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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3592_Библиотеки_им_академика_М_И_Перельмана
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μ
μ
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Similarly, the radial (r) momentum equation is given as
∂
vvv vvvρρ ρ ρ τ ττ∂
() ( ) 1( ) () 1( )
rzr rr zr rr
+
∂
tzr
∂
∂
+
r
∂
rrprzrrrr
2
θθθ
−=−
∂
∂
+
∂
+
∂
∂
−
.
(4.8)
∂
Clearly, for axi-symmetric problems, two additional terms appear in the radial
momentum equation. These terms are
which is essentially an acceleration term (or convection term), and
2
on the left-hand side of equation (4.8),
vρθr/
r/
θθ
on the righthand side of (4.8), which is a viscous force, arising from the tangential stresses due to
the curvature effect.
The non-conservative form of the convective terms is as follows:
⎡
φφφ ρφφρρφ
∂
ρ
⎢
⎣
∂
Non-conservative form of the convective terms
() 1( ) ( ) 1( ) ( )
∂
∂
Conservative form of the convective terms
∂
+
∂
v
+
t
+
rz
r
∂
vv vvρφ ρ φ ρ φ
rz rz
+
∂
() 1( ) ( )
∂
=
∂
Conservative form of the convective terms
⎤
∂
v
⎥
⎦
ztrr
∂
∂
−
∂
∂
+
() 1( )
∂
=
∂
∂
φρ ρφ
−
rrvr
⎡
φ
⎢
⎣
These terms are zero by virtue of the continuity equation
vvρφ ρ φ ρ φ
∂
rz
+
∂
∂
−
∂
() ( ) ()
rz z
+
∂
∂
ρρ ρ
+
∂
∂trrrz
∂
+
v
∂
z
∂
r
∂
+
rz
∂
.
rv
∂
r
r
∂
v
ρ
−
φ
∂
z
⎤
∂
⎥
⎦
∂trrrz tr
(4.9)
Therefore, the convective terms may be expressed either in terms of conservative
or non-conservative forms and the former is preferred in the present formulation as
it is best suited to finite volume formulations. For three-dimensional studies,
corresponding three-dimensional forms of the equations are used.
4.3 Viscoelastic models of diseased blood
Most researchers have considered a Newtonian viscosity model for blood flowing
through blood vessels. Blood flowing through a diseased blood vessel has certain
characteristics which make the viscosity properties of the blood non-Newtonian [2].
4.3.1 Carreau model
The four-parameter non-Newtonian viscosity model proposed by Carreau [1] differs
from other models primarily in the curvature of the curve near the transition points
between the Newtonian plateaus and the power region as follows:
where
λ = 3.131 is the time constant associated with the viscosity that changes with the
shear rate, and n = 0.3568, a = 2.
= 0.056 Pa s
0
an a
(1)
μγ μ μ μ λγ∣∣ = + − + ∣∣
() ( )[1 ( )] ,
∞∞
0
−
is the blood viscosity at a zero shear rate,
4-6
(4.10)
=∞0.0032 Pa s
,
μ
μ
5
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4.3.2 Power-law model
In [23], a viscosity model for blood is proposed, taking into account hematocrit and
total protein minus albumin. It is known as the power-law model, also referred to as
the ‘best three variable model’. It is represented by
τγγ=−k
n 1
(4.11)
and
1
−
n
U
*
μγ γ=
() .
∞
k
1
−
n
l
In [6], Chien et al calculated normal blood sample parameters as k = 14.67 × 10
1
−
n
*
(4.12)
−3
Pa s, and n = 0.7755. If n > 1, the fluid is known as shear-thickening, while fluid is
shear-thinning if n < 1. The characteristic parameter for a power-law-model-based
flow is Re
PL
.
As the power-law model does not have the capability of handling Newtonian
regions of shear-thinning fluids at very low and high shear rates, Cross [7] proposed
a model which can be described as a shear rate dependent viscosity model:
−
μμ
where
= 0.0364 Pa s
0
∣∣ = +
μγ μ
()
()
∞
⎡
⎢
1
⎢
⎣
is the blood viscosity at a very low shear rate,
∞
0
+
,
n
⎤
⎛
⎞
γ
⎥
⎜
⎟
⎥
γ
⎝
⎠
⎦
c
c
(4.13)
=−2.63 s
1
the reference shear rate and n = 1.45 is the model constant.
is
4.3.3 Quemada model
Further considering the viscosity of a concentrated disperse system, Quemada [20]
proposed a model based on shear rate and hematocrits. The system of equations of
shear stress and effective viscosity in tensorial form and dimensionless form are
−
⎛
⎜
τμ
F
⎜
⎝
1
kk
+
1
2
∞
0
1/
γγ
+
γγ
c
2
⎞
/
c
⎟
φγ=−
⎟
⎠
(4.14)
and
2
⎛
1
*
μγ μ
() 1
γ
c
where
F
are
Quemada-model-based flow are Re
as 1060 kg m
−6
10
*
=
c
=×
1.2 10 Pa s
===
c
, the viscosity of plasma (the suspending medium), is
Ul/
∞
3
−
−3
and hematocrit is
1
−
kk1.88 s , 2.07 and 4.33.
∞
, whereas the Newtonian viscosity of blood is considered as 2.02 ×
kg m−2s−1.
⎜
F
⎜
2
⎝
0
QU
+
γγ
∞
*
γγ
. The values of other parameters
+
kk
0
1/
φ = 0.4
The characteristic parameters for a
*
and
. The density of blood is considered
c
−
⎞
*
*
/
c
⎟
,
φ=−
⎟
*
⎠
c
(4.15)
4-7
b
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4.4 CFD modeling of blood flow in a diseased vessel
4.4.1 Laminar flow model
The laminar flow model of the internal flow regime in closed vessels can be modeled
by the semi-implicit pressure linked equation (SIMPLE). It is a procedure for solving
the governing equations of fluid transport. The continuity equation can be
discretized in this form of the laminar flow model by the following set of equations:
−+−+−=JA J A JA JA JA JA[][][]0
ee ww nn sn tt bb
(4.16)
or
N
faces
∑
f
ff
=JA 0,
(4.17)
where fluxes at the control volume faces are
ρρρρρρ======JuJ uJuJuJuJu(), (), (), (), (), (). (4.18)
eewwnnssttbb
To relate the face velocity,
vv
uww,,,,,
ewns t
, to the stored values of velocity at
the cell centers, linear interpolation of cell-centered velocities to the faces will result
in unphysical checker-boarding of pressure. To avoid checker-boarding, the face
values of velocities are not averaged linearly, but using momentum-weighted
averaging, as outlined in [21] using weighting factors based on the
Using this procedure, the face flux for any face of the control volume,
coefficient.
a
P
, may be
f
written as
where
Jf
⎜
f
ff
⎝
++∇·−+∇·
⎛
vv
aa
Pc nc pc nc
ρ=
⎡
dp p r p p r(()) ()
f
⎣
+
,0 ,0 ,1 ,1
+
aa
Pc Pc
,0 ,1
c
c
0
=ˆ+−JJdp p(),
fff
00
()
c
1
cc01
c
11
⎞
⎤
⎟
⎦
⎠
(4.19)
(4.20)
ˆ
=
Jf
f
ff
With reference to figure 4.3,
the normal velocities, respectively, within the two cells on either side of the face
ˆ
The term
contains the influence of velocities in these two adjacent cells andfis a
f
function of the average of the momentum equation.
on either side of face
Figure 4.3. Control volume centroids of each control volume on either side of facef.
⎛
⎜
⎝
vv
aa
Pc nc pc nc
ρ
f
+
,0 ,0 ,1 ,1
+
aa
Pc Pc
,0 ,1
p
C0
.
+∇−∇
dpr pr
(). ().]. (4.21)
[
fc c
and
are the pressures and
p
C1
4-8
⎞
⎟
00 11
are coefficients for the cells
a
P
⎠
nC,0
and
nC,1
are
f
.
′
p
′
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Accordingly for the control volume, as shown in figure 4.3,
ww w
nn n
bb b
Now for a guessed pressure field
*
JJdpp().
ee
=ˆ+−JJdp p
ee e
=ˆ+−JJdp p
=ˆ+−JJdp p
=ˆ+−JJdp p
sss
=ˆ+−JJdp p
tt t
=ˆ+−JJdp p
*
p
=
()
PE
()
WP
()
PN
()
SP
()
PT
().
BP
, the resulting east face flux from equation (4.22a)is
*
ˆ
+−
**
e
PE
a
(4.22 )
b
(4.22 )
c
(4.22 )
d
(4.22 )
e
(4.22 )
f
(4.22 )
(4.23)
This does not satisfy the volume continuity equation (4.15), so a correction
*
added to the face flux
Similarly, a pressure correction′pis added to the guessed pressure
corrected pressure
so that the corrected face fluxebecomes
e
*
=+
JJ J.
e
′
ee
becomes
*
=+′
pp p.
p
The SIMPLE algorithm states that
Substituting
Similarly,
′
Jdp p().
e
in equation (4.22a) we obtain
e
*
=+′−
JJ dp p a( ). (4.27 )
e
e
=+′−
JJ dp p b() (4.27)
w
w
=+′−
JJ dp p c() (4.27)
n
n
=+′−
JJ dp p d() (4.27)
s
=+′−
JJ dp p e() (4.27)
t
t
=
*
*
*
s
*
′
−
e
PE
e
PE
w
WP
n
PN
s
SP
t
PT
′
′
′
′
′
′
(4.24)
*
so that the
(4.25)
(4.26)
is
e
*
=+′−
JJ dp p f( ). (4.27 )
b
b
b
′
BP
4-9
p
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Now putting the value of
⎡
**
JA dA p p JA dA p p
+
( ( )) ( ( ))
⎣
eee
e
⎡
*
JA dA p p JA dA p p
+
( ( )) ( ( ))
⎣
nnn
n
⎡
**
JA dA p p JA dA p p
+
( ( )) ( ( )) 0.
⎣
ttt
t
JJJJJ,,,,,
ewnstb
′
′
−
PE
′
−
PN
′
−
PT
−+′−
′
−
′
−+′−
in the continuity equation (4.15),
eww
w
′
+
sss
s
bbb
b
′
SP
BP
′
WP
⎤
′
−
⎦
⎤
′
⎦
Rearranging equation (4.13) we can write the equation as follows:
′
P
P
′
=
E
E
′
+
W
W
′
+
N
N
′
+
S
S
′
+
+
T
B
T
B
where
= + + +++a dAdA dAdAdAdA
Peewwnnssttbb
=adA
Eee
=adA
Www
=adA
nnn
=adA
sss
=adA
Ttt
=adA
Bbb
′
⎤
⎦
+
=
+ap ap a p ap ap ap ap b,
+
(4.28)
(4.29)
a(4.30 )
b(4.30 )
c(4.30 )
d(4.30 )
e(4.30 )
f(4.30 )
g(4.30 )
Δ
** ** **
bJAJA JAJA JAJA f f
e
e
n
n
t
t
w
w
s
s
b
00
ρρ=++−−−+ −
().
b
PP PP
V
Δ
t
h
(4.30 )
Equation (4.29) represents the discretized continuity equation as an equation for
pressure correction
. From equation (4.29) we obtain the pressure correction
′p
and putting the value of′pin equation (4.26) we obtain the correct pressure field
which satisfies the continuity equation (4.27). Then putting this pressure field in the
momentum equation (4.29), we obtain the correct velocity.
The profiles at a particular blockage show similar trends of distribution. The
agreement between the power-law model and cross model is remarkable, except for
the type II profile at 56% obstruction where the cross model shows a flatter
distribution with relatively high magnitude. It can be observed again from the plots
that the post-stenotic zone for higher blockage endures a long zone of oscillatory
shear stress although the magnitude is relatively less. It is well understood that the
flow structure of a stenosed vessel is dominated by vortical structures and the poststenotic recirculation region [19].
Figure 4.4 shows a contour plot of the time averaged velocity stream function for
the type 1 inlet velocity profile at different degrees of stenosis, considering the powerlaw viscosity model. The comparison is performed at two different Reynolds
numbers in figure 4.4, denoted as Re I and Re II. At low Reynolds numbers,
4-10
′p
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Figure 4.4. Contour view of velocity profiles of blood flowing through mild stenosed vessel at various time
steps.
recirculation bubbles propagate away from the region of stenosis with time,
producing a lower severity of stenosis (at 25%).
The numerical study of aneurysm was performed using two rheological models,
namely the power-law model and Quemada model, to represent the diseased state of
blood in a developing flow condition in a straight blood vessel with severe aneurysm
of 250% of the mean diameter of the vessel [5]. The results for such a situation of
fusiform aneurysms has not been not widely covered. Investigations were performed
within the range 690 ⩽ Re ⩽ 2760. The responses of two different non-Newtonian
viscosity models (namely power-law and Quemada) to different shear rate parameters near the boundaries are compared. Stream lines in the duct of the geometry are
shown in figure 4.4, which reflects the development of vortices, considering the
power-law model as the viscosity model for blood. As the resolution of the flow
increases, the vortex size becomes amplified, signifying the serious affects of the
aneurysm on the flow of blood.
A transient nature of flow development occurs in ducts with certain geometry.
Such a phenomenon occurs due to the development of a vortex, initiating with offset
damping, as depicted in figure 4.5.
4.5 Evaluation of the shear index on the vascular wall
The fluid-mechanical parameters, such as shear stress, vortical structures, pressure
profiles and velocity distribution, can be evaluated from the time-dependent
primitive variable data. Two important parameters need special mention in this
context as they have relevance to the health of arterial tissue, namely the timeaveraged wall shear stress (TAWSS) and the oscillator shear index (OSI) as
described by [22]. TAWSS can be simply integrated and averaged over a time cycle.
Wall shear stress is determined as
u
∂
x
r
wall
×+′
4-11
μ=−
∂
2
RzWSS (1 ( ) ).
(4.31)
1
η
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Figure 4.5. Stream line at different time steps at Re I (for the power-law model).
∂
u
Downstream from the stenosis,
′=Rz() 0
The axial component of the velocity u
Rz
Let
⁎
x
u
x
2
QR
π
/
M
0
⁎
Rz,() ,
()
η===
R
0
, therefore,
is already calculated by equation (4.31).
x
r
be the non-dimensional variables. The
Rz
()
WSS
x
μ=− ∣
∂
r
wall
.
boundary conditions are the following:
The velocity is zero at the wall:
There is symmetry in relation to the z-axis of the tube:
⁎⁎
3
∂
uu
xx
0, and 0 for 0
3
η
∂
∂
η
∂
Let the axial velocity along the z-axis be
Finally, the flow rate is conserved:
The coefficients A
⁎⁎
,0, 4
===−+ ==−
awa a w
012
xx x
of equation (4.16) can also be written as
i
⁎
0, for
x
.
η===
⁎⁎
==
QQQ Rz ud/2()
⁎
6
Q
,0, 3
aaw
2
⁎
(())
Rz
.
η==
⁎⁎
xx
34
for
=
w
M
= 0
1
2
∫
0
⁎
.
⁎
.
η
x
⁎
6
Q
⁎
(())
Rz
2
4-12
(4.32)
1
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⁎⁎ ⁎
uw w
xx x x
⎛
4
⎜
⎝
⁎
6
Q
2
⁎
(())
Rz
⎞
⎛
2
ηη∴=+− + + −
⎟
⎠
⁎
3
w
⎜
⎝
⁎
6
Q
2
⁎
(())
Rz
⎞
4
. (4.33)
⎟
⎠
Here, the polynomial degree is limited to four. Since the flow rate is known as a
function of time, the determination of the A
determination of the velocity along the z-axis, i.e.
by the experimental values. Finally, as
⁎
Rz()
coefficients is equivalent to the
i
⁎
The values of
.
x
downstream from the stenosis, wall
=
⁎
w
x
are given
shear stress can now be obtained:
⁎
∂
u
x
−
=−
η
∂
Qw12 4 .
⁎
⁎
x
(4.34)
Wall shear-stress plots for vessels with 25% and 50% blockages are calculated
using the above equation, and are presented later in this section.
Wall shear stress based descriptors are also used as markers for calculating
various states of the endothelial wall of the vessel. Gradient-based descriptors and
harmonic-based descriptors are calculated in this chapter. The WSS spatial gradient
(WSSG) is a marker of endothelial cell tension. It is calculated from WSS gradient
tensor components parallel and perpendicular to the TAWSS vector (m and n,
respectively) [ 8]:
T
1
=
⎛
⎜
∫
⎝
0
2
⎞
ττ
∂
wm wn
,
∂
⎛
⎟
⎜
+
⎠
⎝
2
⎞
∂
,
⎟
dtWSSG
.
⎠
∂Tmn
(4.35)
Figure 4.6 shows the comparative behavior of WSSG in a diseased vessel under
the in fl uence of variations in flow rates.
The maximum absolute rate of change in WSS magnitude over the cardiac cycle is
also known as the temporal gradient of WSS (WSST), and it is calculated as follows:
WSST max . (4.36)
⎜
⎝
⎛
∂t
⎞
τ=∂
w
⎟
⎠
Table 4.1 shows transient evaluations of WSST in diseased and healthy arteries at
Reynolds numbers Re I, Re II and Re III.
Figure 4.6. Graphical representation of WSSG distribution in diseased blood vessel at different Re.
4-13
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Table 4.1. Transient form of WSST in both diseased and healthy vessels.
Healthy vessel Diseased vessel
Viscosity model Flow rate
Power-law Re I 0.0147 0.0147
Re II 0.0127 0.0187
Re III 0.0094 0.0085
Quemada Re III 0.0122 0.0101
WSST WSST
The behavior of WSST is not similar to that of WSSG for the Quemada model.
Here, the Quemada model shows better responses to WSST, but it also depends on
flow rate. It is observed that WSST decreases with increased flow rate in a healthy
vessel. However, an oscillatory behavior of WSST is found in diseased vessels with
increased flow rate.
Figure 4.7 shows a comparative behavior of WSST in a healthy vessel under the
influence of variations in flow rates. WSST in the axial direction (x-direction) is
found to be more pronounced than WSST in the radial direction (y-direction) at Re I
and Re II. However, with a further increase in flow rate to Re II, an oscillatory
behavior of WSST is observed in both the x- and y-directions. With a further
increase in flow rate to Re III, it is observed that WSST in the radial direction is
greater than that in the axial direction.
The harmonic component of WSS waveforms can be a possible metric of
disturbed flow. This statement is supported by results revealing that endothelial
cells sense and respond to the frequency of the WSS profiles. The time varying WSS
magnitude at each node can be Fourier decomposed, and the dominant harmonic
(DH) is defined as the harmonic with the highest amplitude [12]. DH is calculated as
Table 4.2 shows transient evaluations of DH in diseased and healthy arteries at
Reynolds numbers Re I, Re II and Re III. It can be seen that the value of DH
increases with increased flow rate in the diseased vessel, whereas it behaves in
oscillatory manner in the healthy vessel. However, DH also shows better responses
to the flow when considering the rheological properties of the fluid as in the
Quemada model.
4.5.1 Oscillatory shear index
The OSI can be defined as
where τ
is the time-mean wall shear stress, also known as time-averaged wall
mean
shear stress (TAWSS), and τ
and they are formulated using
ww w00
⎛
1
⎜
1 , (4.38)
=−OSI
2
⎝
is the time-mean magnitude of the wall shear stress,
mag
τω π=≡=Fnw F TDH max( ( )), FFT( ), 2 / . (4.37)
⎞
τ
mean
⎟
τ
⎠
mag
4-14
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Figure 4.7. Graphical representation of WSST distribution in a healthy blood vessel at (a) Re I, (b) Re II and
(c) Re III.
Table 4.2. DH in both diseased and healthy vessels.
Viscosity model Flow rate
Power-law Re I 30.6964 30.6964
Quemada Re III 24.0394 84.4652
Healthy vessel Diseased vessel
DH DH
Re II 5.5557 70.6169
Re III 13.2493 82.7616
4-15
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