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Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
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Figure 3.14. Baseline (left) and follow-up (right) shear stress distribution for the left coronary artery.
Figure 3.15. Left coronary artery for a specific patient. Left: shear stress distribution. Right: plaque
concentration distribution.
Table 3.1. Biomolecular parameters for a specific patient at baseline.
Time Total cholesterol LDL HDL Triglycerides
Baseline 198 100 36 309
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detect luminal narrowing, on the one hand, and generate FE models and process
them, on the other. A number of different patients with significant ISR for the same
time period of six months follow-up were analyzed. The geometry of the patients for
baseline and follow-up has been used for the imaging techniques described in detail.
The flow equations are coupled with the transport equation, applying realistic
boundary conditions for each patient.
Blood flow simulation is described with the Navier–Stokes and continuity
equations. The governing FE equations used in modeling wall tissue deformation
with emphasis on the implementation of nonlinear constitutive models are described.
The coupling of fluid dynamics and solute dynamics at the endothelium is achieved
by the Kedem–Katchalsky equations.
The discrete approach used the DPD method with conservative, dissipative and
random forces and an additional attractive force to the arterial wall. Some of the
initial results are presented for two patients for the left and right coronary artery.
The biomolecular parameters ICAM1, LDL, HDL and triglycerides are used for
both patients to calculate plaque concentration. The baseline geometry is used to
calculate the shear stress distribution. Follow-up CT studies after 24 months are
compared to the simulation results. A good agreement is achieved. This methodology could be validated with retrospective studies from the literature if at least two
points in time are available (baseline and follow-up).
The presented methodologies represent patient-specific modeling tools which can
be used for clinical treatment decisions. The obtained results are very valuable
because they make it possible to visualize and present the spatial distribution of
biomechanical quantities, which is practically impossible to obtain without modeling. An additional aim of this chapter was to validate our computer simulation
model for plaque progression in patients with stented coronary arteries.
The stress distribution of the artery wall and stent during expansion of occluded
zones is analyzed. The shear stress distribution before and after stent deployment is
also compared. A better understanding of stent deployment procedures and arterial
wall responses, as well as optimal stent design, can be obtained using computer
simulation.
Acknowledgment
This study was funded by grants from the Serbia III41007, ON174028 and EC
HORIZON2020 689 068 SMARTool project.
References
[1] De Backer G G 2009 The global burden of coronary heart disease Medicographia 31 343–8
[2] Herrington W, Lacey B, Sherliker P, Armitage J and Lewington S 2016 Epidemiology of
atherosclerosis and the potential to reduce the global burden of atherothrombotic disease
Circ. Res.
[3] Meads C, Cummins C, Jolly K, Stevens A, Burls A and Hyde A 2000 Coronary artery stents
in the treatment of ischaemic heart disease: a rapid and systematic review Health Technol.
Assess 4 1–153
118 535–46
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[4] Serruys P W et al 1994 A comparison of balloon-expandable-stent implantation with balloon
angioplasty in patients with coronary artery disease N. Engl. J. Med.
[5] Fischman D L et al 1994 A randomized comparison of coronary-stent placement and
balloon angioplasty in the treatment of coronary artery disease N. Engl. J. Med.
[6] Pache J et al 2003 Intracoronary stenting and angiographic results: strut thickness effect on
restenosis outcome (ISAR-STEREO-2) trial J. Am. Coll. Cardiol.
[7] Lee C H et al 2007 Sirolimus-eluting, bioabsorbable polymer-coated constant stent (Cura) in
acute ST-elevation myocardial infarction: a clinical and angiographic study (CURAMI
Registry) J. Invasive Cardiol. 19 182–5
[8] Costa R A et al 2006 Angiographic results of the first human experience with the Biolimus
A9 drug-eluting stent for de novo coronary lesions Am. J. Cardiol.
[9] Yuk S H, Oh K S, Park J, Kim S J, Kim J H and Kwon I K 2012 Paclitaxel-loaded poly
(lactide-co-glycolide)/poly(ethylene vinyl acetate) composite for stent coating by ultrasonic
atomizing spray Sci. Technol. Adv. Mater.
[10] Kim S J et al 2011 Development of a biodegradable sirolimus-eluting stent coated by
ultrasonic atomizing spray J. Nanosci. Nanotechnol.
[11] Farooq V, Gogas B D and Serruys P W 2011 Restenosis—delineating the numerous causes
of drug-eluting stent restenosis Circ. Cardiovasc. Interv.
[12] Fischman D L et al 1994 A randomized comparison of coronary-stent placement and
balloon angioplasty in the treatment of coronary artery disease N. Engl. J. Med.
[13] Morice M C et al 2002 A randomized comparison of a sirolimus-eluting stent with a
standard stent for coronary revascularization N. Engl. J. Med.
[14] Newman V S, Berry J L, Routh W D, Ferrario C M and Dean R H 1996 Effects of vascular
stent surface area and hemodynamics on intimal thickening J. Vasc. Interv. Radiol.
[15] Murata T, Hiro T, Fujii T, Yasumoto K, Murashige A, Kohno M, Yamada J, Miura T and
Matsuzaki M 2002 Impact of the cross-sectional geometry of the post-deployment coronary
stent on in-stent neointimal hyperplasia: an intravascular ultrasound study Circ. J.
[16] Murphy J and Boyle F 2008 Assessment of the effects of increasing levels of physiological
realism in the computational fluid dynamics analyses of implanted coronary stents 3rd Annual
Int. IEEE EMBS Conf.
[17] DePaola N, Gimbrone M A J, Davies P F and Dewey C F 1992 Vascular endothelium
responds to fluid shear stress gradients Arterioscler. Thromb.
[18] Balossino R, Gervaso F, Migliavacca F and Dubini G 2008 Effects of different stent designs
on local hemodynamics in stented arteries J. Biomech.
[19] Simard T, Hibbert B, Ramirez F D, Froeschl M, Chen Y X and O’Brien E R 2014 The
evolution of coronary stents: a brief review Can. J. Cardiol.
[20] Lüscher T F et al 2007 Drug-eluting stent and coronary thrombosis: biological mechanisms
and clinical implications Circulation
[21] Kimura T et al 2010 Comparisons of baseline demographics, clinical presentation, and long-
term outcome among patients with early, late, and very late stent thrombosis of sirolimuseluting stents: observations from the Registry of Stent Thrombosis for Review and
Reevaluation (RESTART) Circulation
[22] Mori H, Joner M, Finn A V and Virmani R 2016 Malapposition: is it a major cause of stent
thrombosis? Eur. Heart J.
[23] Armstrong E J et al 2012 Clinical presentation, management, and outcomes of angiographically
documented early, late, and very late stent thrombosis JACC Cardiovasc. Interv.
37 1217–9.
13 025005
11 5689–97
4 195–205
12 1254–7
41 1053–61
30 35–45
115 1051–8
122 52–61
331 489–95
331 496–501
41 1283–8
98 443–6
331 496–501
349 1315–23
7 387–93
66 489–93
5 131–40
3-24
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[24] Tada T et al 2013 Risk of stent thrombosis among bare-metal stents, first-generation drug-
eluting stents, and second-generation drug-eluting stents: results from a registry of 18,334
patients JACC Cardiovasc. Interv.
[25] He Y, Duraiswamy N, Frank A O and Moore J E Jr 2005 Blood flow in stented arteries: a
parametric comparison in three-dimensions J. Biomech. Eng.
[26] Natarajan S and Mokhtarzadeh-Dehghan M R 2000 A numerical and experimental study of
periodic flow in a model of a corrugated vessel with application to stented arteries Med. Eng.
Phys.
22 555–66
[27] LaDisa J F, Guler I, Olson L E, Hettrick D A, Kersten J R, Warltier D C and Pagel P S 2003
Three-dimensional computational fluid dynamics modeling of alterations in coronary wall
shear stress produced by stent implantation Ann. Biomed. Eng.
[28] Seo T, Schachter L G and Barakat A I 2005 Computational study of fluid mechanical
disturbance induced by endovascular stents Ann. Biomed. Eng.
[29] Tortoriello A and Pedrizzetti G 2004 Flow-tissue interaction with compliance mismatch in a
model stented artery J. Biomech.
[30] Berry J L, Santamarina A, Moore J E, Roychowdhury S and Routh W D 2000 Experimental
and computational flow evaluation of coronary stents Ann. Biomed. Eng.
[31] LaDisa J F, Olson L E, Douglas H E, Warltier D C, Kersten J R and Pagel P S 2006
Alterations in regional vascular geometry produced by theoretical stent implantation
influence distributions of wall shear stress: analysis of curved artery using 3D computational
fluid dynamics models Biomed. Eng. Online
[32] LaDisa J F, Olson L E, Guler I, Hettrick D A, Kersten J R, Warltier D C and Pagel P S 2005
Circumferential vascular deformation after stent implantation alters wall shear stress evaluated
with time-dependent 3D computational fluid dynamics models J. Appl. Physiol.
[33] Rajamohan D, Banerjee R K, Back L H, Ibrahim A A and Jog M A 2006 Developing
pulsatile flow in a deployed coronary stent J. Biomech. Eng.
[34] Nikolsky E et al 2003 Stent deployment failure: reasons, implications, and short- and long-
term outcomes Catheter Cardiovasc. Interv.
[35] Andreou I et al 2016 Atherosclerotic plaque behind the stent changes after bare-metal and
drug-eluting stent implantation in humans: Implications for late stent failure? Atherosclerosis
252 9–14
[36] Everett K D et al 2016 Structural mechanics predictions relating to clinical coronary stent
fracture in a 5 year period in FDA MAUDE database Ann. Biomed. Eng.
[37] Voros S et al 2011 Prospective validation of standardized, 3-dimensional, quantitative
coronary computed tomographic plaque measurements using radiofrequency backscatter
intravascular ultrasound as reference standard in intermediate coronary arterial lesions:
results from the ATLANTA (assessment of tissue characteristics, lesion morphology, and
hemodynamics by angiography with fractional flow reserve, intravascular ultrasound and
virtual histology, and noninvasive computed tomography in atherosclerotic plaques) I study
JACC Cardiovasc. Interv.
[38] de Graaf M A et al 2013 Automatic quantification and characterization of coronary
atherosclerosis with computed tomography coronary angiography: cross-correlation with
intravascular ultrasound virtual histology Int. J. Cardiovasc. Imaging
[39] Athanasiou L et al 2016 Three-dimensional reconstruction of coronary arteries and plaque
morphology using CT angiography—comparison and registration with IVUS BMC Med.
Imaging
16 9
4 198–208
6 1267–74
22 637–47
31 972–80
33 444–56
37 1–11
28 386–98
5 40
98 947–57
128 347–59
59 324–8
44 391–403
29 1177–90
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Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
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[40] Bourantas C V et al 2008 ANGIOCARE: an automated system for fast three-dimensional
coronary reconstruction by integrating angiographic and intracoronary ultrasound data
Catheter Cardiovasc. Interv.
[41] Plissiti M E, Fotiadis D I, Michalis L K and Bozios G E 2004 An automated method for
lumen and media-adventitia border detection in a sequence of IVUS frames Trans. Inf.
Technol. Biomed.
[42] de Jaegere P et al 1998 Intravascular ultrasound-guided optimized stent deployment.
Immediate and 6 months clinical and angiographic results from the Multicenter
Ultrasound Stenting in Coronaries Study (MUSIC Study) Eur. Heart J.
[43] Magnus P C et al 2015 Optical coherence tomography versus intravascular ultrasound in the
evaluation of observer variability and reliability in the assessment of stent deployment: the
OCTIVUS study Catheter Cardiovasc. Interv.
[44] Zhang Y et al 2012 Comparison of intravascular ultrasound versus angiography-guided
drug-eluting stent implantation: a meta-analysis of one randomised trial and ten observational studies involving 19,619 patients EuroIntervention
[45] Brugaletta S et al 2012 Comparison of in vivo eccentricity and symmetry indices between
metallic stents and bioresorbable vascular scaffolds: insights from the ABSORB and SPIRIT
trials Catheter Cardiovasc. Interv.
[46] Wang A et al 2013 Automatic stent strut detection in intravascular optical coherence
tomographic pullback runs Int. J. Cardiovasc. Imaging
[47] Nakazawa G et al 2009 Incidence and predictors of drug-eluting stent fracture in human
coronary artery a pathologic analysis J. Am. Coll. Cardiol.
[48] Scheinert D et al 2005 Prevalence and clinical impact of stent fractures after femoropopliteal
stenting J. Am. Coll. Cardiol.
[49] Woo J et al 2010 Nonlinear registration of serial coronary CT angiography (CCTA) for
assessment of changes in atherosclerotic plaque Med. Phys.
[50] Marquering H A et al 2008 Coronary CT angiography: IVUS image fusion for quantitative
plaque and stenosis analyses Proc. SPIE 6918 69181G
[51] Kojic M, Filipovic N, Stojanovic B and Kojic N 2008 Computer Modeling in Bioengineering:
Thеoretical Background, Examples and Software (Chichester: Wiley)
[52] Filipovic N, Mijailovic S, Tsuda A and Kojic M 2006 An implicit algorithm within the
arbitrary Lagrangian–Eulerian formulation for solving incompressible fluid flow with large
boundary motions Comput. Methods Appl. Mech. Eng.
[53] Taylor C A, Hughes T J R and Zarins C K 1998 Finite element modeling of blood flow in
arteries Comput. Methods Appl. Mech. Eng.
[54] Nanfeng S, Nigel W, Alun H, Simon T X and Yun X 2006 Fluid-wall modelling of mass
transfer in an axisymmetric stenosis: effects of shear-dependent transport properties Ann.
Biomed. Eng.
[55] Filipovic N, Rosic M, Tanaskovic I, Milosevic Z, Nikolic D, Zdravkovic N, Peulic A, Fotiadis
D and Parodi O 2012 ARTreat project: three-dimensional numerical simulation of plaque
formation and development in the arteries IEEE Trans. Inf. Technol. Biomed.
[56] Filipovic N 2013 PAK-Athero, Finite Element Program for Plaque Formation and
Development (Serbia: University of Kragujevac)
[57] Boryczko K, Dzwinel W and Yuen D 2003 Dynamical clustering of red blood cells in
capillary vessels J. Mol. Model.
8 131–41
34 1119–28
72 166–75
19 1214–23
86 229–35
8 855–65
79 219–28
29 29–38
54 1924–31
45 312–5
37 885–96
195 6347–61
158 155–96
16 272–8
9 16–33
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Vascular and Intravascular Imaging Trends, Analysis, and
Challenges, Volume 1
Stent applications
Petia Radeva and Jasjit S Suri
Chapter 4
Current status of computational fluid dynamics
for modeling of diseased vessels
Arindam Bit, Himadri Chattopadhyay and Jasjit Suri
Cardiac disorders in association with vascular anomalies are some of the most
common diseases, frequently affecting many populations worldwide due to rapid
changes in human lifestyles. There is an immense need for early detection of these
kinds of disorders using minimally invasive techniques, in order to intervene with
early treatment of the disorder, preventing late stage complexity. Computational
fluid dynamics (CFD) is one of the few non-invasive techniques for the diagnosis of
vascular disorders. At the same time, it provides tools for treatment planning and
management, with the most efficient parametric combination for therapeutic
solutions. In this chapter, comprehensive descriptions of a few case studies are
provided, which reflect the capabilities of this technique to perform modeling of
diseased vessels as well as modeling of treatment protocols.
4.1 Introduction
4.1.1 Disease vessel classification
Blood vessels are important components of the circulatory system. A blood vessel
experiences sustainable stress conditions in the presence of the continuous flow of
blood through its lumen. Cardiovascular diseases (CVDs), such as coronary artery
disease and stroke, are among the largest causes of death and disability in the
industrialized world. Data from European CVD statistics shows that each year 4
million deaths in Europe occur due to CVD, which is 47% of the total mortality rate
[14]. The blood vessels distribute blood to different organs and supply them with
nutrition. The arteries, far from being just ordinary conduits, adapt to varying flow
and pressure conditions by enlarging or shrinking to meet changing hemodynamic
demands. In this chapter, different types of diseases associated with the vascular
doi:10.1088/2053-2563/ab01fach4 4-1 ª IOP Publishing Ltd 2019
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network are first described. This is followed by a description of the constitutive
equation for transport of blood across the vascular network. Diseased rheological
conditions of the transported blood are discussed thereafter. The following part of
the chapter deals with the numerical modeling aspects of the transported blood
across a diseased vascular network. This is followed by a comprehensive discussion
of the various factors and parameters to determine the wall-effect in the presence of
shear stress. Finally, the chapter concludes with descriptions of various shear-stress
parameters and their influence in the determination of pathological conditions of
vascular networks.
4.1.1.1 Stenosed vessel
Diseases in blood vessels are broadly classified as stenoses (figure 4.1) and aneurysms
(figure 4.2). Coronary heart disease and stroke are clinical symptoms of atherosclerosis, which is an inflammatory disease in which high plasma concentrations of
cholesterol, in particular those of low-density lipoprotein (LDL) cholesterol, are
among the principal risk factors.
More precisely, atherosclerosis is a pathological process promoted by infl ammation of the inner arterial wall (intima), initiated by an excess of LDL in the blood.
The LDL particles, as well as high-density lipoproteins (HDL), become oxidized by
ongoing chemical reactions within the body and form ox-LDL, contributing to the
atherosclerotic process when the ox-LDL concentration exceeds a threshold. At this
stage, endothelial cells activate the immune system (monocytes and T cells) to deal
with the problem. Once in the intima, the incoming immune cells instantaneously
differentiate into active macrophages, which absorb ox-LDL by phagocytosis. This
reaction transforms macrophages into foam cells that should be removed by the
immune system, yielding the secretion of a pro-inflammatory signal contributing to
the recruitment of new monocytes, and consequently this starts a chronic inflammatory reaction (auto-amplification phenomenon). The inflammation process
involves the proliferation and the migration of smooth muscle cells to create a
fibrous cap over the lipid deposit, isolating it from the blood flow. The fibrous cap
changes the geometry of the vessel and modifies the blood flow. The occurrence of a
heart attack or stroke is thus attributed to the degradation and rupture of the cap,
the formation of a blood clot in the lumen and the subsequent obstruction of the
artery forming stenoses.
Figure 4.1. Stenosed blood vessel.
4-2
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Figure 4.2. Details of an aneurysm.
4.1.1.2 Aneurismal vessel
An aneurysm is a localized dilation, bulging or ballooning of a blood vessel to
greater than 1.5 times its original diameter. Aneurysms are typically either fusiform
or saccular in shape, occurring most frequently in the abdominal aorta, but can also
occur in the thoracic aorta, inter-cranial arteries/capillaries and coronary arteries.
They result from weakening of an arterial wall section owing to a variety of genetic,
biomechanical, biochemical and hemodynamic factors, such as hereditary conditions, atherosclerosis, inflammation, infection, hypertension, lung disease, smoking
and obesity. Thus, the exact causes and sequence of events are not yet fully
understood, but are commonly a result of multiple factors. When aneurysm rupture
occurs in the cerebral circulation, the clinical manifestation is usually a stroke.
However, massive bleeding and circulatory collapse are the results of ruptured
abdominal and thoracic aortic aneurysms. In the United States, more than two
million people are diagnosed annually with aneurysms and approximately 18 000
Americans are killed by all types of aneurysms every year. A notable victim of
aneurysm is none other than Albert Einstein. Although aneurysms can occur in any
blood vessel, most, i.e. about 75%, are asymptomatic. When symptoms do occur,
they generally result from compression of adjacent structures and can be felt in
several ways, such as a pulsating sensation or severe pain, depending on the location
of the aneurysm. Figure 4.2 shows the most common form of aneurysm, i.e.
saccular.
Thoracic aortic aneurysms (TAAs) occur in the chest. The rupture of a TAA can
cause rapid blood loss and death. Clinically, they are not as common as abdominal
aortic aneurysms. Hereditary conditions (e.g. Marfan’s syndrome) are believed to be
the main cause of TAAs. An aortic dissection results from a tear between the tissue
layers of the aorta, which is caused by blood flow pumped from the heart, and may
acutely dilate causing an aneurysm. The tear usually occurs in the thoracic aorta.
Clinically, the aorta is seldom dilated before the dissection occurs. Risk factors include
high blood pressure, particularly high diastolic blood pressure, and hereditary
disorders such as Marfan’sandEhlers–Danlos syndrome.
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Abdominal aortic aneurysms (AAAs) are located below the renal arteries and
above the iliac bifurcation. Extension into one or both of the vessels supplying blood
to the legs occurs in 40% of patients afflicted with AAAs. Furthermore, as many as
15% of patients with AAAs may also develop aneurysms further down their leg
involving the arteries in their groin (femoral arteries) or behind their knees (popliteal
arteries). Seventy-five percent of abdominal aortic aneurysms occur in people over
60 years of age. The AAA shapes are irregular, including bulge, prism, zigzag, boat
and cylinder. Typically, an AAA is fusiform and asymmetric because the spinal
column limits posterior expansion and hence causes anterior bulging.
A brain aneurysm is a weak ballooning in the brain’s blood vessels. Clinically, it is
also called a cerebral or intracranial aneurysm. Brain aneurysms usually occur at a
branch of the brain arteries. If the brain aneurysm reaches a certain size (i.e. >2 cm),
the aneurysm may generate pressure on the surrounding tissue and cause progressive
problems, and if an aneurysm ruptures and bleeds, stroke or death may follow.
Unfortunately, 60% of people with ruptured brain aneurysms die within a year.
Compared with aortic aneurysms, heart (or coronary) aneurysms are quite rare.
They may occur within the first two weeks after a severe heart attack when much of
the heart muscle in the left ventricle (main pumping chamber of the heart) may be
dead. The dysfunctional muscle and scar tissue may stretch and dilate to form an
aneurysm. Symptoms may include chest pain or pressure, pain in the jaw or arms,
trouble breathing, or fainting spells. Rupture of a ventricular aneurysm is usually
fatal. Once an aneurysm has reached a critical diameter, with the growth rate and/or
other parameter values indicating possible rupture, the treatment options are either
open surgery or endovascular repair.
4.2 Constitutive equation of blood flow in a diseased vessel
4.2.1 Mass conservation equation
The equation for mass conservation [3] in a general form is given by equation (4.1)as
u
() 1( )
∂
where,
ρρ ρ∂
t
∂
,
, etc, are defined as follows:
x
x
2
1
==xxzfor Cartesian coordinates
1
+
xrrux
∂
1
1
for cylindrical axi-symmetric coordinates
==xyrfor Cartesian coordinates
2
for cylindrical axi-symmetric coordinates
moreover,
=r 1 for Cartesian coordinates.
The mass conservation equation for problems involving two-dimensional,
Cartesian geometry may be written as
∂
2
+
4-4
=
∂
0,
2
(4.1)
v
v
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∂
u
() ()
∂
+
t
∂
xy
∂
vρρ ρ∂
+
=
∂
0,
(4.2)
where x and y are the coordinate directions, and u andvare the velocity components
in those coordinate directions, respectively.
For cylindrical, axi-symmetric geometries, the same equation reads [4]
() 1( )
∂
vvρρ ρ∂
zr
+
tzrrr
∂
∂
where r is the radial coordinate and z is the axial coordinate.
∂
+
=
0,
∂
and
r
z
respective velocity components.
4.2.2 Momentum conservation equations
The x momentum equation in Cartesian coordinates may be written as
∂
()
u
() () () ()
∂
t
∂
uu
∂
+
+
x
∂
u
∂
vρρ ρ τ
=−
y
∂
∂
p
∂
xx y
∂
xx
+
∂
τ
yx
+
.
∂
Similarly, the y momentum equation in Cartesian coordinates may be written as
() () ()
vvvvρρ ρ
∂
∂
u
∂
+
xypyx y
∂
∂
+
∂
=−
∂
() ()
∂
∂
ττ
+
∂
∂
xy yy
+
∂t
.
In equations (4.4) and (4.5), the viscous stresses appear explicitly and for
Newtonian fl uids they are given as follows:
⎡
xx
⎢
⎣
⎡
⎢
yy
⎣
2
u
∂
−∇·
3
x
∂
2
v
∂
−∇·
3
∂
y
⎤
() (4.6)
vτμ=
⎥
⎦
⎤
() (4.6)
vτμ=
⎥
⎦
(4.3)
are the
(4.4)
(4.5)
a2
b2
It may be noted here that for incompressible flows, the divergence of velocities,
, and hence, the expressions for stresses, as given in equation (4.6), assume
v
·=0
a much simpler form.
In cylindrical coordinates, for axi-symmetric problems the axial (z) momentum
equation is given as
() ( ) 1( ) () 1()
vvv vvρρ ρ τ τ∂
zzz rz zz zr
∂
⎡
u
∂
∂
⎢
yx
⎣
∂
=−
4-5
ττμ==
yx xy
∂
+
∂
+
∂
⎤
∂
v
+
⎥
∂
⎦
p
∂
∂
+
zzrrr
∂
+
∂
c.(4.6)
∂
.
(4.7)
∂tzrrr
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