as the power-law paradigm
(Enderle et al. (2000)).
In physiological flows, there are other important factors that can be accounted
for such as the effects of vessel tapering together with the geometry of stenosis. It
has been pointed out that most of the vessels could be considered as long and narrow,
slowly tapering cones. Besides, the process of systolic and diastolic also affects the
vessels segment because it makes the vessel segment converges and diverges. Thus
the effects of vessel tapering together with the non-Newtonian behaviour of the
streaming blood seem to be equally important, hence deserves special attention.
In the next sections, we present the research background for the project
followed by the objectives, scope, significance of research and outline of the
dissertation.
1.2
Research Background
A number of researchers have studied the flow of non-Newtonian fluids with
various perspectives. Ronald L. Fournier (1998) explained about the field of
rheology concerns the deformation and flow behaviour of fluids, the prefix rheo is
from Greek and refers to something that flows because of the particulate nature of
blood. He expected the rheological behaviour of blood to be some what more
complex than a simple fluid such as water. He mentioned that in order to understand
the flow behaviour of blood, one must first define the relationship between shear
stress and the shear rate. Ishikawa et al. (1998) found that the non-Newtonian
4
pulsatile flow through a stenosed tube is different from Newtonian flow. The nonNewtonian property strengthens the peaks of wall shear stress and wall pressure,
weakens the strength of the vortex and reduces the vortex size and separated region.
Therefore, he concluded that non-Newtonian flow is more stable than Newtonian
flow.
Chakravarty and Mandal (1994) studied the unsteady flow behaviour of blood
in an artery under stenotic condition analytically, by considering blood to be a nonNewtonian fluid and by properly accounting for blood viscoelasticity while the
geometry of the stenosis was chosen to be overlapping to some extent, depending on
time. Chakravarty et al. (1996) investigated the effect of a single cycle of body
acceleration on unsteady non-Newtonian blood flow past a time-dependent arterial
stenosis. Mandal (2005) pointed out that in some disease conditions, for example,
patients with severe myocardial infarction, cerebrovascular diseases and
hypertension, blood exhibits non-Newtonian properties. Gijsen et al. (1999) studied
the impact of non-Newtonian properties of blood on the velocity distribution. They
made a comparison between the non-Newtonian fluid model and a Newtonian fluid
at different Reynolds numbers. Comparison reveals that the character of flow of the
non-Newtonian fluid is simulated quite well by using the appropriate Reynolds
number. Cheng Tu and Michel Deville (1995) noticed that for non-Newtonian flow
through 75% stenosis, the influence of the geometrical disturbance affects the flow
over a longer axial range.
John Enderle et al. (2000) pointed out those significant attempts to define
such non-Newtonian behaviour, however did not appear until the 1960s, when
variable-shear rotational viscometers were introduced. Since then, literally dozens of
constitutive models have been proposed that attempt to relate shear stress to shear
rate in the fluid. They said, the most practical of these is an empirical power law
formulation that generalizes Newton’s law of viscosity. R. Manica and A.L. de
Bortoli (2003) presented the simulation of incompressible non-Newtonian flow
through channels with sudden expansion using the Power Law model. The Power
Law model is applied to predict pseudoplastic (shear thinning) and dilatant (shear
thickening) behaviour in such expansions. They pointed out that a better
understanding of non-Newtonian flow through sudden expansions should lead to
5
both the design and development of hydrodynamically more efficient process and to
an improved quality control of the final products.
The effect of vessel tapering is another important factor that should be
considered. Chakravarty and Mandal (2000) formulated the problem on tapered
blood vessel segment having overlapping stenosis. The problem is modelled
mathematically as a thin elastic tube with a circular section containing an
incompressible Newtonian fluid representing blood. Jeffords and Knisley (1956) and
Bloch (1962) pointed out that most of the vessels could be considered as long and
narrow, slowly tapering cones (Chakravarty and Mandal (2000)).
Inside a normal artery, red and white blood cells and other particles can flow
freely to the peripheral organs. The walls of the inner linings of arteries are smooth
and uniform in thickness. As an initial study, Formaggia et al. (2003) and Lee and
Xu (2002) observed blood flow behaviour in non-stenotic vessel or a normal artery.
Over time, however, the stenosis can build up within the artery walls. Quite a good
number of theoretical studies related to blood flow through stenosed arteries have
been carried out recently, Misra and Chakravarty (1989), Chakravarty (1987) and
Chakravarty and Datta (1987). Most of the studies carried out so far have been
focused on the presence of mild or single stenosis as discussed by Chakravarty et al.
(1995, 1996, 2000), Chakravarty and Mandal (1997, 2000), Taylor et al. (1998), Lee
and Xu (2002) and Mandal (2005). Moayeri and Zendehboodi (2003) found that
once a mild stenosis is developed, the resulting flow disorder plays an important role
in the further development of the disease.
In order to update resemblance to the in vivo situation, some studies have
been investigated an overlapping stenosis in blood vessel segment subject to the
pulsatile pressure gradient. Chakravarty and Mandal (1996), noted that the problem
becomes more acute in the presence of an overlapping stenosis in the artery instead
of having a mild stenosis as considered by aforesaid researchers. The study has been
extended by Chakravarty and Mandal (2000) to include the time-dependent geometry
of an overlapping stenosis present in a tapered artery. However, these studies
considered a Newtonian model for blood flow. Beside the mild and overlapping
6
stenosis, Chakravarty and Sannigrahi (1999) gave special attention to multistenoses
which appear in the artery.
There are different methods of solution in solving the problem of blood flow
in normal and stenosed artery. Some researchers are solving analytically and some
of them use numerical methods. Gerrald and Taylor (1977) used the finite difference
method to solve the problem of blood flow in a normal artery. The finite difference
method based on the central difference approximation has been employed by
Chakravarty and Mandal (1994, 1997) and Mandal (2005). Misra and Pal (1999)
observed the blood motion using Crank Nicolson implicit finite difference method.
Runge-Kutta formula has been used by Chakravarty and Mandal (1996, 2000),
Chakravarty et al. (1995, 1996, 2000) and Chakravarty and Sannigrahi (1999).
Beside the finite difference scheme, the finite element method has also been
employed. Sud and Sekhon (1986) used the finite element model of flow in the
normal branched arterial system subject to externally imposed periodic body
acceleration and the relevance works have been extended by Sud and Sekhon (1987)
by considering a stenosed artery. Formaggia et. al. (2003) presented a finite element
Taylor-Galerkin scheme combined with operator splitting techniques in order to
carry out several test cases.
1.3
Objectives of Research
The main objective of this research is to develop a mathematical model for
non-Newtonian blood flow through a tapered stenotic artery.
The specific objectives are:
1. To derive the governing equations of blood flow, comprising the equation
of continuity and the equation of motion in terms of the viscous shear
stress.
7
2. To formulate the geometry of mild stenosis.
3. To carry out the radial coordinate transformation on the governing
equations.
4. To solve the governing equations numerically using a finite difference
scheme.
1.4
Scope of Research
This research takes into consideration the stenotic blood flow through the
tapered artery to be incompressible, unsteady, two-dimensional and axisymmetric
under laminar flow condition. The flowing blood is treated as a non-Newtonian fluid
that is characterized by the generalized Power-law model and is observed through a
mild stenosis. The discussion of this problem follows from the work of (Mandal
(2005)).
1.5
Significance of Study
The benefits of this study are:
1. The development of a more realistic mathematical model to describe
blood flow.
2. The development of a numerical package for the computation and
simulation of bio-fluid problems.
8
1.6
Outline of Dissertation
This dissertation is divided into six chapters including this introductory
chapter. Section 1.2 – 1.5 present the research background, objectives, scope and
significance of research.
Chapter II presents the derivation of the governing equations. First, we show
the derivation of continuity equation and then the derivation of the equation of
motion in terms of the viscous stress tensor, τ . After that, both equations will be
converted to cylindrical coordinates. The derivations of these formulae are given in
Appendix A. Then, we show the derivation of the mathematical model. The last
section in this chapter states the boundary conditions. The next chapter contains a
discussion on the geometry of stenosis with their mathematical formulation. Then,
we will show how to formulate the geometry of mild stenosis in a non-tapered and
tapered artery.
The following chapter presents the transformation of the governing equations
using the radial coordinate transformation. Then, the derivations of the radial
velocity component, vr ( x, z , t ) and the solution of the axial velocity component,
vz ( x, z , t ) are shown using the finite difference method. In the same section, the
volumetric flow rate, the resistance and the wall shear stress will be determined.
Next, we will show the numerical procedure to programme the finite difference
method using MATLAB programming. The complete program is given in Appendix
B. Lastly, in this chapter we state some comments about the numerical results.
Chapter V discusses the numerical results. This chapter will be divided into
eight sections including the introduction. In Section 5.2, we discuss the results for
the axial and radial velocities at different taper angles under stenotic conditions and
also at taper angle with comparison between flow through stenosis, non stenosis and
steeper stenosis. Next section, the results and discussion for axial and radial velocity
at different times and at different type of fluid in the same times are given. Section
9
5.4 will be presented the results for axial and radial velocity at different axial
positions. The following section will illustrate the results of the variation flow rate,
resistance and wall shear stress with time. Last sections we will state some
comments about the results obtained. Finally, Chapter VI will conclude the research
problem and list out several suggestions for future research.
CHAPTER II
DERIVATION OF THE GOVERNING EQUATIONS
2.1
Introduction
In Section 2.2, we will derive the equation of continuity and the equations of
motion in terms of the viscous stress tensor, τ . Then, in Section 2.3 we will show
how these governing equations are converted into cylindrical coordinates. In the
following section, the governing equations will be reduced to the two-dimensional
equations with the assumptions of axisymmetric, unsteady flow and incompressible
fluid. The non-Newtonian fluid is characterized by the generalized Power-law
model. Lastly in this chapter, we will state the boundary conditions and the pressure
gradient.
2.2
The Equation of Continuity
The governing equations of fluid mechanics are a complex set of nonlinear
partial differential equations that describe the flow of fluid such as liquid and gases.
11
In this research, the equation of continuity and then the equation of motion in terms
of the viscous stress tensor, τ will be derived in order to govern the movement of
blood in a stenos artery.
This equation is developed by writing a mass balance over a stationary
volume element ∆x ∆y ∆z through which the fluid is flowing (see Figure 2.2.1):
⎧ Rate of mass ⎫ ⎧ Rate of ⎫ ⎧ Rate of ⎫
⎨
⎬=⎨
⎬−⎨
⎬
⎩ accumulation ⎭ ⎩ mass in ⎭ ⎩ mass out ⎭
(2.2.1)
z
•
y
( x + ∆x , y + ∆ y , z + ∆ z )
( ρ vx ) x
( ρ vx ) x +∆x
∆z
∆y
( x, y , z )
•
∆x
x
Figure 2.2.1: Region of Volume ∆x ∆y ∆z Fixed in Space through which a
Fluid is Flowing.
Here, for convenience we adopt a Cartesian coordinates system, where the velocity, v
and density, ρ are functions of ( x, y, z ) space and time, t. Fixed in this ( x, y, z )
space is an infinitesimally small element of sides dx, dy and dz (refer Figure 2.2.1).
There is mass flow through this fixed element as shown in Figure 2.2.1. Consider
the pair of faces perpendicular to the x axis. The rate of mass flowing in through the
face at x is
( ρ vx ) x ∆y ∆z .
The rate of mass flowing out through the face at x + ∆x is
(2.2.2)
12
( ρ vx ) x+∆x ∆y ∆z .
(2.2.3)
The pair of faces perpendicular to the y axis. The rate of mass flowing in through the
face at y is
( ρv )
y
y
∆x ∆z .
(2.2.4)
The rate of mass flowing out through the face at y + ∆y is
( ρv )
y
y +∆y
∆x ∆z .
(2.2.5)
The pair of faces perpendicular to the z axis. The rate of mass flowing in through the
face at z is
( ρ vz ) z ∆x ∆y .
(2.2.6)
The rate of mass flowing out through the face at z + ∆z is
( ρ vz ) z +∆z ∆x ∆y .
(2.2.7)
The rate of mass accumulation within the volume element is
( ∆x ∆y ∆z )( ∂ρ
∂t ) .
(2.2.8)
Substituting equations (2.2.2) – (2.2.8) into equation (2.2.1). Then, the mass balance
becomes
∂ρ ⎞
⎡
⎤
⎟ = ∆y ∆z ⎡⎣( ρvx ) x − ( ρvx ) x+∆x ⎤⎦ + ∆x ∆z ⎢⎣( ρvy ) y − ( ρvy ) y+∆y ⎥⎦
⎝ ∂t ⎠
( ∆x ∆y ∆z ) ⎛⎜
(2.2.9)
+ ∆x ∆y ⎡⎣( ρvz ) z − ( ρvz ) z+∆z ⎤⎦ .
13
By dividing equation (2.2.9) with ( ∆x ∆y ∆z ) , we get
⎡ ρv − ρv
⎤
⎛ ∂ρ ⎞ ⎣⎡( ρvx ) x − ( ρvx ) x+∆x ⎦⎤ ⎣⎢( y ) y ( y ) y+∆y ⎦⎥ ⎣⎡( ρvz ) z − ( ρvz ) z+∆z ⎦⎤
+
+
.
⎜ ⎟=
∆x
∆y
∆z
⎝ ∂t ⎠
Taking the limit as ∆x , ∆y and ∆z → 0 . Thus, the above equation becomes
⎛ ∂
⎞
∂ρ
∂
∂
= − ⎜ ρ vx + ρ v y + ρ vz ⎟ .
∂t
∂y
∂z
⎝ ∂x
⎠
(2.2.10)
Equation (2.2.10) is the equation of continuity, which describes the rate of change of
density at a fixed point resulting from the changes in the mass velocity vector ρ v .
Then rewrite equation (2.2.10) in vector form:
∂ρ
= − (∇ ⋅ ρv )
∂t
(2.2.11)
Here, ( ∇ ⋅ ρ v ) is called the ‘divergence’ of ρ v , sometimes written as div ρ v . Note
that, the vector ρ v is the mass flux. There are two other forms of the continuity
equation that are commonly used. One of these is the expanded form obtained by
substituting the vector identity:
∇ ⋅ ( ρ v ) = v ⋅∇ρ + ρ∇ ⋅ v
(2.2.12)
The remaining form is obtained by replacing the first two terms in equation (2.2.12)
by the material derivative,
D ρ ∂ρ
=
+ ( v ⋅∇ ) ρ .
Dt
∂t
D ρ Dt is a symbol for the instantaneous time rate of change of density of the fluid
element as it moves through volume ( ∆x ∆y ∆z ) . By definition, this symbol is called
14
the substantial derivative D Dt . Note that, D ρ Dt is the time rate of change of
density of the given fluid element as it moves through space. Thus, equation (2.2.11)
can rewrite as
Dρ
= −ρ (∇ ⋅ v ) .
Dt
(2.2.13)
The continuity equation involves only the fluid density and the fluid velocity. It
applies to all fluids, compressible and incompressible, Newtonian and nonNewtonian and for the whole range of flow speeds. The equation of continuity in
this form describes the rate of change of density as seen by an observer ‘floating
along’ with the fluid. When expanded in Cartesian coordinates, the continuity
equation is given by
⎛ ∂vx ∂v y ∂vz ⎞
⎛ ∂ρ
∂ρ
∂ρ
∂ρ ⎞
+ vx
+ vy
+ vz
+
+
⎟ = 0.
⎜
⎟+ ρ⎜
∂x
∂y
∂z ⎠
⎝ ∂t
⎝ ∂x ∂y ∂z ⎠
where ρ depends only insignificantly on pressure and temperature. With such
‘incompressible’, or more appropriately constant density, ρ fluids the density and
⎛ Dρ
⎞
therefore the volume of each fluid particle remain constant with time ⎜
= 0⎟ .
⎝ Dt
⎠
Then the continuity equation reduces to
∇⋅v = 0 .
(2.2.14)
Thus, in Cartesian coordinates the velocity field must satisfy
∂vx ∂v y ∂vz
+
+
= 0.
∂x ∂y ∂z
(2.2.15)
15
2.3
The Equations of Motion
This section describes the steps taken to derive the equations of motion.
Now, for a volume element ∆x ∆y ∆z , a momentum balance is given by
⎧ Rate of ⎫ ⎧ Rate of ⎫ ⎧ Rate of ⎫ ⎧Sum of forces⎫
⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪
⎨ momentum ⎬ = ⎨momentum⎬ − ⎨momentum⎬ + ⎨ acting on ⎬
⎪accumulation⎪ ⎪ in
⎪ ⎪ out
⎪ ⎪ system ⎪
⎩
⎭ ⎩
⎭ ⎩
⎭ ⎩
⎭
(2.3.1)
z
•
y
( x + ∆x , y + ∆ y , z + ∆ z )
(τ xx ) x
(τ xx ) x +∆x
∆z
∆y
( x, y , z )
•
∆x
x
Figure 2.3.1: Volume Element ∆x ∆y ∆z with Arrows Indicating the Direction in
which the x-component of Momentum is Transported through the Surfaces.
However, in order to consider the unsteady-state behaviour, we will allow the fluid to
move through all the six faces of volume element in any arbitrary direction.
Equation (2.3.1) is a vector equation with components in each of the three coordinate
directions x, y, and z.
2.3.1
x - Component of Momentum
Let us consider the rates of flow of the x-component of momentum flowing
into and out of the volume element shown in Figure (2.3.1). Momentum flows into
and out of the volume element by two mechanisms. First, by convection (that is by
16
virtue of the bulk fluid flow) and then by molecular transfer (that is by virtue of the
velocity gradients or the shear and normal stress distributions acting on the surface,
also imposed by the outside fluid “ tugging” or “pushing” on the surface by means of
friction).
By convection, the rate at which the x-component of momentum enters the faces at x,
ρ vx vx x ∆y ∆z .
(2.3.2)
The rate at which it leaves at x + ∆x is,
ρ v x vx
x +∆x
∆y ∆z .
(2.3.3)
The rate at which it enters the faces at y,
ρ v y vx y ∆x ∆z .
(2.3.4)
The rate at which it leaves at y + ∆y is,
ρ v y vx
y +∆y
∆x ∆z .
(2.3.5)
The rate at which it enters the faces at z,
ρ vz vx z ∆x ∆y .
(2.3.6)
The rate at which it leaves at z + ∆z is,
ρ vz vx
z +∆z
∆x ∆y .
(2.3.7)
17
Then the convection flow of x-momentum must be considered across all six faces
(2.3.2) - (2.3.7) and that the net convective x-momentum flow into the volume
element is
∆y ∆z ( ρ vx vx x − ρ vx vx
x +∆x
) + ∆x ∆ z ( ρ v v
y x y
− ρ v y vx
y +∆y
)
(2.3.8)
+ ∆x ∆y ( ρ vz vx z − ρ vz vx
z +∆z
).
By molecular transport, the rate at which the x-component of momentum enters the
faces at x,
τ x x x ∆y ∆z .
(2.3.9)
The rate at which it leaves at x + ∆x is,
τ xx
x +∆x
∆y ∆z .
(2.3.10)
The rate at which it enters the faces at y,
τ y x y ∆x ∆z .
(2.3.11)
The rate at which it leaves at y + ∆y is,
τ yx
y +∆y
∆x ∆z .
(2.3.12)
The rate at which it enters the faces at z,
τ z x z ∆x ∆y .
The rate at which it leaves at z + ∆z is,
(2.3.13)
18
τ zx
z +∆z
∆x ∆y .
(2.3.14)
Note that, these momentum fluxes may be considered as stresses. τ xx is a normal
stress on the x-face. τ yx is the x directed tangential (or shear) stress on the y face
resulting from viscous forces. In most cases, the only important forces will be those
arising from the fluid pressure, p and the gravitational force per unit mass, g . The
resultant of these forces in the x-direction is
∆y ∆z ( p x − p x +∆x ) + ρ g x ∆x ∆y ∆z
(2.3.15)
Pressure in a moving fluid is defined by the equation of state p = p ( ρ , T ) . The rate
of accumulation of x momentum with the element is
⎛ ∂ρ vx ⎞
∆x ∆y ∆z ⎜
⎟
⎝ ∂t ⎠
(2.3.16)
Substituting equations (2.3.8), (2.3.9) - (2.3.16) into equation (2.3.1). Therefore
equation (2.3.1) becomes,
⎛ ∂ρ vx ⎞
∆x ∆y ∆z ⎜
⎟ = ∆y ∆z ( ρ vx vx x − ρ vx vx
⎝ ∂t ⎠
+ ∆x ∆y ( ρ vz vx z − ρ vz vx
(
+ ∆x ∆z τ yx − τ yx
y
y +∆y
z +∆z
x +∆x
) + ∆y ∆z (τ
) + ∆x ∆y (τ
zx z
) + ∆ x ∆z ( ρ v v
xx x
− τ zx
+ ∆y ∆z ( p x − p x +∆x ) + ρ g x ∆x ∆y ∆z.
Dividing the entire result with ∆x ∆y ∆z , then we get
y x y
− τ xx
z +∆z
x +∆x
)+
)
− ρ v y vx
y +∆y
)
19
∂ρ vx ⎛ ρ vx vx x − ρ vx vx
= ⎜⎜
∂t
∆x
⎝
⎛ τ xx x − τ xx
+ ⎜⎜
∆x
⎝
x +∆x
x +∆x
⎞ ⎛ ρ v y vx y − ρ v y vx
⎟⎟ + ⎜
∆y
⎠ ⎜⎝
⎞ ⎛ τ yx y − τ yx
⎟⎟ + ⎜
∆y
⎠ ⎜⎝
⎛ p − p x +∆x
+ ⎜⎜ x
∆x
⎝
y +∆y
⎞ ⎛ τ −τ
⎟ + ⎜ zx z zx
⎟ ⎝
∆z
⎠
y +∆y
z +∆z
⎞ ⎛ ρv v − ρv v
z x
⎟+⎜ z x z
⎟ ⎝
∆z
⎠
z +∆z
⎞
⎟
⎠
⎞
⎟+
⎠
⎞
⎟⎟ + ρ g x .
⎠
Taking the limit as ∆x , ∆y and ∆z → 0 gives the x-component of the equation of
motion as
⎛ ∂
⎞
∂
∂
∂
( ρ vx ) = − ⎜ ρ vx v x + ρ v y v x + ρ v z v x ⎟
∂t
∂y
∂z
⎝ ∂x
⎠
(2.3.17)
⎛ ∂
⎞ ∂p
∂
∂
− ⎜ τ xx + τ yx + τ zx ⎟ − + ρ g x .
∂y
∂z ⎠ ∂x
⎝ ∂x
2.3.2
y - Component of Momentum
Now, consider the rates of flow of the y-component of momentum into and
out of the volume element shown in Figure (2.3.1). By convection, the rate at which
the y-component of momentum enter the faces at x, x + ∆x , y, y + ∆y , z and z + ∆z
are the same as equations (2.3.2) – (2.3.7) but the third term changes to v y .
ρ vx v y x ∆y ∆z
(2.3.18)
ρ vx v y
(2.3.19)
x +∆x
∆y ∆z
ρ v y v y y ∆x ∆z
(2.3.20)
20
ρ vy vy
y +∆y
∆x ∆z
(2.3.21)
ρ vz v y z ∆x ∆y
(2.3.22)
ρ vz v y
(2.3.23)
z +∆z
∆x ∆y
Then the convection flow of y-momentum must be considered across all six faces
(2.3.18) - (2.3.23) and that the net convective y-momentum flow into the volume
element is
(
∆y ∆z ρ vx v y − ρ vx v y
x
(
x +∆x
) + ∆x ∆z ( ρ v v
+ ∆x ∆y ρ vz v y − ρ vz v y
z
y y y
z +∆z
− ρ vy vy
y +∆y
)
).
(2.3.24)
By molecular transport, the rate at which the y-component of momentum enter the
faces at x, x + ∆x , y, y + ∆y , z and z + ∆z are the same as equations (2.3.9) –
(2.3.14) but the second term in the subscript of τ changes from x to y.
τ x y x ∆y ∆z
(2.3.25)
τxy
(2.3.26)
x +∆x
∆y ∆z
τ y y y ∆x ∆z
(2.3.27)
τ yy
(2.3.28)
y +∆y
∆x ∆z
τ z y z ∆x ∆y
(2.3.29)
τzy
(2.3.30)
z +∆z
∆x ∆y
21
The resultant of these forces in the y-direction is
(
)
∆x ∆z p y − p y +∆y + ρ g y ∆x ∆y ∆z .
(2.3.31)
The rate of accumulation of y momentum with the element is
⎛ ∂ρ v y
∆x ∆y ∆z ⎜
⎝ ∂t
⎞
⎟.
⎠
(2.3.32)
Substituting equations (2.3.24),(2.3.25) - (2.3.32) into equation (2.3.1). Therefore
equation (2.3.1) can rewrite as
⎛ ∂ρ v y ⎞
∆x ∆y ∆z ⎜
⎟ = ∆y ∆z ρ vx v y x − ρ vx v y
⎝ ∂t ⎠
(
(
+ ∆x ∆y ρ vz v y − ρ vz v y
(
z
+ ∆x ∆z τ yy − τ yy
y
(
y +∆y
z +∆z
x +∆x
) + ∆y ∆z (τ
) + ∆x ∆y (τ
zy z
) + ∆x ∆z ( ρ v v
xy x
− τ zy
y y y
− τ xy
z +∆z
x +∆x
− ρ vy vy
y +∆y
)
)
)+
)
+ ∆y ∆z p y − p y +∆y + ρ g y ∆x ∆y ∆z.
Dividing the entire result with ∆x ∆y ∆z . Then we obtain
⎛ ρ vx v y − ρ vx v y
x
=⎜
⎜
∂t
∆x
⎝
∂ρ v y
⎛ τ xy − τ xy
x
+⎜
⎜
∆x
⎝
x +∆x
x +∆x
⎞ ⎛ ρ vy vy − ρ vy vy
y
⎟+⎜
⎟ ⎜
∆y
⎠ ⎝
⎞ ⎛ τ yy − τ yy
y
⎟+⎜
⎟ ⎜
∆y
⎠ ⎝
⎛ p y − p y +∆y
+⎜
⎜
∆y
⎝
⎞
⎟ + ρ gy.
⎟
⎠
y +∆y
⎞ ⎛ τ −τ
⎟ + ⎜ zy z zy
⎟ ⎜
∆z
⎠ ⎝
y +∆y
z +∆z
⎞ ⎛ ρv v − ρv v
z y
⎟+⎜ z y z
⎟ ⎜
∆z
⎠ ⎝
⎞
⎟+
⎟
⎠
z +∆z
⎞
⎟
⎟
⎠
22
Taking the limit as ∆x , ∆y and ∆z → 0 . Thus, the y-component of the equation of
motion is
⎛ ∂
⎞
∂
∂
∂
ρ v y ) = − ⎜ ρ vx v y + ρ v y v y + ρ vz v y ⎟
(
∂t
∂y
∂z
⎝ ∂x
⎠
(2.3.33)
⎛ ∂
⎞ ∂p
∂
∂
− ⎜ τ xy + τ yy + τ zy ⎟ − + ρ g y .
∂y
∂z ⎠ ∂y
⎝ ∂x
2.3.3
z - Component of Momentum
Consider the rates of flow of the z-component of momentum into and out of
the volume element shown in Figure (2.3.1). By convection, the rate at which the zcomponent of momentum enter the faces at x, x + ∆x , y, y + ∆y , z and z + ∆z are the
same as equations (2.3.2) – (2.3.7) and (2.3.18) – (2.3.23) but the third term changes
to vz .
ρ vx vz x ∆y ∆z
(2.3.34)
ρ vx vz
(2.3.35)
x +∆x
∆y ∆z
ρ v y vz y ∆x ∆z
(2.3.36)
ρ v y vz
(2.3.37)
y +∆y
∆x ∆z
ρ vz vz z ∆x ∆y
(2.3.38)
ρ vz vz
(2.3.39)
z +∆z
∆x ∆y
23
Then the convection flow of z-momentum must be considered across all six faces
(2.3.34) - (2.3.39) and that the net convective z-momentum flow into the volume
element is
∆y ∆z ( ρ vx vz x − ρ vx vz
x +∆x
) + ∆x ∆ z ( ρ v v
y z y
− ρ v y vz
y +∆y
)
(2.3.40)
+ ∆x ∆y ( ρ vz vz z − ρ vz vz
z +∆z
).
By molecular transport, the rate at which the z-component of momentum enter the
faces at x, x + ∆x , y, y + ∆y , z and z + ∆z are the same as equations (2.3.9) –
(2.3.14) but the second term in the subscript of τ changes from x to z.
τ x z x ∆y ∆z
(2.3.41)
τ xz
(2.3.42)
x +∆x
∆y ∆z
τ y z y ∆x ∆z
(2.3.43)
τ yz
(2.3.44)
y +∆y
∆x ∆z
τ z z z ∆x ∆y
(2.3.45)
τ zz
(2.3.46)
z +∆z
∆x ∆y
The resultant of these forces in the z-direction is
∆x ∆z ( p z − p z +∆z ) + ρ g z ∆x ∆y ∆z .
The rate of accumulation of z momentum with the element is
(2.3.47)
24
⎛ ∂ρ vz
∆x ∆y ∆z ⎜
⎝ ∂t
⎞
⎟.
⎠
(2.3.48)
Substituting equations (2.3.40),(2.3.41) - (2.3.48) into equation (2.3.1). Therefore
equation (2.3.1) can rewrite as
⎛ ∂ρ vz
∆x ∆y ∆z ⎜
⎝ ∂t
⎞
⎟ = ∆y ∆ z ( ρ v x v z x − ρ v x v z
⎠
+ ∆x ∆y ( ρ vz vz z − ρ vz vz
(
+ ∆x ∆z τ yz − τ yz
y
y +∆y
z +∆z
x +∆x
) + ∆y ∆z (τ
) + ∆x ∆y (τ
zz z
) + ∆x ∆z ( ρ v v
xz x
− τ zz
y z y
− τ xz
z +∆z
x +∆x
− ρ v y vz
y +∆y
)
)
)+
+ ∆y ∆z ( p z − p z +∆z ) + ρ g z ∆x ∆y ∆z.
Dividing the entire result with ∆x ∆y ∆z . Then we get
∂ρ vz ⎛ ρ vx vz x − ρ vx vz
= ⎜⎜
∂t
∆x
⎝
⎛ τ xz − τ xz
+ ⎜⎜ x
∆x
⎝
x +∆x
x +∆x
⎞ ⎛ ρ v y vz y − ρ v y vz
⎟⎟ + ⎜
∆y
⎠ ⎜⎝
⎞ ⎛ τ yz y − τ yz
⎟⎟ + ⎜
∆y
⎠ ⎜⎝
⎛ p − p z +∆z
+⎜ z
∆z
⎝
y +∆y
⎞ ⎛ τ −τ
⎟ + ⎜ zz z zz
⎟ ⎝
∆z
⎠
y +∆y
z +∆z
⎞ ⎛ ρv v − ρv v
z z
⎟+⎜ z z z
⎟ ⎝
∆z
⎠
z +∆z
⎞
⎟+
⎠
⎞
⎟ + ρ gz.
⎠
Taking the limit as ∆x , ∆y and ∆z → 0 . Thus, the y-component of the equation of
motion is
⎞
⎟
⎠
25
⎛ ∂
⎞
∂
∂
∂
( ρ vz ) = − ⎜ ρ v x vz + ρ v y vz + ρ v z vz ⎟
∂t
∂y
∂z
⎝ ∂x
⎠
(2.3.49)
⎛ ∂
⎞ ∂p
∂
∂
− ⎜ τ xz + τ yz + τ zz ⎟ − + ρ g z .
∂y
∂z ⎠ ∂z
⎝ ∂x
The quantities ρ vx , ρ v y , ρ vz are the components of the mass velocity vector ρ v .
Similarly with g x , g y , g z which are the components of the gravitational acceleration
g . Furthermore,
∂p ∂p ∂p
, ,
are the components of a vector ∇p , pvx vx , pvx v y ,
∂x ∂y ∂z
pvx vz , pv y vx , pv y v y , pv y vz , pvz vx , pvz v y , pvz vz are the nine components of the
convective momentum flux pv v . Similarly, τ xx ,τ xy ,τ xz ,τ yx ,τ yy ,τ yz ,τ zx ,τ zy ,τ zz are the
nine components of τ known as the “stress tensor”. Equations (2.3.17), (2.3.33) and
(2.3.49) take up so much space; it is convenient to combine them to give the single
vector equation. Therefore, the rate of increase of momentum per unit volume is
equal to the gravitational force on the element per unit volume minus the rate of
momentum gained by convection per unit volume minus the pressure force on an
element per unit volume and minus the rate of momentum gained by viscous transfer
per unit volume which can be written as
∂
( ρ v ) = − ⎡⎣∇ ⋅ ρ v v ⎤⎦ − ∇p − ⎡⎣∇ ⋅τ ⎤⎦ + ρ g .
∂t
(2.3.50)
Equations (2.3.18), (2.3.35) and (2.3.52) may be rearranged (using the equation of
continuity). Thus we get
ρ
∂τ yx ∂τ zx ⎞
Dvx
∂p ⎛ ∂τ
= − − ⎜ xx +
+
⎟ + ρ gx ,
∂x ⎝ ∂x
∂y
∂z ⎠
Dt
ρ
Dv y
Dt
=−
∂p ⎛ ∂τ xy ∂τ yy ∂τ zy ⎞
−⎜
+
+
⎟ + ρ gy ,
∂y ⎝ ∂x
∂y
∂z ⎠
(2.3.51)
(2.3.52)
26
ρ
∂τ yz ∂τ zz ⎞
Dvz
∂p ⎛ ∂τ
= − − ⎜ xz +
+
⎟ + ρ gz .
∂z ⎝ ∂x
∂y
∂z ⎠
Dt
(2.3.53)
All components (2.3.51) – (2.3.53) add together and can be rewritten into the vector
equation. Therefore we obtain
p
Dv
= −∇ρ − ( ∇ ⋅τ ) + ρ g .
Dt
(2.3.54)
Equation (2.3.54) means that a small volume element moving with the fluid is
accelerated because of the forces acting upon it. It is also a statement of Newton’s
second law in the form of mass time’s acceleration and equal to the sum of forces.
This equation is the description of the changes taking place in an element following
the fluid motion.
2.3.4
The Power-law Model
In this section we wish to perform a similar generalization of the nonNewtonian model. We are now in a position to rewrite the non-Newtonian model in
a form that allows us to describe flow in complex geometries. In the generalized
Power-law model the stress tensor, τ formulae is
⎧⎪
τ = − ⎨m
⎪⎩
n −1
1
(∆ : ∆)
2
⎫⎪
⎬∆
⎭⎪
(2.3.55)
where ∆ is the symmetrical “rate of deformation tensor” with Cartesian components
⎛ ∂v
∆ ij = ⎜ i
⎜
⎝ ∂x j
⎞ ⎛ ∂v j
⎟⎟ + ⎜
⎠ ⎝ ∂xi
⎞
⎟.
⎠
27
For n = 1 , it reduces to Newton’s law of viscosity with m = µ . The constant of
proportionality µ is called the viscosity of the fluid. Thus equation (2.3.55) reduces
to the following form:
τ = −µ ∆ .
This is known as Newton’s law of viscosity and a fluid that behaves in this fashion
are termed Newtonian fluids. For a value of n less than unity the behaviour is
pseudoplastic, where as for n greater than unity the behaviour is dilatants. In the
Power-law model, the expression of
1
1
( ∆ : ∆) = ∑i
2
2
∑
j
1
( ∆ : ∆ ) can rewrite as
2
⎡⎛ ∂ v ∂ v j
⎢⎜⎜ i +
⎢⎣⎝ ∂ x j ∂ xi
⎤
⎞ 2
⎟⎟ − ( ∇ ⋅ v ) δ ij ⎥
⎥⎦
⎠ 3
2
in which i and j take on the values x, y, z and δ ij = 1 for i = j . Below is given the
term
1
( ∆ : ∆ ) in Cartesian coordinates:
2
⎡⎛ ∂v ⎞ 2 ⎛ ∂v y ⎞ 2 ⎛ ∂v ⎞ 2 ⎤ ⎡ ∂ v y ∂v ⎤ 2
1
( ∆ : ∆ ) = 2 ⎢⎜ x ⎟ + ⎜ ⎟ + ⎜ z ⎟ ⎥ + ⎢ + x ⎥
2
∂y ⎦
⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠ ⎥⎦ ⎣ ∂x
2
⎡ ∂v ∂v y ⎤ ⎡ ∂vx ∂vz ⎤ 2 ⎡ ∂vx ∂v y ∂vz ⎤
+⎢ z +
+
+
+
+
⎥ ⎢
⎥ .
⎥ − ⎢
⎣ ∂y ∂z ⎦ ⎣ ∂z ∂x ⎦ 3 ⎣ ∂x ∂y ∂z ⎦
2
2
With the foregoing development in mind, we are now can expressed the components
of τ for non-Newtonian fluids in Cartesian coordinates as
⎧
⎪
τ xx = −2 ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
n −1
2
⎫
⎪ ⎛ ∂vx
⎬⎜
⎪⎭ ⎝ ∂x
⎞
⎟,
⎠
(2.3.56)
28
τ yy
1
⎧
⎪ ⎡1
⎤ 2
= −2 ⎨ m ⎢ ( ∆ ⋅ ∆ ) ⎥
⎦
⎪⎩ ⎣ 2
⎧
⎪
τ zz = −2 ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
⎫ ∂v
⎪⎛ y
⎬⎜
⎪⎭ ⎝ ∂y
⎞
⎟,
⎠
(2.3.57)
⎫
⎪ ⎛ ∂vz
⎬⎜
⎪⎭ ⎝ ∂z
⎞
⎟,
⎠
(2.3.58)
n −1
n −1
2
⎧
⎪
= − ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
⎧
⎪
τ yz = τ zy = − ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
τ xy = τ yx
n −1
2
n −1
2
⎫
⎪ ⎛ ∂ v y ∂vx
+
⎬⎜
∂
x
∂y
⎝
⎪⎭
⎞
⎟,
⎠
(2.3.59)
⎫ ∂v
⎪ ⎛ y ∂vz ⎞
+
⎬⎜
⎟
∂
∂y ⎠
z
⎝
⎪⎭
(2.3.60)
⎫
⎪⎛ ∂vz ∂vx
+
⎬⎜
∂
x
∂z
⎝
⎪⎭
(2.3.61)
and
1
⎧
⎪ ⎡1
⎤ 2
τ zx = τ xz = − ⎨m ⎢ ( ∆ ⋅ ∆ ) ⎥
⎦
⎪⎩ ⎣ 2
n −1
⎞
⎟.
⎠
Actually, the normal stresses should contain one additional term; for example,
equation (2.3.56) should be
⎧
⎪
τ xx = −2 ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
n −1
2
⎫
⎪⎛ ∂vx
⎬⎜
⎪⎭ ⎝ ∂x
⎞ ⎛2
⎞
⎟ + ⎜ µ − κ ⎟ (∇ ⋅ v )
⎠
⎠ ⎝3
in which κ is the ‘bulk viscosity’. The bulk viscosity is identically zero for low
density monatomic gases and is probably not too important in dense gases and
liquids.
29
2.4
The Governing Equations in Cylindrical Coordinates
The continuity equation and the equation of motion have to be converted to
cylindrical coordinate order to govern the blood flow in artery under stenotic
conditions. Cylindrical coordinates will be denoted by (r , θ , z ) as shown in figure
3.3.1. The coordinate r is a distance that is perpendicular from the z – axis. The point
P in this figure has coordinate (r , θ , z ) using r = 0 on the z – axis and θ = 0 on the x –
axis.
z - axis
P
θ
x - axis
r
z
y
y - axis
Figure 2.4.1: Cylindrical Coordinates.
The relationship between Cartesian coordinates and cylindrical coordinates are:
x = r cos θ
,
y = r sin θ
, z=z
(2.4.1)
~ ~ ~
The unit vectors i , j , k of Cartesian coordinate are related to the cylindrical
coordinates, rˆ, θˆ, k , as
rˆ = i cos θ + j sin θ , θˆ = −i sin θ + j cosθ , k = k
(2.4.2)
The length element is given by
ds 2 = dr 2 + r 2 dθ 2 + dz 2
(2.4.3)
30
By using the equations (2.4.1) - (2.4.3), now introduce some formulae in polar
coordinate (refer to the derivation in Appendix A)
∇φ = r̂
∂φ θˆ ∂φ ∂φ
+
+k
∂r r ∂θ
∂z
(2.4.4)
∇⋅v =
1 ∂
1 ∂ vθ ∂vz
+
.
( r vr ) +
r ∂r
r ∂θ ∂ z
(2.4.5)
and
Substituting equations (2.4.4) and (2.4.5) into equations (2.2.14), (2.3.51) – (2.3.53)
and (2.3.56) – (2.2.61) in order to convert this equations from Cartesian coordinate to
cylindrical coordinate. Therefore the equations of motion in cylindrical coordinate
( r ,θ , z )
in terms of τ are:
r component,
ρ
τ
Dvr
∂p ⎡ 1 ∂
1 ∂
∂
=− −⎢
( rτ rr ) +
(τ θ r ) + (τ zr ) − θθ
Dt
r ∂θ
r
∂r ⎣ r ∂r
∂z
⎤
⎥ + ρ gr .
⎦
Dvr ⎛ ∂vr
∂vr vθ ∂vr
∂vr vθ 2 ⎞
or by substituting
=⎜
+ vr
+
+ vz
−
⎟ into the above
∂r r ∂θ
∂z
Dt ⎝ ∂t
r ⎠
equation,
⎛ ∂vr
∂v v ∂v
∂v v 2 ⎞
+ vr r + θ r + vz r − θ ⎟
r ⎠
∂r r ∂θ
∂z
⎝ ∂t
ρ⎜
(2.4.6)
=−
θ component,
τ
∂p ⎡ 1 ∂
1 ∂
∂
−⎢
( rτ rr ) +
(τ θ r ) + (τ zr ) − θθ
r ∂θ
r
∂r ⎣ r ∂r
∂z
⎤
⎥ + ρ gr .
⎦
31
ρ
Dvθ
τ −τ
∂
1 ∂p ⎡ 1 ∂ 2
1 ∂
=−
− ⎢ 2 ( r τ rθ ) +
(τ θθ ) + (τ zθ ) − θ r rθ
Dt
r ∂θ ⎣ r ∂r
r ∂θ
∂z
r
or by substituting
⎤
⎥ + ρ gθ .
⎦
Dvθ ⎛ ∂vθ
∂v v ∂v
∂v v v ⎞
=⎜
+ vr θ + θ θ + vz θ − r θ ⎟ into the above
Dt ⎝ ∂t
∂r
r ∂θ
∂z
r ⎠
equation,
∂v v ∂v
∂v v v ⎞
⎛ ∂vθ
+ vr θ + θ θ + vz θ − r θ ⎟
r ∂θ
r ⎠
∂r
∂z
⎝ ∂t
ρ⎜
(2.4.7)
=−
τ −τ
1 ∂p ⎡ 1 ∂ 2
1 ∂
∂
− ⎢ 2 ( r τ rθ ) +
(τ θθ ) + (τ zθ ) − θ r rθ
r ∂θ ⎣ r ∂r
r ∂θ
r
∂z
⎤
⎥ + ρ gθ .
⎦
z component,
ρ
Dvz
∂p ⎡ 1 ∂
1 ∂
∂
=− −⎢
( rτ rz ) +
(τ θ z ) + (τ zz )⎤⎥ + ρ g z .
Dt
∂z ⎣ r ∂r
r ∂θ
∂z
⎦
or by substituting
Dvz ⎛ ∂vz
∂v v ∂v
∂v ⎞
=⎜
+ vr z + θ z + vz z ⎟ into the above equation,
∂r r ∂θ
∂z ⎠
Dt ⎝ ∂t
∂v v ∂v
∂v ⎞
⎛ ∂vz
+ vr z + θ z + vz z ⎟
∂r r ∂θ
∂z ⎠
⎝ ∂t
ρ⎜
(2.4.8)
=−
1 ∂
∂p ⎡ 1 ∂
∂
−⎢
( rτ rz ) +
(τ θ z ) + (τ zz )⎤⎥ + ρ g z .
r ∂θ
∂z ⎣ r ∂r
∂z
⎦
Equation of continuity in cylindrical coordinates is
1 ∂
1 ∂
∂
( ρ rvr ) +
( ρ vθ ) + ( ρ vz ) = 0 .
r ∂r
r ∂θ
∂z
(2.4.9)
The components of stress tensor for Newtonian fluids in cylindrical coordinates are
32
1
⎧
⎪ ⎡1
⎤ 2
τ rr = −2 ⎨m ⎢ ( ∆ ⋅ ∆ ) ⎥
⎦
⎪⎩ ⎣ 2
⎧
⎪
= −2 ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
1
⎧
⎪
τ zz = −2 ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎦⎥
1
τ θθ
n −1
⎫
⎪ ⎛ ∂vr
⎬⎜
⎪⎭ ⎝ ∂r
n −1
2
n −1
2
1
τ rθ = τ θ r
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎥⎦
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎦⎥
1
τ θ z = τ zθ
⎧
⎪
= − ⎨m
⎪⎩
⎧
⎪
τ zr = τ rz = − ⎨m
⎪⎩
⎡1
⎤
⎢⎣ 2 ( ∆ ⋅ ∆ ) ⎦⎥
(2.4.10)
⎫
⎪⎛ 1 ∂vθ vr
+
⎬⎜
θ
r
r
∂
⎝
⎪⎭
⎫
⎪ ⎛ ∂vz
⎬⎜
⎪⎭ ⎝ ∂z
⎧
⎪
= − ⎨m
⎪⎩
⎞
⎟,
⎠
n −1
2
(2.4.11)
⎞
⎟,
⎠
(2.4.12)
⎫
⎪ ⎛ ∂ ⎛ vθ ⎞ 1 ∂vr
⎬⎜ r ⎜ ⎟
⎪⎭ ⎝ ∂r ⎝ r ⎠ r ∂θ
⎞
⎟,
⎠
(2.4.13)
⎫
⎪⎛ ∂vθ 1 ∂vz ⎞
+
⎬⎜
⎟
⎪⎭ ⎝ ∂z r ∂θ ⎠
(2.4.14)
⎫
⎪⎛ ∂vz ∂vr ⎞
+
⎬⎜
⎟.
∂
∂
r
z
⎝
⎠
⎪⎭
(2.4.15)
n −1
2
⎞
⎟,
⎠
and
Now, the term
1
n −1
2
1
( ∆ : ∆ ) which use in equations (2.4.10) – (2.4.15) is
2
⎡⎛ ∂vr ⎞2 ⎛ 1 ∂vθ vr ⎞ 2 ⎛ ∂vz ⎞ 2 ⎤ ⎡ ∂ ⎛ vθ
1
∆
∆
=
+ ⎟ +⎜
:
2
(
) ⎢⎜ ⎟ + ⎜
⎟ ⎥ + ⎢r ⎜
2
⎢⎣⎝ ∂r ⎠ ⎝ r ∂θ r ⎠ ⎝ ∂z ⎠ ⎥⎦ ⎣ ∂r ⎝ r
⎡ ∂v ∂v ⎤ ⎡ ∂v ∂v ⎤
+⎢ z + θ ⎥ +⎢ r + z⎥ .
⎣ ∂θ ∂z ⎦ ⎣ ∂z ∂r ⎦
2
2
⎞ 1 ∂vr ⎤
⎟+
⎥
⎠ r ∂θ ⎦
2
33
Equations (2.4.6) – (2.4.8) may be used for describing non-Newtonian flow. In order
to use these equations, however the relations between the components of τ and the
various velocity gradients are needed. It follows that, equations (2.4.10) – (2.4.15)
have to replace the expressions by other relations appropriate for the non-Newtonian
fluid of interest.
2.5
Derivation of the Mathematical Model
Let us consider the stenotic blood flow in the tapered artery to be twodimensional, unsteady, axisymmetric and the fluid is treated as a non-Newtonian
fluid characterized by the generalized Power-Law model. Thus, the governing
equations for the z and r components of momentum, (2.4.6) and (2.4.8) together with
equation of continuity, (2.4.9) will reduce to the following forms
∂ vz
∂v
∂v
1∂p 1
+ vr z + vz z = −
−
ρ ∂z ρ
∂t
∂r
∂z
⎡1 ∂
⎤
∂
⎢ r ∂ r ( rτ rz ) + ∂ z (τ zz ) ⎥ ,
⎣
⎦
(2.5.12)
⎤
∂ vr
∂v
∂v
∂
1 ∂ p 1 ⎡1 ∂
+ vr r + vz r = −
− ⎢
( rτ rr ) + (τ rz )⎥
ρ ∂ r ρ ⎣r ∂ r
∂t
∂r
∂z
∂z
⎦
(2.5.13)
∂ vr vr ∂ vz
+ +
=0,
∂r r ∂z
(2.5.14)
and
where the relationship between the shear stress and the shear rate in case of two
dimensional motions are as follows:
34
1
⎧
2
2
2 2
2
⎡
⎤
⎪⎪ ⎛ ∂ v ⎞ ⎛ v ⎞ ⎛ ∂ v ⎞ ⎛ ∂ v ∂ v ⎞
τ zz = −2 ⎨m ⎢⎜ r ⎟ + ⎜ r ⎟ + ⎜ z ⎟ + ⎜ r + z ⎟ ⎥
⎪ ⎢⎣⎝ ∂ r ⎠ ⎝ r ⎠ ⎝ ∂ z ⎠ ⎝ ∂ z ∂ r ⎠ ⎥⎦
⎪⎩
n −1
⎫
⎪⎪ ⎛ ∂ vz ⎞
⎬⋅⎜
⎟,
z
∂
⎝
⎠
⎪
⎪⎭
(2.5.15)
1 n−1 ⎫
⎧
2
2
2 2
2
⎪⎪ ⎡⎛ ∂ v ⎞ ⎛ v ⎞ ⎛ ∂ v ⎞ ⎛ ∂ v ∂ v ⎞ ⎤ ⎪⎪ ⎛ ∂ v ∂ v ⎞
τ rz = −2 ⎨m ⎢⎜ r ⎟ + ⎜ r ⎟ + ⎜ z ⎟ + ⎜ r + z ⎟ ⎥ ⎬⋅ ⎜ z + r ⎟
⎪ ⎢⎣⎝ ∂ r ⎠ ⎝ r ⎠ ⎝ ∂ z ⎠ ⎝ ∂ z ∂ r ⎠ ⎥⎦ ⎪ ⎝ ∂ r ∂ z ⎠
⎩⎪
⎭⎪
(2.5.16)
1
⎧
2
2
2 2
2
⎪⎪ ⎡⎛ ∂ v ⎞ ⎛ v ⎞ ⎛ ∂ v ⎞ ⎛ ∂ v ∂ v ⎞ ⎤
τ rr = −2 ⎨m ⎢⎜ r ⎟ + ⎜ r ⎟ + ⎜ z ⎟ + ⎜ r + z ⎟ ⎥
⎪ ⎣⎢⎝ ∂ r ⎠ ⎝ r ⎠ ⎝ ∂ z ⎠ ⎝ ∂ z ∂ r ⎠ ⎦⎥
⎩⎪
(2.5.17)
and
n−1
⎫
⎪⎪ ⎛ ∂ v ⎞
r
⎬⋅⎜
⎟.
z
∂
⎝
⎠
⎪
⎭⎪
Here vz ( r , z , t ) and vr ( r , z , t ) represents the axial and the radial velocity
components respectively, p is the pressure and ρ , the density of blood.
2.6
The Boundary Conditions
The differential equations that are derived from the conservation laws are
subject to some boundary conditions. Specifically, the equations of motion in terms
of the viscous shear stress are of the form that requires the velocity vector to be given
on all surfaces bounding the flow domain. Furthermore, the ongoing analysis will be
based on the boundary conditions, which have connection between the motion of the
flowing blood and the arterial wall.
There is no radial flow along the axis of the artery and the axial velocity
gradient of the streaming blood in that sense may be assumed to be equal to zero.
35
That means, there is no shear rate of fluid along the axis. These may be expressed
mathematically as
vr ( r , z , t ) = 0 ,
∂ vz ( r , z , t )
= 0 and τ rz = 0 on r = 0 .
∂r
(2.6.1)
The velocity boundary conditions on the arterial wall are taken as
vr ( r , z , t ) =
∂R
∂t
and vz ( r , z , t ) = 0 on r = R ( z , t ) .
(2.6.2)
Also, it is assumed that no flow takes places when the system is at rest, that means
vr ( r , z , t ) = 0 and vz ( r , z , t ) = 0 .
2.6.1
(2.6.3)
The Pressure Gradient
Since the lumen radius, R is sufficiently smaller than the wavelength, λ of
the pressure wave i.e. R λ << 1 , equation (2.5.13) simply reduces to
∂ p ∂ r = 0 (Pedley (1980)) and thus equation (2.5.13) can be omitted. It is
reasonable and convenient to assume that the pressure is independent of radial
coordinate and hence the pressure gradient ∂ p ∂ z appearing in equation (2.5.12),
the form of which has been taken from (Burton, 1966) for human being as
−
∂p
= AD + A1 cos ω t , t > 0
∂z
(2.6.4)
where AD is the constant amplitude of the pressure gradient, A1 is the amplitude of
the pulsatile component giving rise to systolic and diastolic pressure.
CHAPTER III
THE GEOMETRY OF STENOSIS
3.1
Introduction
This chapter presents the various geometries of stenosis i.e. mild (single)
stenosis, overlapping stenosis, multistenosis and other geometries that have been
considered in the literature. The formulation of the geometry of stenosis in the nontapered artery as well as the formulation of the geometry of stenosis in the tapered
blood vessel segment is shown.
3.2
The Geometry of Stenosis
Stenosis is defined as a partial occlusion of the vessels caused by abnormal
growth of tissues or the deposition of cholesterol as substances on the arterial wall.
There are many geometry of stenosis that can be considered and formulated
mathematically. For example:
37
3.2.1
Constant Radius (non stenosis)
R( z) = a
where R ( z ) is the radius of the arterial segment in the constricted region and a, the
constant radius of the normal artery in the non-stenotic region.
3.2.2
The Geometry of Mild Stenosis
{
}
2
R ( z , t ) ⎧⎪ ⎡1 − q1 ( t ) l01 ( z − d ) − ( z − d ) ⎤ , d ≤ z ≤ d + l0
⎣
⎦
=⎨
a
⎪⎩
1,
otherwise
where the z-axis is taken along the axis of the artery, R( z , t ) denotes the radius of the
tapered arterial segment in the stenotic region, lD , the length of the stenosis, d, the
location of the stenosis and q1 (t ) is the time-variant parameter which is given by
q1 (t ) = 1 − b ( cos ω t − 1) e − bω t ; ω = 2π f p .
Figure 3.2.1: The Geometry of Mild Stenosis
3.2.3 The Geometry of a Cosine-Shaped Stenosis
38
⎧
⎧ τm
⎫
⎪q1 ( t ) ⎨a − ⎡⎣1 + cos (π ( z − z1 ) / z0 ) ⎤⎦ ⎬ , d ≤ z ≤ d + 2 z0
2
R ( z, t ) = ⎨
⎩
⎭
⎪
q1 ( t ) a,
otherwise
⎩
Figure 3.2.2: The Geometry of a Cosine-Shaped Stenosis
3.2.4 The Bell-Shaped Stenosis
R ( z ) = RD ⎡1 − δ e −σ z ⎤
⎣
⎦
2
where 0 ≤ δ < 1 is a measure of the degree of contraction, σ is length.
Figure 3.2.3: The Geometry of Bell-Shaped Stenosis
39
3.2.5 The Geometry of Overlapping Stenosis
⎧
⎡ 32τ m ⎧ 11
⎤
R (z, t ) ⎪q1 (t )⎢1 −
(z − d )l 03 − 47 (z − d )2 l 02 + (z − d )3 l 0 − 1 (z − d )4 ⎫⎬⎥, d ≤ z ≤ d + 3 l 0
4 ⎨
=⎨
48
3
2
al 0 ⎩ 32
⎭⎦
⎣
a
⎪
(
)
q
t
otherwise
,
⎩
1
Figure 3.2.4: The Geometry of Two Overlapping Stenosis.
⎧⎡
τm
⎪
R( z , t ) ⎪⎢1 −
= ⎨⎢ 5005al06
a
⎪⎢⎣
⎪⎩
⎧ 668662
(z − d )l05 − 370281(z − d )2 l04 + 743344(z − d )3 l03 ⎫⎪⎤⎥
⎪
9
⎨
⎬⎥ a1 (t ), d ≤ z ≤ d + 2l0
⎪− 698476( z − d )4 l 2 + 307584(z − d )5 l − 51264( z − d )6 ⎪⎥
0
0
⎩
⎭⎦
a1 (t ),
otherwise
Figure 3.2.5: The Geometry of Three Overlapping Stenosis.
40
3.2.6 The Irregular Geometry
Figure 3.2.6: The Irregular Stenosis.
3.2.7 The Geometry of Multi-Stenosis
1
Dimensionless Radial Position
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Dimensionless Axial
Position
Figure 3.2.7: The Geometry of Multi-Mild Stenosis
41
Dimensionless Radial Position
1
0.8
0.6
0.4
0.2
0
5
0
15
10
Dimensionless Axial Position
20
Figure 3.2.8: The Geometry of Multi Irregular Stenosis.
1
Dimensionless Radial Position
0.8
0.6
0.4
0.2
0
0
5
15
10
Dimensionless Axial Position
20
Figure 3.2.9: The Smooth Profile used to Approximate Multi Irregular
Stenosis.
3.3
Formulation of The Geometry of Mild Stenosis in a non-Tapered Artery
In the present research, the geometry of mils stenosis is considered together
with the effects of vessel tapering (refer Figure 3.3.1).
42
r
a
τm
l
d+ 0
2
R = mz + a
z
Figure 3.3.1: The Geometry of Mild Stenosis in non-Tapered (φ = 0 ) Artery.
The geometry of mild stenosis which is time-independent is constructed
mathematically as
R ( z ) = A + B( z − d ) + C (z − d ) .
2
(3.3.1)
At z = d and R = a , equation (3.3.1) becomes
a = A + B ( 0) + C ( 0) .
2
Therefore
A= a.
(3.3.2)
At z = d + l D , R = a and using equation (3.3.2), equation (3.3.1) becomes
a = a + B ( lD ) + C ( lD ) .
2
Simplify the above equation, we get
0 = B ( lD ) + C ( lD ) .
2
43
Thus
B = −ClD .
At z = d +
(3.3.3)
lD
, R = a − τ m and using equation (3.3.2) and (3.3.3), equation (3.3.1)
2
becomes
a −τ m = a − C
lD 2
l2
+C D .
2
4
We then find
−τ m = −C
lD 2
,
4
or
C=
4τ m
.
lD 2
(3.3.4)
Substituting equation (3.3.4) into equation (3.3.3), we obtain
B=−
4τ m
.
lD
(3.3.5)
Now, substituting equations (3.3.2), (3.3.4) and (3.3.5) into equation (3.3.1), then we
get
R( z) = a −
or
4τ m
4τ
2
( z − d ) + 2m ( z − d )
lD
lD
44
⎛ 4τ ( z − d ) ⎞ ⎡ ( z − d ) ⎤
− 1⎥ .
R( z) = a + ⎜ m
⎟⎢
lD
⎝
⎠ ⎣ lD
⎦
(3.3.6)
The geometry of mild stenosis in a non-tapered artery is given by equation (3.3.6).
3.4
Formulation of the Geometry of Mild Stenosis in a Tapered Artery
The geometry of stenosis in the tapered blood vessel segment that is
considered in this study is the mild stenosis. The tapered blood vessel segments
having a mild stenosis in its lumen is modelled as a thin elastic tube with a circular
cross section containing an incompressible non-Newtonian fluid characterized by the
generalized Power-Law model. Let (r , θ , z ) be the coordinates of a material point in
the cylindrical polar coordinates system where the z – axis is taken along the axis of
the artery while r , θ are taken along the radial and the circumferential directions,
respectively. The geometry of mild stenosis is shown as in Figure 3.4.1 for a tapered
artery.
r
R = mz + a
τm
m = tan φ
y = τ m sec φ
y
φ
a
d
d+
l0
2
z
d + lD
Figure 3.4.1: The Geometry of Mild Stenosis in a Tapered (φ > 0) Artery.
45
Here τ m sec φ is taken to be the critical height of the stenosis for the tapered artery
appearing at z = d +
lD
+ τ m sec φ and m ( tan φ ) represents the slope of the tapered
2
vessel. At z = d and R = md + a then equation (3.3.1) becomes
md + a = A + B (0) + C (0) .
2
Therefore
A = md + a .
(3.4.1)
At z = d + l D , R = m (d + l D ) + a and using equation (3.4.1) then equation (3.3.1)
becomes
m (d + l D ) + a = md + a + B (l D ) + C (l D ) .
2
Thus we obtain
B = m − ClD .
At z = d +
(3.4.2)
l ⎞
lD
⎛
, R = m ⎜ d + D ⎟ + a − τ m sec φ and using equation (3.4.1) and (3.4.2)
2
2⎠
⎝
then equation (3.3.1) becomes
l ⎞
l
l
⎛
m ⎜ d + D ⎟ + a − τ m sec φ = md + a + B D + C D .
2⎠
2
4
⎝
2
Simplify the above equation, then we get
2
ml
l
l
md + D + a − τ m sec φ = md + a + B D + C D .
2
2
4
Therefore
46
mlD
l
l2
− τ m sec φ = B D + C D .
2
2
4
(3.4.3)
Substituting equation (3.4.2) into equation (3.4.3),thus we obtain
2
2
ml D
ml
l
l
− τ m sec φ = D − C D + C D .
2
2
2
4
We then find
2
l
τ m sec φ = C D .
4
Therefore
C=
4τ m sec φ
.
lD 2
(3.4.4)
Substituting equation (3.4.4) into equation (3.4.2), equation (3.4.2) becomes
B = m−
4τ m sec φ
.
lD
(3.4.5)
Now, substituting equation (3.4.1), (3.4.4) and (3.4.5) into equation (3.3.1), then
equation (3.3.1) can rewrite as
⎛ 4τ sec φ ⎞
⎛
4τ sec φ ⎞
⎟ ( z − d )2
⎟⎟ ( z − d ) + ⎜ m 2
R ( z ) = md + a + ⎜⎜ m − m
⎟
⎜
lD
⎠
⎝
⎠
⎝ lD
⎛ 4τ sec φ ⎞
⎛ 4τ m sec φ ⎞
2
= md + a + m z − md − ⎜ m
⎟(z − d ) + ⎜
⎟(z − d )
2
lD
lD
⎝
⎠
⎝
⎠
47
⎛ 4τ sec φ ( z − d ) ⎞
= m z + a −⎜ m
⎟ ⎡⎣lD − ( z − d ) ⎤⎦ .
lD
⎝
⎠
(3.4.6)
In order to check whether R ( z ) in the tapered blood vessel segment is correct then it
has to compare with R ( z ) in the non-tapered (φ = 0) blood vessel segment. From
Figure 3.3.1, we get
l ⎞
⎛
R ⎜ d + D ⎟ = a −τ m .
2⎠
⎝
Let φ = 0 and m = 0 . At z = d +
(3.4.7)
lD
, then we obtain
2
l ⎞
⎛
R (z ) = R ⎜ d + D ⎟ .
2⎠
⎝
Then we find
4τ (1) ⎛
l ⎞
l
l
⎛
⎞⎡ ⎛
⎞⎤
R ⎜ d + D ⎟ = a − m2 ⎜ d + D − d ⎟ ⎢l D ⎜ d + D − d ⎟⎥
2⎠
2
2
lD ⎝
⎝
⎠⎣ ⎝
⎠⎦
=a−
2τ m ⎛ lD ⎞
⎜ ⎟
lD ⎝ 2 ⎠
= a −τ m .
(3.4.8)
Since equation (3.4.7) equal to equation (3.4.8), thus R ( z ) in this problem is correct.
This research consider the time-variant parameter a1 (t ) which is given by
a1 (t ) = 1 − b (cos ω t − 1) e − bω t
(3.4.9)
48
where ω = 2 π f p . Therefore, the geometry of the time-variant mild stenosis arterial
segment for different taper angle (see Figure 3.4.2) is writen mathematically as
⎧⎡
⎤
4τ m secφ ( z − d )
(lD − (z − d ))⎥ a1 (t ) ; d ≤ z ≤ d + lD
⎪⎢m z + a −
2
lD
⎪⎣⎢
⎦⎥
⎪
R (z, t ) = ⎨
⎪(m z + a) a (t )
; otherwise
1
⎪
⎪
⎩
(3.4.10)
Equation (3.4.10) is the equation used in the present research.
φ >0
r
φ =0
φ<0
0
z
d
τm
lD
Figure 3.4.2: The Geometry of Mild Stenosis for Different Angles of Tapering.
The different shapes of the artery as shown in the Figure 3.4.2, where φ is the angle
of tapering, φ < 0 , is the converging tapering, φ = 0 is non-tapered artery and φ > 0
is the diverging tapering will be explored.
CHAPTER IV
SOLUTION PROCEDURE
4.1
Introduction
In Section 4.2 we will introduce a radial coordinate transformation to
transform the governing equations. Thereafter, in Section 4.3 we will carry out the
derivation of the radial velocity component, vr ( x, z , t ) . Then, in the following
section we will solve the axial velocity component using a finite difference scheme
to convert the partial differential equations that govern the physical phenomenon into
a system of algebraic equations. In the last section, we will derive the volumetric
flow rate (Q ) , the resistance to flow (∧ ) and the wall shear stress (τ w ) .
4.2
Transformation of the Governing Equations using Radial Coordinate
Transformation
Usually, any problem that involves the coupling of fluid mechanics and
50
vessel wall mechanics, R( z , t ) could be derived as a part of the solution instead of
having its specific form as input. R( z , t ) is known explicitly and hence our attention
will be centered on the haemodynamic factor only. Now, let us introduce a radial
coordinate transformation, given by
x=
r
R( z , t )
(4.2.1)
which has the effect of immobilizing the vessel wall in the transformed coordinate x.
The equations of motion, (2.5.12) and (2.5.14), the relationship between the shear
stress and shear rate, (2.5.15) and (2.5.16) and also the initial and the boundary
conditions, (2.6.1) - (2.6.3) will be transformed. The radial coordinate
transformation (4.2.1) can be rewritten as,
r = xR ( z , t )
(4.2.2)
Consider vz and vr as functions of x, z and t, where x are functions of r and R and R
are functions of z and t. The derivatives with respect to r, z and t are thus as follows:
∂
∂ ∂x
=
⋅
∂r ∂x ∂r
(4.2.3)
∂
∂
∂ ∂x
=
+
⋅
∂z ∂z ∂x ∂z
(4.2.4)
∂
∂
∂ ∂x
= +
⋅
∂t ∂t ∂ x ∂t
(4.2.5)
4.2.1 Transformation the z Momentum
Replacing the forms of
∂
∂
and
given by (4.2.3) and (4.2.5) in equation
∂r
∂t
51
(2.5.12), we find
⎛ ∂ vz ∂ x ⎞
⎛ ∂ vz ∂ vz ∂ x ∂ R ⎞
∂ v z ⎛ ∂ vz ∂ x ∂ R ⎞
+⎜
⋅
⋅
⋅
+
⋅
⋅
⎟ + vr ⎜
⎟ + vz ⎜
⎟
∂t ⎝ ∂ x ∂ R ∂t ⎠
⎝ ∂x ∂r ⎠
⎝ ∂z ∂x ∂R ∂z ⎠
(4.2.6)
=−
1 ∂p 1
−
ρ ∂z ρ
⎡1 ∂
∂ τ xz ∂ x ⎤
∂x ∂
⎢ r ∂ x ( rτ xz ) ∂ r + ∂ z (τ zz ) + ∂ x ∂ z ⎥ .
⎣
⎦
Substituting equations (4.2.1) and (4.2.2) into equation (4.2.6), we obtain
⎛ ∂ v ∂ ⎛ r ⎞⎞
⎛ ∂v ∂v ∂ ⎛ r ⎞ ∂ R⎞
∂ vz ⎛ ∂vz ∂ ⎛ r ⎞ ∂ R ⎞
+ ⎜ ⋅ ⎜ ⎟ ⋅ ⎟ + vr ⎜ z ⋅ ⎜ ⎟ ⎟ + vz ⎜ z + z ⋅ ⎜ ⎟ ⋅ ⎟
∂t ⎝ ∂ x ∂ R ⎝ R ⎠ ∂t ⎠
⎝ ∂ x ∂ r ⎝ R ⎠⎠
⎝ ∂ z ∂ x ∂R⎝ R⎠ ∂ z ⎠
=−
∂τ ⎞ 1 ∂τ
x ∂τ xz ∂ R ⎤
1 ∂p 1⎡ 1 ⎛
− ⎢ ⎜ Rτ xz + xR xz ⎟ + zz −
⎥.
ρ ∂ z ρ ⎣ xR ⎝
∂x ⎠ R ∂z R ∂x ∂z ⎦
Then we find
⎛ ∂ vz ⎛ 1 ⎞ ⎞
⎛ ∂ vz ∂ vz ⎛ r ⎞ ∂ R ⎞
∂ v z ⎛ ∂ vz ⎛ r ⎞ ∂ R ⎞
+⎜
+
⎟ + vr ⎜
⎟
⎜ 2 ⎟⋅
⎜ ⎟ ⎟ + vz ⎜
⎜ 2 ⎟⋅
∂t ⎝ ∂ x ⎝ R ⎠ ∂t ⎠
⎝ ∂ x ⎝ R ⎠⎠
⎝ ∂z ∂x ⎝R ⎠ ∂z ⎠
=−
1 ∂p 1
−
ρ ∂z ρ
⎡ 1
1 ∂ τ xz ∂ τ zz x ∂ τ xz ∂ R ⎤
⎢ xR τ xz + R ∂ x + ∂ z − R ∂ x ∂ z ⎥ .
⎣
⎦
After simplification the left hand side, we obtain
∂ vz
x ∂ v z ∂ R vr ∂ v z
∂ vz
x ∂ vz ∂ R
+ ⋅
⋅
+
⋅
+ vz
− vz
⋅
⋅
∂t
R ∂ x ∂t
R ∂x
∂z
R ∂x ∂z
=−
1 ∂ p 1 ⎡ 1
1 ∂ τ xz ∂ τ zz
x ∂ τ xz ∂ R ⎤
τ xz +
−
+
−
.
⎢
ρ ∂ z ρ ⎣ xR
∂z
R ∂x
R ∂ x ∂ z ⎥⎦
The above equation can be rearranged to get
52
⎡ x ∂ R vr
∂ vz
∂ vz
x ∂ R ⎤ ∂ vz
1 ∂ p
= ⎢ ⋅
−
+ vz ⋅
− vz
−
⎥
R
R ∂z⎦ ∂x
ρ ∂z
∂t
∂z
⎣R ∂t
(4.2.7)
−
1 ⎡ 1
1 ∂ τ xz ∂ τ zz
x ∂ τ xz ∂ R ⎤
.
τ xz +
+
−
⎢
R ∂x
R ∂ x ∂ z ⎥⎦
ρ ⎣ xR
∂z
Equation (4.2.7) cannot be solved directly. We will show later how to get the axial
velocity component, vz ( x, z , t ) .
4.2.2 Transformation of the Continuity Equation
Replacing
∂
∂
,
given by (4.2.3) and (4.2.5) into equation (2.5.14) we obtain
∂r ∂t
∂ vr ∂ x vr ∂ vz ∂ vz ∂ x ∂ R
⋅
+
+
+
⋅
⋅
= 0.
∂ x ∂ r xR ∂ z ∂ x ∂ R ∂ z
Substituting
∂x 1
=
∂r R
,
(4.2.8)
∂x
r
= − 2 into equation (4.2.8), we have
∂R
R
∂ vr ⎛ 1 ⎞ vr ∂ vz ∂ vz ⎛ r ⎞ ∂ R
⋅⎜ ⎟ +
+
+
= 0,
⎜− ⎟
∂ x ⎝ R ⎠ xR ∂ z ∂ x ⎝ R 2 ⎠ ∂ z
and after simplifying it, we get
1 ∂ vr vr ∂ vz x ∂ vz ∂ R
+
+
−
⋅
=0.
R ∂ x xR ∂ z R ∂ x ∂ z
4.2.3
Transformation of the Normal Stress, (τ zz )
Beside that, the shear stress (τ rz ) , the normal stress (τ zz ) and also the boundary
(4.2.9)
53
conditions must be transformed too using radial coordinate transformation.
Substituting equations (4.2.3) and (4.2.5) into equation (2.5.15), we get
1 n−1 ⎫
⎧
2
2
2 2
2
⎪⎪ ⎡⎛ ∂ v ∂ x ⎞ ⎛ v ⎞ ⎛ ∂ v ∂ v ∂ x ⎞ ⎛ ∂ v ∂ v ∂ x ∂ v ∂ x ⎞ ⎤ ⎪⎪
τzz =−2⎨m ⎢⎜ r ⋅ ⎟ + ⎜ r ⎟ + ⎜ z + z ⋅ ⎟ + ⎜ r + r ⋅ + z ⋅ ⎟ ⎥ ⎬
⎪ ⎢⎣⎝ ∂ x ∂ r ⎠ ⎝ xR ⎠ ⎝ ∂ z ∂ x ∂ z ⎠ ⎝ ∂ z ∂ x ∂ z ∂ x ∂ r ⎠ ⎥⎦ ⎪
⎩⎪
⎭⎪
(4.2.10)
⎛ ∂v ∂v ∂ x ⎞
⋅ ⎜ z + z ⋅ ⎟.
⎝ ∂z ∂x ∂z ⎠
Substituting
∂x 1
∂x
x ∂R
=−
into equation (4.2.10), τ zz now becomes
= and
∂z
∂r R
R ∂z
1 n−1⎫
⎧
2
2
2
⎪⎪ ⎡⎛ 1 ∂v ⎞ ⎛ v ⎞2 ⎛ ∂v x ∂ R ∂v ⎞ ⎛ ∂v x ∂ R ∂v 1 ∂v ⎞ ⎤2 ⎪⎪
z
τzz =−2⎨m ⎢⎜ ⋅ r ⎟ +⎜ r ⎟ +⎜ z − ⋅ ⋅ z ⎟ +⎜ r − ⋅ ⋅ r +
⎟⎥ ⎬
∂
∂
∂
∂
∂
∂
∂
∂
R
x
xR
z
R
z
x
z
R
z
x
R
x
⎝
⎠
⎢
⎝
⎠
⎝
⎠
⎝
⎠ ⎥⎦ ⎪
⎪ ⎣
⎪⎩
⎪⎭
(4.2.11)
⎛ ∂v x ∂ R ∂ v ⎞
⋅⎜ z − ⋅ ⋅ z ⎟.
⎝ ∂z R ∂z ∂x ⎠
4.2.4
Transformation of the Shear Stress (τ xz ) ,
Replacing
∂
∂
and
given by (4.2.3) and (4.2.5) in equation (2.5.16), we have
∂r
∂t
1 n−1 ⎫
⎧
2
2
2 2
2
⎪⎪ ⎡⎛ ∂v ∂ x ⎞ ⎛ v ⎞ ⎛ ∂v ∂v ∂ x ⎞ ⎛ ∂v ∂v ∂ x ∂v ∂ x ⎞ ⎤ ⎪⎪
τxz =−2⎨m ⎢⎜ r ⋅ ⎟ +⎜ r ⎟ +⎜ z + z ⋅ ⎟ +⎜ r + r ⋅ + z ⋅ ⎟ ⎥ ⎬
⎪ ⎢⎣⎝ ∂ x ∂ r ⎠ ⎝ xR ⎠ ⎝ ∂ z ∂ x ∂ z ⎠ ⎝ ∂ z ∂ x ∂ z ∂ x ∂ r ⎠ ⎥⎦ ⎪
⎪⎩
⎪⎭
(4.2.12)
⎛ ∂v ∂ x ∂v ∂v ∂ x ⎞
⋅⎜ z ⋅ + r + r ⋅ ⎟.
⎝ ∂ x ∂r ∂ z ∂ x ∂ z ⎠
54
Substituting
∂x 1
∂x
x ∂R
=−
into the above equation (4.2.12), we get
= and
∂z
∂r R
R ∂z
1 n−1 ⎫
⎧
2
2
2 2
2
⎡
⎤
⎪⎪ ⎛ 1 ∂ v ⎞ ⎛ v ⎞ ⎛ ∂ w x ∂ R ∂ v ⎞ ⎛ ∂ v x ∂ R ∂ v 1 ∂ v ⎞
⎪⎪
z
τxz =−2⎨m ⎢⎜ ⋅ r ⎟ +⎜ r ⎟ + ⎜ − ⋅ ⋅ z ⎟ + ⎜ r − ⋅ ⋅ r +
⎟⎥ ⎬
⎪ ⎢⎣⎝ R ∂ x ⎠ ⎝ xR ⎠ ⎝ ∂ z R ∂ z ∂ x ⎠ ⎝ ∂ z R ∂ z ∂ x R ∂ x ⎠ ⎥⎦ ⎪
⎪⎩
⎪⎭
(4.2.13)
⎛ 1 ∂ vz ∂ vr x ∂ R ∂ vr ⎞
⋅⎜
+ − ⋅ ⋅ ⎟.
⎝ R ∂x ∂z R ∂z ∂x ⎠
4.2.5
Transformation of the Boundary Conditions
Replacing
∂
∂
and
in equations (2.6.1) – (2.6.2), with the forms given by (4.2.3)
∂r
∂t
and (4.2.5), we obtain
vr ( x, z , t ) = 0 ,
vr ( x, z , t ) =
∂R
∂t
∂ vz ( x, z , t )
= 0 and τ xz = 0 on x = 0 ,
∂r
(4.2.10)
and vz ( x, z , t ) = 0 on x = 1
(4.2.11)
and
vr ( x, z , t ) = 0 and vz ( x, z , t ) = 0
4.3
(4.2.12)
Derivation of the Radial Velocity Component, vr ( x, z , t )
In order to get the radial velocity component, vr ( x, z , t ) we have to consider
55
equation (4.2.5). Multiplying equation (4.2.5) by xR , then we get
x
∂ vr
∂v
∂v ∂ R
+ vr + xR z − x 2 z ⋅
= 0.
∂x
∂z
∂x ∂z
Next, integrate the equation with respect to x from the limits 0 → x , we obtain
∫
x
0
x
x
x
x
∂ vr
∂v
∂v ∂ R
dx + ∫ vr dx + ∫ xR z dx − ∫ x 2 z ⋅
dx = 0 .
0
0
0
∂x
∂z
∂x ∂z
Then we find
x
x
x
0
0
0
xvr − ∫ vr dx + ∫ vr dx + R ∫ x
x
∂ vz
∂R⎡ 2 x
dx −
x vz − ∫ 2 xvz dx ⎤ = 0 .
0
⎥⎦
0
∂z
∂ z ⎣⎢
Simplification of the above equation gives
x
xvr + R ∫ x
0
∂ vz
∂R 2
∂R x
2 xvz dx = 0 .
dx −
x vz +
∂z
∂z
∂ z ∫0
which upon arranging gives
vr = −
1 ∂R 2
1 ∂R x
R x ∂ vz
+
−
2 xvz dx ,
x
dx
x
v
z
x ∫0 ∂ z
x ∂z
x ∂ z ∫0
and then
vr = x
∂R
2 ∂R x
R x ∂v
vz − ∫ x z dx −
xvz dx .
∂z
x 0 ∂z
x ∂ z ∫0
(4.3.1)
Equation (4.3.1) takes the following form by making use of the boundary condition
(4.2.11) as
56
∂R
∂R
2 ∂R 1
R 1 ∂v
= (1)
xvz dx = 0 .
( 0 ) − ∫ 0 x z dx −
∂t
∂z
1
∂z
1 ∂ z ∫0
and after simplifying it, we get
∂R
∂R 1
R 1 ∂ vz
x
dx
=
+
2
xvz dx
1 ∫0 ∂ z
∂t
∂ z ∫0
−
or
1
−∫ x
0
∂ vz
1 ∂R 2 ∂R 1
+
dx =
xvz dx
∂z
R ∂ t R ∂ z ∫0
(4.3.2)
Since the choice of f(x) is arbitrary, let f(x) be of the form,
(
)
f ( x ) = −4 x 2 − 1 ,
which f(x) satisfying
∫ xf (x ) dx = 1 .
1
(4.3.3)
0
We choose f(x) to be of the above form because we want to simplify equation (4.3.2).
From the left hand side of equation (4.3.3) and substituting the form of f(x), we have
∫
xf ( x ) dx = ∫ x ⎡⎣ −4 ( x 2 − 1) ⎤⎦ dx = 1 .
0
0
1
1
From the derivation of vr ( x, z , t ) , equation (4.3.2) becomes
1
−∫ x
0
1
∂ vz
1 ∂R
2 ∂R 1
dx = ∫ xf ( x )
dx +
xvz dx
0
R ∂t
R ∂ z ∫0
∂z
57
1
⎡2 ∂R
⎤
1 ∂R
= ∫ x⎢
vz +
f ( x ) ⎥ dx .
0
R ∂t
⎣R ∂ z
⎦
(4.3.4)
Comparing the left hand side and right hand side of equation (4.3.4), we can write
−
∂ vz 2 ∂ R
1 ∂R
=
vz +
f ( x) .
∂z R ∂z
R ∂t
(
)
Rearranging and substituting f ( x ) = −4 x 2 − 1 into the above equation, we obtain
∂ vz
2 ∂R
4
∂R
vz + ( x 2 − 1)
=−
.
R ∂z
R
∂z
∂t
(4.3.5)
Then, substituting equation (4.3.5) into equation (4.3.1), gives
vr ( x, z , t ) = x
∂R
∂ R⎤
R x ⎡ 2 ∂R
4
2 ∂R x
vz − ∫ x ⎢ −
vz + ( x 2 − 1)
dx −
xvz dx
⎥
∂z
x 0 ⎣ R ∂z
R
∂t ⎦
x ∂ z ∫0
The above expression can be simplified to
vr ( x, z , t ) = x
=x
∂R
2 ∂R x
4 x
∂R
2 ∂R x
vz +
xvz dx − ∫ ( x3 − x )
dx −
xvz dx
∫
∂z
x ∂z 0
x 0
∂t
x ∂ z ∫0
4 x
∂R
∂R
w − ∫ ( x3 − x )
dx
0
x
∂z
∂t
x
4
∂R
4 ∂R ⎡x
x2 ⎤
=x
vz −
⎢ − ⎥
x ∂t ⎣4
2 ⎦0
∂z
⎡∂ R
⎤
∂R
⎡⎣ 2 − x 2 ⎤⎦ ⎥ .
= x⎢
w−
∂t
⎣∂z
⎦
(4.3.6)
58
Equation (4.3.6) is a new sum of the radial velocity component that we have to
choose.
4.4
Discretization of the Axial Velocity Component, vz ( x, z , t )
The finite difference scheme for solving equation (4.2.3) is based on the
central difference approximations for all the first spatial derivatives in the following
manner:
∂ vz ( vz )i, j +1 − ( vz )i, j −1
∂ vz ( vz )i+1, j − ( vz )i−1, j
=
= ( vz ) fx and
=
= ( vz ) fz .
2∆x
∂x
2∆z
∂z
k
k
k
k
(4.4.1)
while the time derivative in equation (4.2.3) is approximated by
∂ v z ( v z )i , j − ( v z )i , j
=
.
∂t
2∆t
k +1
k
(4.4.2)
Similarly, the derivatives for vr , τ zz and τ xz are
∂ vr ( vr )i, j +1 − ( vr )i, j −1
=
= ( vr ) fx
2∆x
∂x
k
k
and
∂ vr ( vr )i+1, j − ( vr )i−1, j
=
= ( vr ) fz .
2∆z
∂z
k
k
(4.4.3)
59
∂ τ zz (τ zz )i , j +1 − (τ zz )i , j −1
=
= (τ zz ) fx ,
∂x
2∆x
k
k
∂ τ zz (τ zz )i +1, j − (τ zz )i −1, j
=
= (τ zz ) fz
∂z
2∆z
k
k
and
∂ τ xz (τ xz )i , j +1 − (τ xz )i , j −1
=
= (τ xz ) fx .
∂x
2∆x
k
k
(4.4.4)
The discretization of vz ( x, z , t ) is written as vz ( x j , zi , tk ) where it can be written as
( v z )i , j .
k
Here, we define
x j = ( j − 1) ∆x ;
j = 1, 2,… N + 1 where xN +1 = 1.0 .
z i = (i − 1) ∆z ; i = 1, 2,… M + 1 .
t k = (k − 1) ∆t ; k = 1, 2,… .
Using equations (4.4.1) – (4.4.4), equation (4.2.3) may be transformed to the
following difference equations:
( v z )i , j
k +1
= ( v z )i , j
k
⎪⎧
+ ∆t ⎨
⎪⎩
− ( v z )i , j
k
k
⎡ x j ⎛ ∂ R ⎞ k u ik, j
xj ⎛ ∂ R ⎞ ⎤
k
⎢ k ⋅⎜
⎟ − k + ( v z )i , j k ⋅ ⎜
⎟ ⎥
Ri ⎝ ∂ z ⎠ i ⎥
⎢⎣ Ri ⎝ ∂ t ⎠ i Ri
⎦
(( v ) )
z
fz
1 ⎛∂ p⎞
− ⎜
⎟
i, j
ρ⎝ ∂z ⎠
k
k +1
−
(( v ) )
z
fx
k
i, j
k
1⎡ 1
1
k
+ k ⎡(τ xz ) fx ⎤
τ
⎢
k ( xz )i , j
⎦i, j
Ri ⎣
ρ ⎢⎣ x j Ri
60
k
k
k ⎛∂R⎞
x
+ ⎡(τ zz ) fz ⎤ − kj ⎡(τ xz ) fx ⎤ ⎜
⎣
⎦i, j R ⎣
⎦i, j ∂ z ⎟
⎝
⎠i
i
⎤ ⎫⎪
⎥⎬ .
⎥⎦ ⎭⎪
(4.4.5)
Thus, equations (4.2.7) and (4.2.9) have their discretized form as
(τ zz )i, j
k
⎧ ⎡
⎪ ⎛ 1
= −2 ⎨m ⎢⎜ k ⋅ ( vr ) fx
⎢
⎪ ⎢⎝ Ri
⎩ ⎣
(
)
2
k
2
⎞ ⎛ ( vr )i, j ⎞ ⎛
⎟ + ⎜ ( vz )
⎟ +⎜
k
fz
i, j
x
R
⎜
⎠ ⎝ j i ⎟⎠ ⎝⎜
k
(
)
k
xj ⎛ ∂ R ⎞
− k⎜
⎟ ⋅ ( vz ) fx
i, j
Ri ⎝ ∂ z ⎠i
k
k
⎛
xj ⎛ ∂ R ⎞
k
k
k ⎞
1
⎟
u
w
+ ⎜ ( u fz ) − k ⎜
⋅
+
⋅
(
)
(
)
⎟
fx i , j
fx i , j
i, j
⎜
⎟
Ri ⎝ ∂ z ⎠i
Rik
⎝
⎠
⎛
⋅ ⎜ ( vz ) fz
⎜
⎝
(
)
k
xj ⎛ ∂ R ⎞
− k⎜
⎟ ⋅ ( vz ) fx
ij
Ri ⎝ ∂ z ⎠i
k
⎧ ⎡
k
⎪ ⎢⎛ 1
(τ xz )i, j = −2⎨m ⎢⎜ k ⋅ ( vr ) fx
⎪ ⎢⎝ Ri
⎩ ⎣
(
)
(
2
)
(
(4.4.6)
)
k
xj ⎛ ∂ R ⎞
− k ⎜ ⎟ ⋅ ( vz ) fx
i, j
Ri ⎝ ∂ z ⎠i
k
⎛
x ⎛∂R⎞
k
k
k ⎞
1
u
w
+ ⎜ ( u fz ) − kj ⎜
⋅
+
⋅
( fx )i, j ⎟⎟
⎟ ( fx )i , j
i, j
⎜
Ri ⎝ ∂ z ⎠i
Rik
⎝
⎠
(
)
k
xj ⎛ ∂ R ⎞
− k⎜
⎟ ⋅ ( vr ) fx
i, j
Ri ⎝ ∂ z ⎠i
k
(
⎫
⎪
⎪
⎬
⎪
⎪⎭
⎞
⎟.
ij ⎟
⎠
k
⎛
⋅ ⎜ ( vr ) fz
⎜
⎝
⎤
⎥
⎥
⎦
1 n −1
2
)
k
k
2
⎞ ⎛ ( vr )i, j ⎞ ⎛
⎟ + ⎜ ( vz )
⎟ +⎜
k
fz
i, j
x
R
⎜
⎟
⎠ ⎝ j i ⎠ ⎜⎝
k
2
(
2
⎞
⎟
i, j ⎟
⎠
k
)
k
i, j
+
(
2
1
⋅ ( vz ) fx
Rik
⎤
⎥
⎥
⎦
)
(
1 n −1
2
)
2
⎞
⎟
i, j ⎟
⎠
k
⎫
⎪
⎪
⎬
⎪
⎪⎭
⎞
⎟.
i, j ⎟
⎠
k
(4.4.7)
The discretization forms for boundary conditions (4.2.10) – (4.2.12) are
( vr )i , j = 0
k
,
( vz )i ,1 = ( vz )i,2
k
k
,
(τ xz )i ,1 = 0 ,
k
(4.4.8)
61
k
⎛∂R⎞
( vz )i, N +1 = 0 , ( vr )i, N +1 = ⎜ ⎟
⎝ ∂ t ⎠i
k
(4.4.9)
and
( vr )i , j = 0
1
,
( vz )i , j = 0
1
(4.4.10)
By making use of (4.4.6) and (4.4.7) and the prescribed conditions (4.4.8) – (4.4.10),
the difference equation (4.4.5) will be solved. After obtaining the axial velocity
component, then the radial velocity can be calculated directly from equation (4.3.6).
Rewrite equation (4.3.6) in discretized form, we obtain
k
⎡⎛ ∂ R ⎞ k
⎤
⎛∂R⎞
k +1
2
= x j ⎢⎜
⎟ ( v z )i , j − ⎜
⎟ ⎡⎣ 2 − x j ⎤⎦ ⎥ .
⎢⎣⎝ ∂ z ⎠i
⎥⎦
⎝ ∂ t ⎠i
( vr )i , j
k +1
(4.4.11)
After obtaining discretized forms of the axial and the radial velocity of the streaming
blood, they can be used to obtain the volumetric flow rate (Q ) , the resistance to
flow (∧ ) , the wall shear stress (τ w ) from the following relations.
4.5
Discretized forms of the Blood Flow Characteristics
The Flow Rate (Q )
The volumetric flow rate, (Q ) is
R
Q = ∫ 2 π r vz dr .
0
62
Using the radial coordinate transformation x =
r
, we get
R
r = xR and dr = Rdx .
Then we have
1
Q = ∫ 2 π xR vz Rdx .
(4.5.1)
0
In the discretized form equation (4.5.1) can be written as
Qik = 2 π ( Rik )
2
∫ x (v )
1
k
z i, j
0
dx j
(4.5.2)
The Resistance to flow (∧ )
The equation of resistance to flow is,
∧=
L (∂ p ∂ z )
.
Q
(4.5.3)
In the discretized form, equation (4.5.3) can rewrite as
∧ =
k
i
L(∂ p ∂ z )
Qik
k
.
The Wall Shear Stress (τ w )
(4.5.4)
63
The wall shear stress is defined as
⎛ ∂ vz ∂ vx ⎞
+
⎟.
⎝ ∂x ∂z ⎠
τw = µ ⎜
In cylindrical coordinate we can write the wall shear stress as
⎛ ∂ vz ∂ vr ⎞
+
⎟.
⎝ ∂r ∂z ⎠
τw = µ ⎜
Replacing
(4.5.5)
∂
∂
and
given by (4.2.3) and (4.2.5) into equation (4.5.5), then
∂r
∂t
equation (4.5.5) becomes
⎛ ∂ vz ∂ x ∂ vr ∂ vr ∂ x ⎞
⋅
+
+
⋅
⎟.
⎝ ∂x ∂r ∂z ∂x ∂z ⎠
τw = µ ⎜
Substituting
∂x 1
∂x
x ∂R
= and
into the above equation above, we have
=−
∂r R
∂z
R ∂z
⎛ 1 ∂ vz ∂ vr x ∂ R ∂ vr ⎞
+
−
⋅
⎟.
⎝R ∂x ∂z R ∂z ∂x ⎠
τw = µ ⎜
(4.5.6)
⎛
⎛ ∂ R ⎞⎞
Multiplying equation (4.5.6) with cos ⎜ tan −1 ⎜
⎟ ⎟ , gives
z
∂
⎝
⎠
⎝
⎠
⎛ 1 ∂ vz ∂ vr x ∂ R ∂ vr
+
−
⋅
⎝R ∂x ∂z R ∂z ∂x
τw = µ ⎜
⎛ −1 ⎛ ∂ R ⎞ ⎞
⎞
⎟ cos ⎜ tan ⎜
⎟⎟.
⎠
⎝ ∂ z ⎠⎠
⎝
(4.5.7)
Note that, tan ( ∂ R ∂ z ) come from the projection of wall shear stress on the arterial
wall. In fact ( ∂ R ∂ z ) is the gradient (slope) of the arterial wall (refer to Cavalcanti
(1995)).
64
In the discretized form, equation (4.5.6) can be written as
(τ w )i
k
4.6
⎡1
= µ ⎢ k ( vz ) fx
⎢⎣ Ri
(
) + ((v ) )
k
i, j
r
k
fz i , j
−
xj
Rik
(( v ) )
r
fx
k
k
⎡
⎛∂R⎞ ⎤
⎛∂R⎞ ⎤
⎜
⎟ ⎥ × cos ⎢arctan ⎜
⎟ ⎥ . (4.5.8)
i, j ∂ z
⎢⎣
⎝
⎠i ⎥⎦ x=1
⎝ ∂ z ⎠i ⎥⎦
k
The Numerical Procedure
The MATLAB programming language has been chosen to develop the
numerical algorithm. The purpose of this numerical computation is to approximate
the axial and radial velocity of the flowing blood and then to approximate the
volumetric flow rate, the resistance to flow and the wall shear stress by making use
of the relationship between shear stress and shear rate together with the prescribed
conditions throughout the arterial segment under consideration. For this purpose, the
desired quantities of major physiological significance with the following parameter
values have been made use of (McDonald (1974)):
a = 0.08cm , L = 5cm , l = 1.6cm , d = 2cm , b = 0.1, m = 0.1735 P,
µ = 0.035P , n = 0.639, ρ = 1.06 g cm −3 , f p = 1.2 Hz ,
A = 10 g cm −2 s −2 , A1 = 0.2 A , ∆ x = 0.025 and ∆ z = 0.1 .
(4.6.1)
The iterative method has been found to be quite effective in solving the equation
numerically for different time periods. The results appeared to converge with an
accuracy of the order ~ 10−7 when the time step was chosen to be ∆ t = 0.00001 .
The procedures for programming are given in Appendix B.
65
4.7
Some Comments
It must be mentioned that if we had chosen the values suggested by Mandal
(2005) namely instead of those given in (4.6.1),
a = 0.8mm , L = 50mm , l = 16mm , d = 20mm , b = 0.1, m = 0.1735 P,
µ = 0.035P , n = 0.639, ρ = 1.06 ×103 kg m −3 , f p = 1.2 Hz ,
A = 100 kg m −2 s −2 , A1 = 0.2 A , ∆ x = 0.025 and ∆ z = 0.1
(4.7.1)
we obtain the results shown in Figures 4.7.1 – 4.7.2.
Here, if we change the unit g in the parameter values (4.6.1) into unit kg, we
were not able to get any results as the programs cannot be executed. The results
obtained when we change the parameter values (4.6.1) into unit g and mm are also
shown in Figures 4.7.1 – 4.7.2.
0
0
0.2
0.4
0.6
0.8
present
Mandal (2005)
kg mm
-0.05
g mm
u (mm/sec)
-0.1
-0.15
-0.2
-0.25
x
Figure 4.7.1 Radial Velocity Profile for φ = 0
1
66
5
present ( x 0.1 )
Mandal (2005)
4
kg mm ( x 10000000 )
g mm ( x 0.1 )
3
w (mm/sec)
2
1
0
0
0.2
0.4
0.6
0.8
1
-1
-2
-3
x
Figure 4.7.2 Axial Velocity Profile for φ = 0
The results for the variation of blood flow characteristics are illustrated as the
following Figures 4.7.3 – 4.7.5.
4
2
Flow Rate (mm3/sec)
0
0
0.5
1
1.5
2
2.5
3
present ( x 0.1 )
-2
Mandal (2005)
kg mm ( x 100000 )
g mm ( x 0.1 )
-4
-6
-8
t (sec)
Figure 4.7.3 Variation of the Rate of Flow for φ = 0
67
6
4
2
0
Resistance (N-s/m5)
( x1000000000 )
0
0.5
1
1.5
2
2.5
3
-2
-4
-6
-8
-10
present
Mandal (2005)
kg mm ( x 0.0000001 )
g mm
-12
t (sec)
Figure 4.7.4 Variation of the Resistance for φ = 0
0.06
0.04
Wall Shear Stress (N/m2)
0.02
0
0
0.5
1
1.5
2
2.5
-0.02
-0.04
-0.06
-0.08
present ( x 0.1 )
Mandal (2005)
kg mm (x 1000000000000)
g mm ( x 10 )
-0.1
t (sec)
Figure 4.7.5 Variation of the Wall Shear Stress for φ = 0
3
68
In conclusion, we choose the unit g and cm in the parameter values (4.6.1)
rather than kg and mm or g and mm (4.7.1) as in Mandal (2005). From the figures, it
is obvious that the units given in (4.6.1) are the exact values and note the results
obtained by Mandal differ from the present results by a factor of 10−1 .
CHAPTER V
NUMERICAL RESULTS AND DISCUSSION
5.1
Introduction
This chapter discusses the computed results obtained. Section (5.2) discusses
the effects of stenosis and taper angles on the axial and radial velocities. The values
of axial and radial velocities at different times are plotted. It follows in Section (5.4)
with the values of axial and radial velocity plotted at different axial positions. Then,
next section shows and explains the variation of the rate of flow, the resistance to
flow the wall shear stress with time. The last section will discuss the numerical
difficulties.
5.2
Effect of Tapering on Axial and Radial Velocity.
5.2.1
Different Taper Angle under Stenotic Conditions
Figure (5.2.1) and Figure (5.2.2) illustrates how the constricted arterial
70
tapering with varied taper angle influences the patterns of the flow-field at t = 0.45s.
Figure (5.2.1) shows the result for the axial velocity profile for different taper angles.
The curves decrease from their individual maxima and finally drop to zero. As we
can see, the axial flow velocity shifts towards the origin when φ = −0.2D (converging
(
)
(
tapering). Meanwhile, for a non-tapered φ = 0D and positively tapered φ = 0.2D
)
artery, the axial flow velocity is shifted away from the origin. From the graph we
can see the prominent curve that refers to the axial flow velocity for a positive
(
)
tapered φ = 0.2D artery having higher individual maxima compared to the nontapered and converging tapered artery. Figure (5.2.2) represents the result for the
radial velocity for the same cases as in Figure (5.2.1). The radial velocity also shifts
towards the origin when φ = −0.2D , while they shift away from the origin for a non-
(
)
tapered and diverging tapering φ = 0.2D . All the curves appear to decline from zero
on the axis, as one move away from it and finally increase towards the wall to reach
some finite values on the wall surface, which clearly reflects the presence of wall
motion in the present model.
7
phi = 0.2 ( x 0.1 )
6
phi = 0 ( x 0.1 )
phi = -0.2 ( x 0.1 )
w (mm/sec)
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
x
Figure 5.2.1: Axial Velocity Profiles for Different Taper Angles at t = 0.45s
(τ m
= 0.4 a , d = 20mm , lD = 16mm , z = 28mm ) .
71
0
0
0.2
0.4
0.6
0.8
1
-0.02
phi = 0.2
phi = 0
-0.04
phi = -0.2
u (mm/sec)
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
x
Figure 5.2.2: Radial Velocity Profiles for Different Taper Angles at t = 0.45s
(τ m
5.2.2
= 0.4 a , d = 20mm , lD = 16mm , z = 28mm )
Effect of Tapering and Stenosis on Axial and Radial velocity.
Figure (5.2.3) illustrates the results for the axial velocity profile for the nonNewtonian rheology characterized by the generalized Power-Law model at a specific
location of z = 28mm in the stenotic region of tapered artery at t = 0.45s. There are
5 distinct curves with different perspectives and distinguishable marks. The curves
decrease from their individual maxima and finally drop to zero on the wall surface.
Examining the behavior of the present figure, observe that the axial velocity profile
(
)
assumes a flat shape in the presence of a converging tapering φ = −0.1D and
(
)
parabolic shape for non-tapered φ = 0 D artery under stenotic condition. Referring
to the observation of Chakravarty and Mandal (2000), this result qualitatively agrees
with the Newtonian rheology of flowing blood through a tapered artery. The two
(
)
curves (φ = −0.1) and φ = 0 D at top of the present figure, show the effect of
72
tapering without stenosis (τ m = 0) where again the vessel tapering diminishes the
flow velocity significantly.
16
phi = 0, tm = 0 ( x 0.1 )
14
phi = -0.1, tm = 0 ( x 0.1 )
phi = 0.1 ( x 0.1 )
phi 0 ( x 0.1 )
12
phi = -0.1 ( x 0.1 )
w (mm/sec)
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
x
Figure 5.2.3: Axial Velocity Profiles at z = 28mm for t = 0.45s
(τ m
= 0.4 a , d = 20mm , lD = 16mm )
Thus, one can conclude that the axial flow velocity reduces to some extent
with vessel tapering which does not depend on the presence of any arterial
constriction or not. The third curve from the top that corresponds to the result for a
(
)
diverging tapering φ = 0.1D is different from all the remaining curves and become
higher than those non-tapered and converging tapering as anticipated.
Figure (5.2.4) shows the results of the radial velocity component varying
radially at the same critical location of z = 28mm and t = 0.45s. The curves are
found to be negative which are different from the characteristics of the axial velocity
profiles. From this figure, the effect of tapering on the radial velocity profile can be
seen in the presence of stenosis. In the case of a non-stenosed artery, the maximum
deviation occurs near the wall.
73
0
0
0.2
0.4
0.6
0.8
1
-0.05
u (mm/sec)
-0.1
-0.15
phi = -0.1
phi = 0
-0.2
phi = 0.1
phi = -0.1, tm = 0
phi = 0, tm=0
-0.25
x
Figure 5.2.4: Radial Velocity Profiles at z = 28mm for t = 0.45s
(τ m
5.3
= 0.4 a , d = 20mm , lD = 16mm )
Axial and Radial Velocity at Different Times.
Figure (5.3.1) displays the variation of the axial velocity profiles at the same
axial positions of z = 28mm in tapered artery (φ = −0.1) for different times spread
over a single cardiac period. All the results visualized in the present figure are
directly responsible for the pulsatile pressure gradient produced by the heart. Note
that, the rate of decrease as the axial velocity in the systolic phase over a single
cardiac cycle, as anticipated. Whenever time increases from 0.1s to 0.45s, the curves
shift towards the origin. Eventually, at t = 0.7s, the curve is shifted away from the
origin. At time 0.45s and 0.7s, the axial velocity increases back because the process
happened in the second phase, when the heart relaxes and becomes full of blood once
more which is called diastole.
74
8
7
t = 0.45s, Newtonian ( x 0.1)
t = 0.1s ( x 0.1)
t = 0.7s ( x 0.1)
t = 0.3s ( x 0.1)
6
t = 0.45s ( x 0.1)
w (mm/sec)
5
4
3
2
1
0
0
0.25
0.5
0.75
1
x
Figure 5.3.1: Axial Velocity Profiles for Different Times at z = 28mm
(τ
m
= 0.4 a , d = 20mm , lD = 16mm , φ = −0.1D
)
The top of the figure is shown the result of the axial velocity at t = 0.45s, but
the flowing is treated as a Newtonian (µ = 0.035) . From the observation of the result
obtained by Newtonian model, the streaming blood is much higher than the nonNewtonian values. In this case, the different value is under the influence of shear
rate. If the flowing blood is treated as Newtonian, it has the high shear rate flow,
which increases the axial velocity. Thus, the non-Newtonian characteristics of the
flowing blood affect the axial velocity profile that can be estimated by the relevant
curves.
The results indicating the unsteady behavior of the flow over a single cycle is
presented in Figure (5.3.2). It is interesting to note that, the radial velocity profile
assumes a positive value with time advancement from 0.1s to 0.3s in the systolic
phase where in this phase the heart muscle has contracted fully and blood is squeezed
out. From 0.45s to 0.7s, the radial velocity profile assumes to continue with negative
values in the diastolic phase where the heart relaxes and becomes full of blood once
more.
75
0.25
0.2
0.15
0.1
u (mm/sec)
0.05
0
0
0.2
0.4
0.6
0.8
1
-0.05
-0.1
t = 0.1s
-0.15
t = 0.3s
t = 0.45s
-0.2
t = 0.45s, Newtonian
t = 0.7s
-0.25
x
Figure 5.3.2: Radial Velocity Profiles for Different Times at z = 28mm
(τ
m
= 0.4 a , d = 20mm , lD = 16mm , φ = −0.1D
)
This typical nature of the curves reflects very closely the radial motion of the
arterial wall for a single cardiac cycle. This figure also illustrates the result of the
flowing blood that having Newtonian rheology. This points out that, the Newtonian
characteristic of the flowing blood affects the radial velocity pattern less significantly
than the axial velocity profile.
5.4
Axial and Radial Velocity at Different Axial Positions.
In order to analyze the flow-field intensively along the arterial segment,
Figure (5.4.1) shows the result of the axial velocity profiles for five distinct axial
locations at t = 0.45s for (φ = −0.1) . The axial velocity profile is parabolic at the
upstream (z = 15mm) because this area does not have stenosis while a flattening
trend is followed at the converging section (z = 24mm) where the blood flow starts at
76
a steeper stenosis area. Subsequently, at the specific location (z = 28mm) where this
is the restricted area because it is the area of the critical height of the stenosis and
then the axial velocity becomes much blunter. Finally at (z = 45mm) where there is
the end stenosis area and thus the axial flow velocity gets back again into the
parabolic patterns. This figure visualizes the graph at (z = 28mm) where at this area
it is considered non-stenosed (linear model), and consequently it is observed that the
higher velocity happens here rather than under stenotic consideration. This result
agrees qualitatively with Tu et al. (1992) through their research on the stenotic blood
flow in which the flowing blood is treated as Newtonian fluid.
14
z = 15mm ( x 0.1 )
12
z = 45mm ( x 0.1 )
z = 24mm ( x 0.1 )
w (mm/sec)
10
z = 28mm ( x 0.1 )
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
x
Figure 5.4.1: Axial Velocity Profiles for Different Axial Positions at t = 0.45s
(τ
m
= 0.4 a , d = 20mm , lD = 16mm , φ = −0.1D
)
Figure 5.4.2 shows the result of the radial velocity component at t = 0.45s
which shows that all curves are negative and become concave near the wall except at
the downstream of the stenosis. Note that, at the downstream the back flow occurs
near the wall where the direction of the velocity changes from positive to negative
and that causes separation in the flow field. From the physiological point of view, the
arterial tapering plays an important key in order to characterize the flow phenomena.
77
0
0
0.2
0.4
0.6
0.8
1
-0.05
u (mm/sec)
-0.1
-0.15
z = 28mm
z = 45mm
-0.2
z = 15mm
z = 24mm
-0.25
x
Figure 5.4.2: Radial Velocity Profiles for Different Axial Positions at t = 0.45s
(τ
m
= 0.4 a , d = 20mm , lD = 16mm , φ = −0.1D
5.5
Variation of Blood Flow Characteristics
5.5.1
Variation of the Rate of Flow with Time.
)
Figure 5.6.1 illustrates the results by showing the variation of flow rate at a
specific location of z = 28mm for certain distinct cases stretched over a period of
nearly four cardiac cycles. Note that, the pulsatile nature of the flow rate has been
found to be distributed for all the curves throughout the time scale. In the absence of
constriction, the flow rates are enhanced significantly for the entire time range. The
flow rate for a non-tapered artery having higher magnitude than the flow rate for a
negative tapered artery (φ = −0.1) . However the magnitude of the flow rate for a
diverging tapered artery is every time higher than those of non-tapered and
negatively tapered artery.
78
16
14
Flow Rate (mm3/sec)
12
10
phi = -0.1, tm = 0 ( x 0.1 )
8
phi = 0.1 ( x 0.1 )
phi = -0.1, Newtonian ( x 0.1 )
6
phi = 0 ( x 0.1 )
phi = -0.1 ( x 0.1 )
4
2
0
0
0.5
1
1.5
2
2.5
3
t (sec)
Figure 5.5.1: Variation of the Rate of Flow with Time at z = 28mm
(τ m = 0.4 a , d = 20mm , lD = 16mm )
In addition, the corresponding Newtonian model yields an analogous
behavior with higher magnitudes. If one analyses the relevant curves of the present
figure, thus the effect of taper angle and non-Newtonian rheology of the flowing
blood can be quantified. Therefore, by observing all the results referred to the present
figure, we conclude that the presence of stenosis, the taper angle and non-Newtonian
rheology of the flowing blood certainly bears the potential to influence the flow rate
to a considerable extent.
5.5.2
Variation of the Resistance of Flow with Time.
Figure 5.6.2 indicates how the resistances to flow are influenced by the
unsteady flow behavior of blood as well as by the vessel tapering, the vessel wall
distensibility, the stenosis, the steeper stenosis and by the non-Newtonian rheology
of the streaming blood.
79
7
6
phi = -0.1
phi = 0
5
Resistance (N-s/m5)
x10000000000
phi = -0.1, Newtonian
phi = 0.1
phi = -0.1, tm = 0
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
t (sec)
Figure 5.5.2: Variation of the Resistance of Flow with Time at z = 28mm
(τ m = 0.4 a , d = 20mm , lD = 16mm )
The resistances to flow follow a reverse trend from those of Figure 5.6.1 in a
way that the streaming fluid experiences higher resistance when the rate of flow in
the constricted tapered artery are correspondingly lower and vice-versa. Unlike the
characteristics of the flow rate, one may observe that the flowing blood experiences
much higher resistances to flow in the presence of arterial constriction, in the
absence of vascular wall distensibility and in the presence of non-Newtonian
characteristic of the flowing blood. However, the effects of tapering and the steeper
stenosis on the resistive impedances are not ruled out from the present investigation.
5.5.3
Variation of the Wall Shear Stress with Time.
Finally, the variation of the time-dependent wall shear stress at a specific
location of z = 28mm corresponding to a constricted zone of a tapered artery has
been portrayed in Figure 5.6.3 The wall shear stresses represented by the curves of
the concluding figure appear to be compressive in nature.
80
0
0
0.5
1
1.5
2
2.5
3
-0.02
Wall Shear Stress (N/m2)
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
phi = -0.1, Newtonian ( x 0.1 )
phi = -0.1 ( x 0.1 )
phi = 0 ( x 0.1 )
phi = 0.1 ( x 0.1 )
phi = -0.1 ( x 0.1 )
-0.18
t (sec)
Figure 5.5.3: Variation of the Wall Shear Stress with Time at z = 28mm
(τ m = 0.4 a , d = 20mm , lD = 16mm )
It appears that the cardiac cycle and the rate of decline with negative values
gradually diminishes for the rest of the pulse cycles when the streaming blood is
Newtonian past a tapered artery (φ = −0.1D ) which has a remarkable deviation with
the corresponding non-Newtonian result if one goes through the relevant curves of
the present figure and thereby the effect of non-Newtonian rheology of the flowing
blood on the wall shear stress can be well established. However, for the rest of
curves of the present figure, there is a remarkable variation of the stress
characteristics almost immediately after the onset of the first cardiac where small
fluctuations with some fixed amplitudes keep the stress steady with the advancement
of time. The stress yields all time higher values for a diverging tapering than all
other existing results corresponding to converging tapering and without tapering so
far as their magnitudes are concerned. The deviation in the result for the constricted
artery can be visualized and quantified their effects on the relevant curves of the
present figure. This observation of the present results further highlights the validity
of the present improved mathematical model.
CHAPTER VI
CONCLUSION
6.1
Summary of Research
This chapter contains an overview of the study as well as suggestions for future
research. The investigation considered in this dissertation is focused on the
mathematical model of non-Newtonian behavior of the streaming blood together with
the effects of vessel tapering. The research background, objectives, scope and
significance of the research were presented in Chapter I.
In Chapter II, the derivation of continuity equation and the equations of motion
in term of viscous stress tensor were given. We saw how the equations of motion in
Cartesian coordinates were transformed to the cylindrical coordinate system. The
derivation of the governing equations followed next and then the appropriate boundary
conditions and the pressure gradient were stated. The various forms of stenosis together
with their different mathematical equations were discussed in Chapter III. The
mathematical formulation of the geometry of mild stenosis in a non tapered and tapered
artery was explained.
82
Chapter IV describes the method of solution. Firstly, the transformation of the
governing equations using radial coordinate transform was discussed. Secondly, we
discussed the derivations of the radial velocity component and the solution of the axial
velocity component using the finite difference method based on the central difference
approximation. Further, we also discussed how to determine the volumetric flow rate,
the resistance and the wall shear stress. Lastly, in this chapter we described the
numerical procedure of the finite difference method and some comments.
In Chapter V, we have discussed the numerical results of the problem. We have
discussed the results for axial and radial velocity in different cases and the result for
flow dependence on the pressure gradient. These numerical results corresponding to the
effects of vessel tapering and the non-Newtonian behavior of the flowing blood on the
physiological flow phenomena have been obtained. Furthermore, the axial velocity
obtained by Newtonian model are dramatically much higher than the non-Newtonian
values but the Newtonian characteristic of the flowing blood affects the radial velocity
pattern less significantly than the axial velocity profile. Section 5.5 illustrated the results
of variation of flow rate, resistance and wall shear stress with time. Last section, we
have stated some comments from the results obtained. Unlike the characteristics of the
flow rate, one may observe that the flowing blood experiences much higher resistances
to flow in the presence of arterial constriction and the present of non-Newtonian
characteristics of the flowing blood. The results of wall shear stress have shown that the
Newtonian model of the flowing blood is dramatically much higher than the nonNewtonian values. In the case of the effect of vessel tapering, we also conclude that the
velocity, flow rate and the wall shear stress yields all time higher values for a diverging
tapering than all other existing results corresponding to converging tapering and without
tapering so far as their magnitudes are concerned.
83
6.2
Suggestions for Future Research
The work presented in this dissertation can be extended to several areas of
research especially regarding the bio-magnetic fluid dynamics (BFD). Tzirtzilakis et al.
(2002, 2005) pointed out that BFD is a new area in fluid mechanics which considers the
fluid dynamics of biological fluids in the presence of magnetic fields and mentioned that
the most characteristic bio-magnetic fluid is the blood. Blood can be considered as a
magnetic fluid because the red blood cells contain haemoglobin molecules, a form of
iron oxides, which are present at a uniquely high concentration in the erythrocytes. One
possibility is to extend Tzirtzilakis’s work (2002) on bio-magnetic fluid flow in a
rectangular duct to a BFD flow in cylindrical elastic tube with stenosis, which is more
realistic from the physiological point of view.
84
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89
APPENDICES
Appendix A
1.
Formulae in Polar Coordinates
1.1
Derivation of the equation (2.4.4)
From equation (2.4.2), we get
i = rˆ cos θ − θˆ sin θ
(1.1)
j = rˆ sin θ − θˆ cos θ
(1.2)
k = k
(1.3)
Let φ ( x, y, z ) is a scalar field and ∇ is a vector derivative.
Thus,
∇φ = i
∂φ ∂φ ∂φ
+j
+k
∂x
∂y
∂z
(1.4)
90
From equation (2.4.1), we get
∂x
= cos θ
∂r
,
∂x
= −r sin θ
∂θ
∂y
= sin θ
∂r
,
,
∂z
=0
∂r
∂y
= r co s θ
∂θ
,
∂z
=0
∂θ
∂x
∂y
∂z
=0 ,
=0 ,
=1
∂z
∂z
∂z
(1.5)
(1.6)
(1.7)
As we know that,
∂φ ∂φ ∂x ∂φ ∂y ∂φ ∂z
=
. + . + .
∂r ∂x ∂r ∂y ∂r ∂z ∂r
Substituting this with equation (1.5). Then,
⇒
∂φ
∂φ
∂φ
= cos θ
+ sin θ
∂r
∂x
∂y
(1.8)
∂φ ∂φ ∂x ∂φ ∂y ∂φ ∂z
.
=
+ .
+ .
∂θ ∂x ∂θ ∂y ∂θ ∂z ∂θ
Substituting this with equation (1.6). Then,
⇒
∂φ
∂φ
∂φ
= −r sin θ
+ r co s θ
∂θ
∂x
∂y
∂φ ∂φ ∂x ∂φ ∂y ∂φ ∂z
=
. + . + .
∂z ∂x ∂z ∂y ∂z ∂z ∂z
(1.9)
91
Substituting this with equation (1.6). Then,
⇒
∂φ ∂φ
=
∂z ∂z
(1.10)
Multiply equation (1.8) with r cos θ and equation (1.9) with − sin θ .
Thus,
r cos θ
∂φ
∂φ
∂φ
= r cos 2 θ
+ r cos θ sin θ
∂r
∂x
∂y
(1.11)
− sin θ
∂φ
∂φ
∂φ
= r sin 2 θ
+ r cos θ sin θ
∂θ
∂x
∂y
(1.12)
Adding equation (1.11) and equation (1.12), we get
∂φ
∂φ 1
∂φ
= cos θ
− sin θ
∂x
∂r r
∂θ
(1.13)
Multiply equation (1.8) with r sin θ and equation (1.9) with cos θ .
Thus,
∂φ
∂φ
∂φ
= r cos θ sin θ
+ r sin 2 θ
∂r
∂x
∂y
(1.14)
∂φ
∂φ
∂φ
= −r cos θ sin θ
+ r cos 2 θ
∂θ
∂x
∂y
(1.15)
r sin θ
cos θ
Adding equation (1.14) and equation (1.15), we get
∂φ
∂φ cos θ ∂φ
= sin θ
+
∂y
∂r
r ∂θ
(1.16)
92
Substituting equations (1.1), (1.2), (1.3), (1.10), (1.13) and (1.16) into equation
(1.4).
∂φ 1
∂φ ⎞
⎛
ˆ
∇φ = rˆ cos θ − θˆ sin θ ⎜ cos θ
− sin θ
⎟ + rˆ sin θ − θ cos θ
∂r r
∂θ ⎠
⎝
(
)
(
)
∂φ
k
∂z
∇φ = rˆ cos 2 θ
+
θˆ
r
∂φ ˆ
∂φ rˆ
∂φ
− θ sin θ cos θ
− sin θ cos θ
∂r
∂r r
∂θ
sin 2 θ
∂φ
∂φ ˆ
∂φ
+ rˆ sin 2 θ
+ θ sin θ cos θ
∂θ
∂r
∂r
rˆ
∂φ θˆ
∂φ ∂φ
+ sin θ cos θ
+ co s 2 θ
+k
r
∂θ r
∂θ
∂z
∇φ = r̂
∂φ θˆ ∂φ ∂φ
+
+k
∂r r ∂θ
∂z
(1.17)
From equation (1.17), we can write the vector derivative, ∇ in cylindrical
coordinates as
∇ = r̂
1.2
∂ θˆ ∂ ∂
+
+k
∂r r ∂θ
∂z
Derivation of the equation (2.4.5)
Let v ( x, y, z ) is a vector velocity which can write as
(1.18)
93
v = vr rˆ + vθ θˆ + vz k
where rˆ = rˆ ( r ,θ , z ) , θˆ = θˆ ( r ,θ , z ) , k = k ( r ,θ , z )
⎛ ∂ θˆ ∂ ∂ ⎞
∇ ⋅ v = ⎜⎜ rˆ +
+ k ⎟⎟ ⋅ vr rˆ + vθ θˆ + vz k
∂
∂
∂k ⎠
r
r
θ
⎝
(
)
Then,
⎧∂
⎫
⎧1 ∂
⎫
vr rˆ + vθ θˆ + vz k ⎬ + θˆ ⎨
vr rˆ + vθ θˆ + vz k ⎬
∇ ⋅ v = rˆ ⋅ ⎨
⎩ ∂r
⎭
⎩ r ∂θ
⎭
(
)
(
)
(1.19)
⎧∂
⎫
+ k ⎨ ⋅ vr rˆ + vθ θˆ + vz k ⎬
⎩ ∂k
⎭
(
)
⎡ ∂v
∂rˆ ˆ ∂vθ
∂θˆ ∂vz
∂k ⎤
∇ ⋅ v = rˆ ⋅ ⎢ rˆ r + vr
+θ
+ vθ
+k
+ vz ⎥
∂r
∂r
∂r
∂r
∂r ⎦
⎣ ∂r
⎡ rˆ ∂vr vr ∂rˆ θˆ ∂vθ vθ ∂θˆ k ∂vz vz ∂k ⎤
+ θˆ ⋅ ⎢
+
+
+
+
+
⎥
⎣ r ∂θ r ∂θ r ∂θ r ∂θ r ∂θ r ∂θ ⎦
(1.20)
⎡ ∂v
∂rˆ ˆ ∂vθ
∂θˆ ∂vz
∂k ⎤
+ k ⋅ ⎢ rˆ r + vr
+θ
+ vθ
+k
+ vz
⎥
∂k
∂k
∂k
∂k
∂k ⎦
⎣ ∂k
Note that, rˆ,θˆ, k are not constant vectors and so must be differentiated.
Thus,
∂rˆ
∂θˆ
∂k
=0 ,
=0 ,
=0
∂r
∂r
∂r
(1.21)
∂rˆ ˆ
∂θˆ
∂k
=θ ,
= −rˆ ,
=0
∂θ
∂θ
∂θ
(1.22)
94
∂rˆ
∂θˆ
∂k
=0 ,
=0 ,
=0
∂k
∂k
∂k
(1.23)
rˆ ⋅ k = 0 , θˆ ⋅ k = 0 , rˆ ⋅ θˆ = 0
(1.24)
rˆ ⋅ rˆ = 1 , θˆ ⋅θˆ = 1 , k ⋅ k = 1
(1.25)
Substituting equations (1.21) – (1.22) into equation (1.20). Therefore,
∇⋅v =
∂vr 1 ∂vθ vr ∂vz
+
+ +
∂r r ∂θ r ∂z
∇⋅v =
1 ∂
1 ∂v ∂v
( rvr ) + θ + z .
r ∂r
r ∂θ ∂z
or
95
APPENDICES
Appendix B
1.
The Numerical Algorithm using The MATLAB Programming Language
%Input parameter values (cm unit)
a=0.08; l0=1.6; d=2; b=0.1; m=0.1735; rho=1.06; L=5; miu=0.035;
A0=10; A1=0.2*A0; tm=0; omega=2.4*pi; delx=0.025; delz=0.1;
delt=0.00001; phi=-0.1*pi/180; mm=tan(phi); N=40; M=50; Q=300000;
%x, z & T incremental
for i= 1:M+1
z(i)=(i-1)*delz;
end;
for j= 1:N+1
x(j)=(j-1)*delx;
end;
for k=1:31
T(k)=(k-1)*0.1;
end;
for i= 1:M+1
t=0;
if (z(i)>=d) & (z(i)<=d+l0)
%Define the function of geometry (equation (3.4.10))
R(i)=((mm*z(i)+a)+((tm*(sec(phi))*(z(i)-d))/((tm^2)*(sin(phi))^2-(l0^2)/4))*...
96
(l0-(z(i)-d)))*(1-b*(cos(omega*t)-1)*exp(-b*omega*t));
%Differentiate equation (3.4.10)w.r.t t and define the function of dRdt
dRdt(i)=(b*sin(omega*t)*omega*exp(-b*omega*t)+b^2*(cos(omega*t)-1)*omega*...
exp(-b*omega*t))*(mm*z(i)+a+(tm*(sec(phi))*(z(i)-d)*(l0-z(i)+d))...
/((tm^2)*((sin(phi))^2)-((l0^2)/4)));
%Differentiate equation (3.4.10)w.r.t z and define the function of dRdz
dRdz(i)=(1-b*(cos(omega*t)-1)*exp(-b*omega*t))*(mm+(tm*sec(phi)*(l0-z(i)+d))...
/(tm^2*(sin(phi))^2-l0^2/4)-(tm*sec(phi)*(z(i)-d))/(tm^2*(sin(phi))...
^2-l0^2/4));
else
R(i)=(mm*z(i)+a)*(1-b*(cos(omega*t)-1)*exp(-b*omega*t));
dRdt(i)=(b*sin(omega*t)*omega*exp(-b*omega*t)+b^2*(cos(omega*t)-1)*omega*...
exp(-b*omega*t))*(mm*z(i)+a);
dRdz(i)=(1-b*(cos(omega*t)-1)*exp(-b*omega*t))*mm;
end;
%Boundary condition (4.4.10)
for j=1:N+1
u(i,j)=0;
w(i,j)=0;
end;
end;
***********************************************************************
for k=1:Q
t=(k)*delt;
for i=1:M+1
for j=1:N+1
%Define the functions of wfz anf wfx (equation(4.4.1))which devide
%into 3 different cases using forward, central and backward
%difference approximation. (wfz and ufz are the differentiation of
%axial and radial velocity w.r.t z).
if i==1
wfz(i,j)=(w(i+1,j)-w(i,j))/delz;
ufz(i,j)=(u(i+1,j)-u(i,j))/delz;
elseif i==M+1
wfz(i,j)=(w(i,j)-w(i-1,j))/delz;
ufz(i,j)=(u(i,j)-u(i-1,j))/delz;
97
else
wfz(i,j)=(w(i+1,j)-w(i-1,j))/(2*delz);
ufz(i,j)=(u(i+1,j)-u(i-1,j))/(2*delz);
end;
if j==1
wfx(i,j)=(w(i,j+1)-w(i,j))/delx;
ufx(i,j)=(u(i,j+1)-u(i,j))/delx;
%s is the part of the generalized Power-law model.
s=abs((((1/R(i))*(ufx(i,j)))^2+(wfz(i,j)-(x(j)/R(i)*(dRdz(i))*wfx(i,j)))^2+...
(ufz(i,j)-(x(j)/R(i))*dRdz(i)*ufx(i,j)+(1/R(i))*wfx(i,j))^2)^(1/2));
elseif j==N+1
wfx(i,j)=(w(i,j)-w(i,j-1))/delx;
ufx(i,j)=(u(i,j)-u(i,j-1))/delx;
s=abs((((1/R(i))*(ufx(i,j)))^2+(u(i,j)/(x(j)*R(i)))^2+(wfz(i,j)-(x(j)/R(i)*...
(dRdz(i))*wfx(i,j)))^2+(ufz(i,j)-(x(j)/R(i))*dRdz(i)*ufx(i,j)+(1/R(i))*...
wfx(i,j))^2)^(1/2));
else
wfx(i,j)=(w(i,j+1)-w(i,j-1))/(2*delx);
ufx(i,j)=(u(i,j+1)-u(i,j-1))/(2*delx);
s=abs((((1/R(i))*(ufx(i,j)))^2+(u(i,j)/(x(j)*R(i)))^2+(wfz(i,j)-(x(j)/R(i)*...
(dRdz(i))*wfx(i,j)))^2+(ufz(i,j)-(x(j)/R(i))*dRdz(i)*ufx(i,j)+(1/R(i))*...
wfx(i,j))^2)^(1/2));
end;
%Devide into two cases - avoid deviding with zero.
if s==0
s=0;
else
s=1/(s^0.361);
end;
%Define the function of tzz (equation (4.4.6)).
tzz(i,j)=-2*(m*s)*(wfz(i,j)-(x(j)/R(i))*dRdz(i)*wfx(i,j));
%Define the function of txz(equation(4.4.7)).
txz(i,j)=-1*(m*s)*(ufz(i,j)-(x(j)/R(i))*dRdz(i)*ufx(i,j)+(1/R(i))*wfx(i,j));
end;
end;
98
%Define the functions of tzzfz, tzzfx and txzfx(equation(4.4.4)).
for i=1:M+1
for j=1:N+1
if i==1
tzzfz(i,j)=(tzz(i+1,j)-tzz(i,j))/delz;
elseif i==M+1
tzzfz(i,j)=(tzz(i,j)-tzz(i-1,j))/delz;
else
tzzfz(i,j)=(tzz(i+1,j)-tzz(i-1,j))/(2*delz);
end;
if j==1
tzzfx(i,j)=(tzz(i,j+1)-tzz(i,j))/delx;
txzfx(i,j)=(txz(i,j+1)-txz(i,j))/delx;
elseif j==N+1
tzzfx(i,j)=(tzz(i,j)-tzz(i,j-1))/delx;
txzfx(i,j)=(txz(i,j)-txz(i,j-1))/delx;
else
tzzfx(i,j)=(tzz(i,j+1)-tzz(i,j-1))/(2*delx);
txzfx(i,j)=(txz(i,j+1)-txz(i,j-1))/(2*delx);
end;
end;
end;
for i=1:M+1
for j=2:N
%Define the function of dp/dz as shown in equation (2.6.4)
dPdz=-(A0+A1*cos(omega*(t)));
%Define the function of w(the axial velocity approximation (4.4.5))
w(i,j)= w(i,j)+delt*((-1/rho)*dPdz+((x(j)/R(i))*dRdt(i)-(u(i,j)/R(i))+...
(x(j)/R(i))*dRdz(i)*w(i,j))*wfx(i,j)-w(i,j)*wfz(i,j)-...
(1/rho)*(1/(x(j)*R(i))*txz(i,j)+(1/R(i))*txzfx(i,j)+tzzfz(i,j)-(x(j)/R(i))*dRdz(i)*tzzfx(i,j)));
%Define the function of w(the axial velocity approximation(4.4.11))
u(i,j)=x(j)*(dRdz(i)*w(i,j)+dRdt(i)*(2-x(j)^2));
end;
end;
99
for i=1:M+1
if (z(i)>=d) & (z(i)<=d+l0)
R(i)=((mm*z(i)+a)+((tm*(sec(phi))*(z(i)-d))/((tm^2)*(sin(phi))^2-...
(l0^2)/4))*(l0-(z(i)-d)))*(1-b*(cos(omega*t)-1)*exp(-b*omega*t));
dRdt(i)=(b*sin(omega*t)*omega*exp(-b*omega*t)+b^2*(cos(omega*t)-1)*omega*...
exp(-b*omega*t))*(mm*z(i)+a+(tm*(sec(phi))*(z(i)-d)*(l0-z(i)+d))/((tm^2)*...
((sin(phi))^2)-((l0^2)/4)));
dRdz(i)=(1-b*(cos(omega*t)-1)*exp(-b*omega*t))*(mm+(tm*sec(phi)*(l0-z(i)+d))/...
(tm^2*(sin(phi))^2-l0^2/4)-(tm*sec(phi)*(z(i)-d))/(tm^2*(sin(phi))^2-...
l0^2/4));
else
R(i)=(mm*z(i)+a)*(1-b*(cos(omega*t)-1)*exp(-b*omega*t));
dRdt(i)=(b*sin(omega*t)*omega*exp(-b*omega*t)+b^2*(cos(omega*t)-1)*omega*...
exp(-b*omega*t))*(mm*z(i)+a);
dRdz(i)=(1-b*(cos(omega*t)-1)*exp(-b*omega*t))*mm;
end;
end;
%Boundary condition (4.4.8)&(4.4.9)
for i=1:M+1
for j=1:N+1
w(i,1)=w(i,2); w(i,N+1)=0;
u(i,1)=0; u(i,N+1)=dRdt(i);
end;
end;
%Define the functions of flow rate, Q and resistance, v (equations(4.5.2)&(4.5.4))
for i=1:M+1
for j=1:N+1
F(i,j)= x(j)*w(i,j);
end;
end;
for i=1:M+1
f(i)=0;
end;
for i=1:M+1
100
for j=2:N
f(i)=f(i)+F(i,j);
end;
end;
for i=1:M+1
Q(i)=(2*pi*(R(i))^2)*(delx/2)*((F(i,1)+F(i,N+1))+(2*f(i))); %from trapezium formulae
if Q(i)==0
v(i)=0;
else
v(i)=abs(L*(dPdz))/Q(i);
end;
end;
%Define the function of Wall Shear Stress (equation (4.5.8))
for i= 1:M+1
if i==1
wss(i)=miu*((1/R(i))*(w(i,N+1)-w(i,N))/(delx)+(u(i+1,N+1)-u(i,N+1))/(delz)...
-(1/R(i))*((u(i,N+1)-u(i,N))/(delx))*dRdz(i))*cos(atan(dRdz(i)));
elseif i==M+1
wss(i)=miu*((1/R(i))*(w(i,N+1)-w(i,N))/(delx)+(u(i,N)-u(i-1,N))/(delz)...
-(1/R(i))*((u(i,N+1)-u(i,N))/(delx))*dRdz(i))*cos(atan(dRdz(i)));
else
wss(i)=miu*((1/R(i))*(w(i,N+1)-w(i,N))/(delx) +(u(i+1,N)-u(i-1,N))/(2*delz)...
-(1/R(i))*((u(i,N+1)-u(i,N))/(delx))*dRdz(i))*cos(atan(dRdz(i)));
end;
end;