CHAPTER-II
EFFECTS OF PERIPHERAL LAYER VISCOSITY AND
HEMATOCRIT ON BLOOD FLOW THROUGH THE
ARTERY WITH MULTIPLE MILD STENOSES
2.1 INTRODUCTION
Stenosis in arteries of mammals is a common occurrence and
hemodynamic factors play a significant role in the formation and
proliferation of this disease. In this chapter, an investigation has been done to
find the effects of the shape of stenosis, peripheral layer viscosity and
Hematocrit; on resistance parameter and wall shear stress involving a twofluid layer model. The mathematical model consists a two-layered model of
blood, considering a core region of suspension of all the erythrocytes (small
spherical non-flexible particles), assumed to be a particle fluid suspension
(i.e., a suspension of red cells in plasma) and a peripheral layer of plasma.
The study thus presents a theoretical model for blood, seems to be the only
one of its kind which enables one to observe the simultaneous effects of
Hematocrit and the peripheral layer viscosity on the flow characteristics.
Blood is principally made of a calibrated suspension of red blood cells
(RBC). Depending on the flow, RBCs move alone or in aggregates
(rouleaux), often in capillaries of smaller sizes than their own radius (Guyton
31
Chapter 2: Effects of peripheral layer…
(2006)). Blood is an “intelligent” fluid, as its rheological properties adapt
according to the different situations of the flow. This capacity, which is not
fully understood, comes partly from the significant deformability of RBCs.
The understanding of the flow behavior of a single confined RBC is a first
step to understand the basic rheological properties of the blood in arteries.
Red blood cell aggregation is determined by Hematocrit, which is the most
important determinant of blood viscosity. In patients with clinical signs of the
hyper-viscosity syndrome, hemodilution has proven to be a simple but
effective method to improve tissue perfusion by lowering blood viscosity.
Several studies (Crowell and Smith (1967) and Lee et al. (1994)) have
shown that there is an optimal Hematocrit for tissue perfusion. Hematocrit
also affects thrombosis (Goldsmith et al. (1995)).
Fig.2.1 (a) Normal Red Blood Cells
32
Fig.2.1 (b) Sickle Red Blood Cells
Chapter 2: Effects of peripheral layer…
Fig.2.2
Destructed Red Blood Cells in the artery
McDonald (1960), Whitmore (1963), Fung (1964) and Lew and Fung (1970)
and have used single-phase homogeneous Newtonian viscous fluid, a
classical approach that does not account for the presence of red cells i.e.
Hematocrit. Although, this approach provides satisfactory tools to describe
certain aspects of blood flow in aorta and large arteries, it fails to give an
adequate representation of flow field, especially in the vessels of small
diameter (2400 μm – 8μm) (Srivastava and Srivastava (1983)). The studies
mentioned just above on two-fluid layer modeling have represented blood
either by a single phase Newtonian or non-Newtonian fluid in the core
region. With increasing interest in two-layer model flow and its applications
to blood flow problems and dealing with the problem of microcirculation, the
individuality of red blood cells cannot be ignored. Lih (1969) and Shukla et
al. (1980b) have considered that the viscosity in core region is more than the
peripheral region due to axial migration of blood particles.
33
Chapter 2: Effects of peripheral layer…
Normal and sickled red blood cell are depicted in Fig.2.1 (a), Fig.2.1
(b), respectively. Sickled RBC's (Fig.2.2) can become trapped within the
blood vessels and thus interfere with normal blood flow. This obstruction
can lead to sudden pain anywhere in the body as well as cause damage to
body tissues and organs over time.
Stenosis could affect one or more segments of the human
cardiovascular system. Studies on initiation and growth of stenosis
(atherosclerotic plaques) in the human cardiovascular system have been
carried out from several view-points. Arteriosclerosis is a common disease
which severely influences human health. It has been found that the initiation
and localization of arteriosclerosis is closely related to local hemodynamic
factors. Arteries throughout the body may be affected by hardening, which
causes symptoms because hardened arteries cannot carry enough blood to the
body. Narrowing or hardening of the arteries that feed the heart (the coronary
arteries) can lead to a heart attack. Due to these serious consequences,
attention has been given in studies of blood flow in stenotic region under
different conditions. Smith et al. (2002) have assumed the artery to be
circularly cylindrical in shape. Siouffi et al. (1984) and Tu and Deville
(1996), have investigated the effect of unsteadiness on the flow through
stenosis and bifurcations. Forrester and Young (1970) and Young and Tsai
(1973, a-b) have contributed a lot in developing a mathematical model for
blood flow in atherosclerosis.
34
Chapter 2: Effects of peripheral layer…
However, Shukla et al.
(1980b) have investigated the effect of
Peripheral viscosity on blood flow through the artery with mild stenosis,
without considering the effect of Hematocrit and multiple stenoses. So we are
interested to find the effects of peripheral viscosity and Hematocrit on the
resistance to flow and wall shear stress in the artery with multiple mild
stenoses.
2.2 ASSUMPTIONS
In this chapter following assumptions have been made
1. Blood is assumed to be Newtonian, incompressible and homogeneous fluid.
2. Motion of the fluid is laminar and steady.
3. The inertia term is neglected, as the motion is slow.
4. No body force acts on the fluid.
5. There is no slip velocity at the wall.
6. Cylindrical polar co-ordinates are used.
7. The axis of symmetry of the artery taken as z-axis.
8. A two-layer model consisting of a core region of suspension of all the
erythrocytes (particles) in plasma (fluid) assumed to be a particle-fluid
mixture and a peripheral layer of cell-free plasma (Newtonian fluid).
9. The viscosity in core region is more than the peripheral region.
10. Stenoses are mild.
11. Heights of all the stenosis are much less than the lengths of all stenosis.
12. Heights of all the stenosis are much less than the radius of the annular region
and velging effect in the interphase layer is neglected.
35
Chapter 2: Effects of peripheral layer…
2.3 NOTATIONS
R (z) =R : Radius of the artery with stenosis
R0
: Radius of uniform portion of the artery
R1
: Radius of core region of the artery
L
: Length of the artery
δi
: Height of ith stenosis (i=1,2,3.)
Li
: Length of ith stenosis (i=1, 2, 3.)
di
: Location of ith stenosis (i=1,2,3.)
p
:Pressure in the fluid
p1
: Inlet pressure
p0
: Outlet pressure
(r,θ,z)
: Cylindrical polar co-ordinate system
w
: Velocity of fluid in the z direction
µ
: Viscosity of blood
µ1
: Viscosity of central layer fluid
µ2
: Viscosity of peripheral layer fluid
𝐺
: Pressure gradient

: Shear stress
S
: Wall shear stress for stenosis
Q
: Flow rate
Q core
: Flow rate in the core region
Q peripheral : Flow rate in the peripheral region
λ
: Resistance to flow (resistance parameter)
36
Chapter 2: Effects of peripheral layer…
𝜆̅
: Non-dimensional resistance parameter
2.4 DEVELOPMENT OF THE MODEL
In section one, an investigation has been done to find the effect of
multiple stenosis, peripheral layer viscosity and Hematocrit by considering
the blood is Newtonian fluid when there is significant gap in the consecutive
stenosis, while in section two effect of peripheral layer viscosity and
Hematocrit is considered when there is no significant gap between the
stenosis on the resistance parameter and wall shear stress, respectively.
SECTION ONE:
Let us consider the laminar and steady flow of blood in the artery. The
geometry of multiple stenoses (Fig.2.3) which are assumed to be symmetric
is given by:
𝑅(𝑧)
𝛿1
2𝜋
𝐿1
=1−
[1 + 𝑐𝑜𝑠 ( ) (𝑧 − 𝑑0 − )] ;
𝑅0
2𝑅0
𝐿1
2
=1
𝑑0 ≤ 𝑧 ≤ 𝐿1 + 𝑑 0
;
0 ≤ 𝑧 ≤ 𝑑0
(2.1)
𝑅(𝑧)
𝛿2
2𝜋
𝐿2
= 1−
[1 + 𝑐𝑜𝑠 ( ) (𝑧 − {𝑑0 + 𝐿1 + 𝑑1 + })] ;
𝑅0
2𝑅0
𝐿2
2
𝑑0 +𝐿1 + 𝑑1 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝑑1 + 𝐿2
=1
; 𝑑0 +𝐿1 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝑑1
(2.2)
37
Chapter 2: Effects of peripheral layer…
𝑅(𝑧)
𝛿3
2𝜋
𝐿3
= 1−
[1 + 𝑐𝑜𝑠 ( ) (𝑧 − {𝑑0 + 𝐿1 + 𝑑1 + 𝐿2 + 𝑑2 + })] ;
𝑅0
2𝑅0
𝐿3
2
𝑑0 +𝐿1 + 𝑑1 + 𝐿2 + 𝑑2 ≤ 𝑧
≤ 𝑑0 + 𝐿1 + 𝑑2 + 𝐿2 + 𝑑2 + 𝐿3
=1
𝑜𝑟
; 𝑑0 +𝐿1 + 𝑑1 + 𝐿2 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝑑1 + 𝐿2 + 𝑑2
𝑑0 +𝐿1 + 𝑑1 + 𝐿2 + 𝑑2 + 𝐿3 ≤ 𝑧 ≤ 𝐿
(2.3)
where R (z) is the radius of the artery with stenosis, R0 is constant radius of
the artery, R1 is radius of the core region of the artery L1, L2, L3 are length of
stenosis and δ1, δ2, and δ3 are the height of stenosis respectively. For
simplicity and easy to understand the effects of multiple mild stenosis in the
mathematical model, it is assumed that the heights of all stenosis are much
less than the radius of annular region i.e., δi<< (R0-R1);i=1,2,3.
Fig.2.3 Geometry of multiple stenoses with peripheral layer
38
Chapter 2: Effects of peripheral layer…
It is assumed that there is no flow of blood in θ direction in the artery.
The magnitude of velocity in r - direction is very less in comparison to z direction. Therefore, pressure gradient is the function of z only which caused
the motion of flow.
Therefore, the basic equation governing the flow of blood in the arterial
system is given by:
0= -
dp 1 ¶ é
¶wù
+
êm(r )r ú
dz r ¶ r êë
¶ r úû
(2.4)
,
where, w is axial velocity, p is fluid (blood) pressure and µ(r) is viscosity
function.
The present study deals with in two sections, section one considered
the significant gap between consecutive stenosis while in section two, it is
assumed that there is no significant gap between consecutive stenosis in the
artery. As we discussed above, two cases of mathematical model are studied
to find the effect of Peripheral layer viscosity and Hematocrit on the
resistance parameter and wall shear stress in section one and two,
respectively.
2.5 CASE I: EFFECT OF PERIPHERAL VISCOSITY
In this case an investigation has been done to find the effect of Peripheral
layer viscosity on the resistance parameter and wall shear stress respectively.
Let µ1 and µ2 be the viscosity in core region and peripheral region (Fig.2.3)
respectively defined as given below:
39
Chapter 2: Effects of peripheral layer…
 1
; 0  r  R1
 2
; R1  r  R( z )
 r   
(2.5)
It is assumed that viscosity of the in peripheral layer µ 2 is smaller than the
viscosity µ1 of the core region.
From equation (2.4) and (2.5), we have
1 d  w1 
dp
r

r dr  r  dz
2 d  w2 
dp
r



r dr  r  dz
; 0  r  R1
(2.6-a)
; R1  r  R
(2.6-b)
,
where w1, w2 are the velocity in core region and peripheral region
respectively and dp/dz is pressure gradient. The boundary conditions are
given as:
w1
0
r
; r 0
(2.7-a)
w2  0
; rR
(2.7-b)
w1  w2
; r  R1
(2.7-c)
The (2.7-a) is the maximum velocity at the centerline; (2.7-b) is no- slip
velocity at the wall; (2.7-c) is the continuity of the velocities at the
interphase.
40
Chapter 2: Effects of peripheral layer…
2.5.1 MATHEMATICAL ANALYSIS
Solving equations (2.6-a) and (2.6-b) by using the boundary
conditions (2.7-a), (2.7-b) and (2.7-c), the expression of velocities in the core
and peripheral region respectively are given as:
(2.8-a)
𝜋𝐺
𝑤1 =4𝜇 [𝜇2 𝑟 2 +𝑅12 (1−𝜇2 )−𝑅 2 ]
2
(2.8-b)
𝜋𝐺
𝑤2 =4𝜇 [𝑟 2 −𝑅 2 ]
2
𝜇
where G= dp/dz and 𝜇2 = 𝜇2.
1
It has been seen that from the equation of continuity that flow rate Q is
constant and sum of the flow rates through the core region and in the
peripheral region will also be constant. Therefore, the flow rate through the
artery is the sum of the flow rates through the core region and that in the
peripheral region i.e; Q =Q core + Q peripheral
R1
Q core   2 rw1dr  
0

   
2
G
 2R1   R2  1  2  R12 
82
2  




(2.9-a)
The flow rate through the peripheral region can be obtained as
Q

R
 G  2
 R  R1 2
  2w2 rdr  
R1
 8 2 
peripheral
The expression for the flow rate (Q =Q
thus given by
41
core
+Q
peripheral)

2
(2.9-b)
through the artery is
Chapter 2: Effects of peripheral layer…
 G
Q   
 8 2


 4
4
  R  R1 1   2 


(2.10)
The expression G  dp is given by:
dz
G
dp 82Q
1

dz
  R4  R 4 1   
2
1




(2.11)
Integrating equation (2.11), using the conditions the inlet pressure p=p1 at
z=0, and outlet pressure p=p0 at z=L, we have the pressure drop as
p1  p0 
82Q


1
L
0


 R4  R 4 1   
2
1


dz
(2.12)
2.5.1.1 RESISTANCE PARAMETER
The resistance to flow is defined as follows;

p1  p0
Q
Using equation (2.12), the resistance parameter is given as

82


L
0
1


 R4  R 4 1   
2
1


dz
(2.13)
Using equation (2.1), (2.2), and (2.3) in equation (2.13), we get the
expression of resistance parameter as
2 𝐿−(𝐿1 +𝐿2 +𝐿3 )
𝐼0
𝜆 = 𝜋[
where
+ 𝐺1 +𝐺2 + 𝐺3 ] ,
(2.14)
𝐼0 = 4𝜇 [𝑅04 − (1 − 𝜇̅2 )𝑅14 ]
(2.15)
1
2
42
Chapter 2: Effects of peripheral layer…
𝑑 +𝐿1 𝑑𝑧
𝐴(𝑧)
0
𝐺1 = ∫𝑑 0
(2.16-a)
𝑑 +𝐿 +𝑑 +𝐿2 𝑑𝑧
𝐺2 = ∫𝑑 0+𝐿 1+𝑑 1
0
1
(2.16- b)
𝐴(𝑧)
1
𝑑 +𝐿 +𝑑 +𝐿 +𝑑 +𝐿3 𝑑𝑧
𝐴(𝑧)
0
1
1
2
2
𝐺3 = ∫𝑑 0+𝐿 1+𝑑 1+𝐿 2+𝑑 2
,
(2.16- c)
in which,
1
𝐴(𝑧) = 4𝜇 [𝑅 4 − (1 − 𝜇̅2 )𝑅14 ]
(2.17)
2
The non-dimensional expression of resistance parameter is given as
𝜆=
𝜆𝜋𝑅04
8𝐿𝜇1
= 𝜇2[
𝐿−(𝐿1 +𝐿2 +𝐿3 )
𝐿(1−𝐾4 )
𝑑 +𝐿1
𝐻1 = ∫𝑑 0
𝑤ℎ𝑒𝑟𝑒
1
+ 𝐿 ∑3𝑖=1 𝐻𝑖 ]
𝑑𝑧
,
𝑑 +𝐿 +𝑑 +𝐿2
𝑑𝑧
𝐻2 = ∫𝑑 0+𝐿 1+𝑑 1
1
𝑅 4
[( ) −𝐾4 ]
𝑅0
1
𝑑 +𝐿 +𝑑 +𝐿 +𝑑 +𝐿3
𝐻3 = ∫𝑑 0+𝐿 1+𝑑 1+𝐿 2+𝑑 2
0
1
1
2
𝑅
𝐾 4 = (1 − 𝜇2 ) (𝑅1 )
in which,
,
(2.19- b)
𝑑𝑧
𝑅 4
[( ) −𝐾4 ]
𝑅0
2
(2.18)
(2.19- a)
𝑅 4
[( ) −𝐾4 ]
𝑅0
0
0
,
,
(2.19- c)
4
(2.20)
0
Solving equations (2.19- a), (2.19- b) and (2.19 -c), with the help of equations
(2.1), (2.2), and (2.3), we get
1
𝐻𝑖 = 2𝜋 [𝐼1 − 𝐼2 +𝐼3 −𝐼4 ]
where,
,
2𝜋
1
𝐼1 = ∫0
2𝜋
𝐼3 = ∫0
2π
𝐼4 = ∫0
(2.21-b)
{(𝑎𝑖 −𝐾)+𝑏𝑖 𝑐𝑜𝑠𝜑𝑖 }
2π
𝐼2 = ∫0
(2.21-a)
1
{(𝑎𝑖 +𝐾)+𝑏𝑖 𝑐𝑜𝑠𝜑𝑖 }
(2.21-c)
𝑖
{(𝑎𝑖 −𝑖𝐾)+𝑏𝑖 𝑐𝑜𝑠𝜑𝑖 }
(2.21-d)
𝑖
{(𝑎𝑖 +𝑖𝐾)+𝑏𝑖 𝑐𝑜𝑠𝜑𝑖 }
43
,
(2.21-e)
Chapter 2: Effects of peripheral layer…
in which,
𝛿
𝛿
𝑎𝑖 = 1 − 2𝑅𝑖 , 𝑏𝑖 = 2𝑅𝑖 ; for i = 1,2,3.
0
(2.22)
0
2𝜋
𝐿
𝜑1 = 𝜋 − 𝐿 (𝑧 − {𝑑0 + 21 })
(2.23- a)
1
2𝜋
𝐿
𝜑2 = 𝜋 − 𝐿 (𝑧 − {𝑑0 + 𝑑1 + 𝐿1 + 22 })
(2.23- b)
2
2𝜋
𝐿
𝜑3 = 𝜋 − 𝐿 (𝑧 − {𝑑0 + 𝑑1 + 𝑑2 + 𝐿1 + 𝐿2 + 23 })
(2.23- c)
3
Using the method of calculus of residues, this can be further simplified to
evaluating the integrals, as follows
1
Hi 
 ai  K 
2
 bi

2
1
 ai  K 
2
 bi

2
 
2sin  i 
2
 4a
2
i
K  si
2
2

1
4
(2.24)
where,
2𝐾𝑎
𝑖
2
2
2
2
𝑡𝑎𝑛𝜃𝑖 = 𝑎2 −𝑏2 −𝐾
2 , 𝑠𝑖 = 𝑎𝑖 − 𝑏𝑖 − 𝐾 , for i=1,2,3.
𝑖
(2.25)
𝑖
Thus, the expression of the resistance parameter is given by:
𝜆=
𝜇2 𝐿−(𝐿1 +𝐿2 +𝐿3 )
[ (1−𝐾4 )
𝐿
+ 𝐻1 +𝐻2 +𝐻3 ]
(2.26)
When δ2 and δ3 both are zero, equation (2.26), exhibits the same
results as obtained by Shukla et al. (1980b). It is also noted that when
2 1
and only one stenosis i.e. δ2 =0, δ3 =0 is taken in the artery equation (2.26)
reduces to particular case of Newtonian fluid as obtained by Young (1968).
44
Chapter 2: Effects of peripheral layer…
2.5.1.2 NUMERICAL SIMULATION
The expression of the resistance parameter given by equation (2.26)
with respect to different parameters of stenosis and peripheral layer viscosity
is plotted in Fig.2.4.(Table 2.1) It is seen from Fig.2.4 that as the height of
stenosis, length of stenosis and peripheral layer viscosity increase in the
resistance parameter '
blood vessels, resistance parameter also steadily increases.
2.2
2.1
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
2=0.01,3=0.08
2=0.08,3=0.05
2=0.15,3=0.01
L1=L2=L3=0.8
2=1.0
L1=L2=L3=0.1
2=0.3
L1=L2=L3=0.05
2=0.1
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
height of stenosis 1
Fig.2.4 Variation of the resistance parameter against height of stenosis (δ1)
for different length of stenosis, other height of stenosis and peripheral layer
viscosity
45
Chapter 2: Effects of peripheral layer…
2.5.1.3 WALL SHEAR STRESS
The wall shear stress at the wall can be defined as
w 

 R    r  
r  r  R z 

(2.27)
From equations (2.4) and (2.11), the wall shear stress at maximum height of
stenosis is given as:
 R i 
RQ
 A z 
; i=1, 2, 3.
(2.28)
From equations (2.17) and (2.28), the expressions of nondimensional wall
shear sress for multiple stenoses are given as:
   4  QA  z   
 s i  R03
s
i

i 
1 

R0 



4
i
 1 
 K 
R0 




1
2
4
; i=1, 2, 3.
(2.29)
Thus,the expression of nondimensional wall shear sress at maximum height
of first stenosis at z=d0+L1/2, is given as:
   

1 
1



4
R0 
 


4
 1  1   K 
R0 




s
1
Similarly,
 
2
(2.30)
is nondimensional wall shear stress for second stenosis at
s
2
z=d0+ L1+d1+L2/2, and
 
s
is nondimensional wall shear stress for third
3
stenosis at z= d0+ L1+d1+L2+d2+L3/2 respectively.
Equation (2.29) reduces the same result as obtained by Shukla et al.
(1980b), when there is only one stenosis i.e. δ2 =0, δ3 =0 considered in the
46
Chapter 2: Effects of peripheral layer…
artery. It is also noted that when
2 1
equation (2.29) reduces to particular
case of Newtonian fluid for multiple stenosis.
2.5.1.4 NUMERICAL SIMULATION
The expression of the wall shear stress given by equation (2.29) with
respect to different parameters of stenosis and peripheral layer viscosity is
plotted in Fig. 2.5(Table 2.2). It is seen from Fig.2.5 that as the height of
stenosis and peripheral layer viscosity increase in the blood vessels, wall
shear stress also steadily increases.
1.2
2=1.0
2=0.5
wall shear stress
1.1
1.0
2=0.3
0.9
0.8
2=0.1
0.7
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
height of stenosis 1
Fig.2.5 Variation of the wall shear stress against height of stenosis (δ1) for
different peripheral layer viscosity
47
Chapter 2: Effects of peripheral layer…
Table 2.1
Simulated values of the resistance parameter for heights of stenosis, lengths
of stenosis and peripheral layer viscosity
Resistance parameter
L1=L2=L3=0.05, ̅𝝁𝟐 =0.1
δ1
0.04
0.08
0.12
0.16
0.20
L1=L2=L3=0.1, 𝝁
̅ 𝟐 =0.3
L1=L2=L3=0.8, 𝝁
̅ 𝟐 =1.0
δ2=0.01
δ3=0.08
δ2=0.08
δ3=0.05
δ2=0.15
δ3=0.01
δ2=0.01
δ3=0.08
δ2=0.08
δ3=0.05
δ2=0.15
δ3=0.01
δ2=0.01
δ3=0.08
δ2=0.08
δ3=0.05
δ2=0.15
δ3=0.01
0.312
0.324
0.325
0.331
0.367
0.451
0.454
0.465
0.469
0.492
0.612
0.621
0.633
0.641
0.673
0.732
0.738
0.741
0.744
0.767
0.812
0.823
0.836
0.844
0.881
0.913
0.924
0.945
0.953
1.023
1.523
1.534
1.546
1.562
1.583
1.623
1.635
1.644
1.657
1.693
1.765
1.773
1.786
1.794
1.912
Table 2.2
Simulated values of the wall shear stress for first height of stenosis and
peripheral layer viscosity
Wall shear stress
δ1
0.04
0.08
0.12
0.16
0.20
̅𝝁𝟐 =0.1
0.6125
0.6234
0.6378
0.6456
0.6798
̅𝝁𝟐 =0.3
0.8145
0.8267
0.8378
0.8467
0.8892
48
̅𝝁𝟐 =0.5
0.9156
0.9276
0.9481
0.9594
1.053
̅𝝁𝟐 =0.1
1.1234
1.1343
1.1465
1.1621
1.1831
Chapter 2: Effects of peripheral layer…
2.6 CASE II: EFFECT OF HEMATOCRIT
In this case an investigation has been done to find the effect of
Hematocrit on the resistance parameter and wall shear stress. The relation of
blood viscosity and the Hematocrit which is quite effective is given as
(Einstein (1906)):

 p 1  2.5H1 

 p 1  2.5H 2 
; 0  r  R1
 r   
; R1  r  R( z )
(2.31)
,
where H1, H2 are the Hematocrit in core region and peripheral region
(Fig.2.6) respectively, and µp is the viscosity of plasma.
r
d0
L1
d1
L2
2
1
R(z)
d2
R0
L3
3
H1
R1
z
p10
p0
H2
Fig.2.6 Geometry of multiple stenoses with Hematocrit
49
Chapter 2: Effects of peripheral layer…
2.6.1 RESISTANCE PARAMETER
Similar as defined in case one, the expression of resistance parameter is given
as:
𝜆=
Where,
𝜇2 𝐿−(𝐿1 +𝐿2 +𝐿3 )
[ (1−𝑀4 ) +
𝐿
𝑃1 +𝑃2 +𝑃3 ]
(1+2.5𝐻 )
(2.32)
1+2.5𝐻
𝑅
4
𝜇2 = (1+2.5𝐻2 ) , 𝑀4 = (1 − 1+2.5𝐻2 ) (𝑅1 )
1
1
(2.33)
0
The Pi (i=1, 2, 3) are given as:
Pi 
1
 ai  M 
where,
2
 bi 2
1

 ai  M 
𝛿
2
 bi 2

 
2sin  i 
2
 4a
i
2
M 2  si 2

1
4
𝛿
𝑎𝑖 = 1 − 2𝑅𝑖 , 𝑏𝑖 = 2𝑅𝑖 ; for i = 1,2,3.
0
0
2𝑀𝑎
𝑖
2
2
2
2
𝑡𝑎𝑛𝜃𝑖 = 𝑎2 −𝑏2 −𝑀
2 , 𝑠𝑖 = 𝑎𝑖 − 𝑏𝑖 − 𝑀 ; for i=1, 2, 3.
𝑖
(2.34)
(2.35- a)
(2.35- b)
𝑖
It is also noted that when H2=H1 and only one stenosis i.e. δ2 =0, δ3 =0
is taken in the artery; equation (2.32) resembles to same result as obtained by
Young (1968).
2.6.2 NUMERICAL SIMULATION
The expression of the resistance parameter is given by equation (2.32)
with respect to different parameters of stenosis and Hematocrit is plotted in
Fig.2.7 (Table 2.3). It is seen from Fig.2.7 that as 24.8% Hematocrit (Hb SS,
plasma cell dycrasias diseases), normal Hematocrit (45.0%) and 63.20%
Hematocrit (uncontrolled Hypertension diseases) (Bugliarello and Sevilla
50
Chapter 2: Effects of peripheral layer…
(1970), Shu (1982)), increases the resistance parameter also steadily
increases.
It is clearly visible from Fig.2.7 that as the height of stenosis, length
of stenosis and Hematocrit increase in the blood vessels, resistance parameter
resistance parameter '
also steadily increases.
2.2
2.1
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
2=0.01,3=0.08
2=0.08,3=0.05
2=0.15,3=0.1
L1=L2=L3=0.8
H1=50.00 %
H2=63.20%
L1=L2=L3=0.1
H2=45.00%
L1=L2=L3=0.05
H2=24.80%
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
height of stenosis 1
Fig.2.7
Variation of the resistance parameter against height of stenoses (δ1)
for different lengths of stenoses, other heights of stenoses and Hematocrit
51
Chapter 2: Effects of peripheral layer…
2.6.3 WALL SHEAR STRESS
The wall shear stress at the wall can be defined as
w 

 s    r  
r  r  R z 

(2.36)
From equations (2.4) and (2.11), we find the shear stress as follows:
The wall shear stress at maximum height of stenosis is given as:
 s i 
RQ
 A z 
; i=1, 2, 3.
(2.37)
From equations (2.17) and (2.37), the expressions of nondimensional wall
shear sress for multiple stenoses are given as:
 
s 
i
 s i  R03

41QA  z  
2
 
 1  i 


R

0 

4

i 
1 

R0 

4
M 


; i=1, 2, 3.
(2.38)
Thus,the nondimensional wall shear sress at maximum height of first stenosis
at z=d0+L1/2 is given as:
   
s

  1  1 


R

0 

1
Similarly,
2
 
4

1 
1 

R0 

4
M 


(2.39)
is nondimensional wall shear stress for second stenosis at
s
2
z=d0+ L1+ d1+L2/2, and
 
s
is nondimensional wall shear stress for third
3
stenosis at z= d0+ L1+ d1+L2+L3/2 respectively.
It is also noted that when H2=H1, equation (2.38) resembles to
particular case of Newtonian fluid for multiple stenosis.
52
Chapter 2: Effects of peripheral layer…
2.6.4 NUMERICAL SIMULATION
The expression of the wall shear stress given by equation (2.39) with
respect to height of stenosis and Hematocrit is plotted in Fig.2.8 (Table 2.4).
It is seen from Fig.2.8 that as 24.8% Hematocrit, 28% Hematocrit, 63.20%
Hematocrit) (Hb SS, plasma cell dycrasias diseases and uncontrolled
Hypertension respectively), and normal Hematocrit (45.0%) (Bugliarello and
Sevilla (1970), Shu (1982)) increases the wall shear stress also steadily
increases.
It is clearly visible from Fig.2.8 that as the height of stenosis and
Hematocrit increase in the blood vessels, wall shear stress also steadily
increases.
wall shear stress
1.1
H1=50.00 %
1.0
0.9
0.8
H2=24.80
H2=28.00
H2=43.31
H2=45.00
0.7
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
height of stenosis 1
Fig.2.8 Variation of the wall shear stress against height of stenosis (δ1) for
different Hematocrit
53
Chapter 2: Effects of peripheral layer…
Table 2.3
Simulated values of the resistance parameter for heights of stenosis, lengths
of stenosis and Hematocrit
Resistance parameter
δ1
0.04
0.08
0.12
0.16
0.20
L1=L2=L3=0.05,
L1=L2=L3=0.1,
L1=L2=L3=0.8,
H2=24.80%
H2=45.00%
H2=63.20%
δ2=0.01 δ2=0.08 δ2=0.15 δ2=0.01 δ2=0.08 δ2=0.15 δ2=0.01 δ2=0.08 δ2=0.15
δ3=0.08 δ3=0.05 δ3=0.01 δ3=0.08 δ3=0.05 δ3=0.01 δ3=0.08 δ3=0.05 δ3=0.01
0.413
0.551
0.612
0.733
0.812
0.912
1.523
1.623
1.764
0.424
0.554
0.621
0.738
0.823
0.924
1.534
1.635
1.772
0.426
0.565
0.633
0.741
0.836
0.945
1.546
1.644
1.786
0.435
0.568
0.641
0.746
0.844
0.953
1.562
1.657
1.795
0.464
0.594
0.673
0.765
0.881
1.023
1.583
1.693
1.912
Table 2.4
Simulated values of the wall shear stress for first height of stenosis (δ1) and
Hematocrit
δ1
0.04
0.08
0.12
0.16
0.20
Wall shear stress
H2=24.80%
0.672
0.674
0.677
0.681
0.704
H2=28.00%
0.814
0.826
0.839
0.845
0.888
54
H2=43.31%
0.915
0.924
0.945
0.957
1.062
H2=45.00%
1.123
1.134
1.146
1.162
1.183
Chapter 2: Effects of peripheral layer…
SECTION TWO :
As discussed above, in this section, the resistance parameter and wall shear
stress are calculated by assuming that there is no significant gap between
consecutive stenosis in the artery. It is also considered that equation of
motion and boundary conditions are same as given in section one.
2.7 DEVELOPMENT AND ANALYSIS OF MATHEMATICAL
MODEL
The geometry of symmetric multiple stenosis by assuming d1=0, d2=0
(Fig.2.9) is given as:
𝑅(𝑧)
𝛿1
2𝜋
𝐿1
=1−
[1 + 𝑐𝑜𝑠 ( ) (𝑧 − 𝑑0 − )] ;
𝑅0
2𝑅0
𝐿1
2
𝑑0 ≤ 𝑧 ≤ 𝐿1 + 𝑑 0
=1
;
0 ≤ 𝑧 ≤ 𝑑0
(2.40)
𝑅(𝑧)
𝑅0
𝛿
2𝜋
𝐿
= 1 − 2𝑅2 [1 + 𝑐𝑜𝑠 ( 𝐿 ) (𝑧 − {𝑑0 + 𝐿1 + 22 })] ;
0
2
𝑑0 +𝐿1 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝐿2
=1
; 𝑑0 +𝐿1 ≤ 𝑧 ≤ 𝑑0 + 𝐿1
(2.41)
𝑅(𝑧)
𝛿3
2𝜋
𝐿3
= 1−
[1 + 𝑐𝑜𝑠 ( ) (𝑧 − {𝑑0 + 𝐿1 + 𝐿2 + })] ;
𝑅0
2𝑅0
𝐿3
2
𝑑0 +𝐿1 + 𝐿2 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝐿2 + 𝐿3
=1
; 𝑑0 +𝐿1 + 𝐿2 ≤ 𝑧 ≤ 𝑑0 + 𝐿1 + 𝐿2
𝑑0 +𝐿1 + 𝐿2 + 𝐿3 ≤ 𝑧 ≤ 𝐿
o𝑟
(2.42)
All the usual symbol has already defined in section one. In this
section, it is also assumed that the heights of all stenosis are much less than
55
Chapter 2: Effects of peripheral layer…
the radius of annular region i.e., δi<< (R0-R1);i=1,2,3.Similarly , in this
section, two cases of mathematical model are studied to find the effect of
Peripheral layer viscosity and Hematocrit on the resistance parameter and
wall shear stress respectively.
d0
L1
L2
1
R(z)
L3
2
1
R0
3
R1
p1
p0
2
Fig.2.9 Geometry of multiple stenosis with peripheral layer when d1=0
and d2=0
2.8 CASE I: EFFECT OF PERIPHERAL VISCOSITY
In this case an investigation has been done to find the effect of Peripheral
layer viscosity on the resistance parameter and wall shear stress.
2.8.1 RESISTANCE PARAMETER
The expressions of G1,G2 and G3, in equations (2.16-a), (2.16-b)
and (2.16-c),respectively recalculated by using respective equations (2.40),
(2.41) and (2.42), Gi is given as :
56
Chapter 2: Effects of peripheral layer…
1
Gi 
 ai  K 
2
 bi 2
1

𝛿
 ai  K 
2
 bi 2

 
2sin  i 
2
 4a
i
2
K 2  si 2

1
4
𝛿
In which, 𝑎𝑖 = 1 − 2𝑅𝑖 , 𝑏𝑖 = 2𝑅𝑖 ; for i = 1,2,3.
0
2𝐾𝑎
𝑖
(2.44- a)
0
𝑖
2
2
2
2
𝑡𝑎𝑛𝜃𝑖 = 𝑎2 −𝑏2 −𝐾
2 , 𝑠𝑖 = 𝑎𝑖 − 𝑏𝑖 − 𝐾 ,for
(2.43)
i=1,2,3.
(2.44- b)
𝑖
Substituting the resultant values of G1,G2 and G3 in equation (2.14) ,
recalulated nondimensional resistance parameter as
𝜆=
𝜇2 𝐿−(𝐿1 +𝐿2 +𝐿3 )
[ (1−𝐾4 )
𝐿
+ 𝐺1 +𝐺2 +𝐺3 ]
(2.45)
It is found that as there is no significant gap between the stenosis i.e.
(d1=0 and d2=0) in the artery, the resistance parameter remains same. When
δ2 and δ3 both are zero, equation (2.45), exhibits the same results as obtained
by Shukla et al. (1980b). It is also noted that when
2 1
and only one
stenosis is taken in the artery; equation (2.45) reduces to particular case of
Newtonian fluid as obtained by Young (1968).
2.8.2 WALL SHEAR STRESS
Similarly we can find wall shear stress for this section.
 s i 
RQ
 A z 
; i=1, 2, 3.
(2.46)
From equations (2.17) and (2.46), the nondimensional wall shear sress for
multiple stenoses is given as:
57
Chapter 2: Effects of peripheral layer…
 
s 
i
 s i  R03

41QI  z  

i 
1 

R0 



4
i
 1 
 K 
R0 




2
4
; i=1, 2, 3.
(2.47)
Thus, the expression of nondimensional wall shear sress at maximum height
of first stenosis is given as:
 
s 
1
Similarly,
 s 1  R03

41QI  z  
2

 1  1 

 R0 

  is
s
4

 
1  1 

  R0 
 K4


at z=d0+L1/2
(2.48)
nondimensional wall shear stress for second stenosis at
2
z=d0+ L1 +L2/2, and
 
s
is nondimensional wall shear stress for third
3
stenosis at z= d0+ L1 +L2+L3/2 respectively.
It is found that as there is no significant gap between the stenosis i.e.
(d1=0 and d2=0) in the artery, the wall shear stress remains same. Equation
(2.47) reduces the same result as obtained by Shukla et al. (1980), when
there is only one stenoses i.e. δ2 =0, δ3 =0 considered in the artery. It is also
noted that when
2 1
equation (2.47) reduces to particular case of Newtonian
fluid for multiple stenoses.
58
Chapter 2: Effects of peripheral layer…
2.9 CASE II: EFFECT OF HEMATOCRIT
In this case an investigation has been done to find the effect of Hematocrit on
the resistance parameter and wall shear stress.
2.9.1 RESISTANCE PARAMETER
Similar as case two of section one, the expression of non-dimensional
resistance parameter is given as
𝜆=
𝜇2 𝐿−(𝐿1 +𝐿2 +𝐿3 )
[ (1−𝑀4 )
𝐿
+ 𝑃1 +𝑃2 +𝑃3 ]
(1+2.5𝐻 )
1+2.5𝐻
(2.49)
4
𝑅
In which, 𝜇2 = (1+2.5𝐻2 ) ,𝑀4 = (1 − 1+2.5𝐻2 ) (𝑅1 )
1
1
(2.50)
0
The Pi (i=1, 2, 3) are given as:
Pi 
where,
1
 ai  M 
2
𝛿
 bi
1

 ai  M 
2
2
 bi

2
 
2sin  i 
2

4ai 2 M 2  si 2
𝛿
𝑎𝑖 = 1 − 2𝑅𝑖 , 𝑏𝑖 = 2𝑅𝑖 ; for i = 1,2,3.
0
0
2𝑀𝑎
𝑖
2
2
2
2
𝑡𝑎𝑛𝜃𝑖 = 𝑎2 −𝑏2 −𝑀
2 , 𝑠𝑖 = 𝑎𝑖 − 𝑏𝑖 − 𝑀 , for i=1,2,3.
𝑖

1
4
(2.51)
(2.52-a)
(2.52- b)
𝑖
It is verified that as there is no significant gap between the stenosis
i.e. (d1=0 and d2=0) in the artery, the resistance parameter remains same. It is
also noted that when H2=H1 and only one stenosis i.e. δ2 =0, δ3 =0 is taken in
the artery; equation (2.49) resembles to same result as obtained by Young
(1968).
59
Chapter 2: Effects of peripheral layer…
2.9.2 WALL SHEAR STRESS
The wall shear stress at the wall can be defined as
w 

 s    r  
r  r  R z 

(2.53)
From equations (2.4) and (2.11), the wall shear stress at maximum height of
stenosis is given as:
 s i 
RQ
 A z 
for i=1, 2, 3.
(2.54)
Thus,the nondimensional wall shear sress for multiple stenosis are given as:
 
s

i
 s i  R03
41QA  z  
2
 
 1  i 

R0 


4

i 
1 

R0 

 M4


for i=1, 2, 3.
(2.55)
The nondimensional wall shear sress is at maximum height of first stenosis is
given as:
 
s 
1
Similarly,
 
 s 1  R03

41QA  z  
2

  1  1 

R0 


4

 
1  1 
R0  at z=d0+L1/2

 M4


(2.56)
is nondimensional wall shear stress for second stenosis at
s
2
z=d0+ L1+ L2/2, and
 
s
is
nondimensional wall shear stress for third
3
stenosis at z= d0+ L1 +L2+L3/2 respectively.
It is seen that as there is no significant gap between the stenosis i.e.
(d1=0 and d2=0) in the artery, the wall shear stress remains same. It is also
noted that when H2=H1, equation (2.55) resembles to particular case of
Newtonian fluid for multiple stenosis.
60
Chapter 2: Effects of peripheral layer…
2.10 CONCULSION
Medically, stenosis refers to the narrowing of an artery due to blockage. The
build-up of fatty tissues and calcium inside blood vessel walls can become
sufficient to interfere with blood flow at a particular point in the arteries, such
as in coronary (the vessels that supply the heart muscle with blood) and
cerebral (the vessels that supply the brain with blood) arteries. When either
of these arteries are narrowed or blocked, blood and oxygen cannot flow
freely to the heart muscle or brain and a stenosis develops. However RBCs is
one of the main constituents of blood, which is responsible for many of the
blood properties and diseases in blood while flowing through the circulatory
system. Thus, we are interested to find the link between shape of stenosis,
peripheral layer viscosity and Hematocrit on resistance parameter and wall
shear stress on blood flow in artery.
Our study reveals that as the height of stenosis, length of stenosis,
peripheral viscosity and Hematocrit get increased, the wall shear stress and
resistance parameter also increase. It is also verified that as there is no
significant gap between the stenosis i.e. (d1=0 and d2=0) in the artery, the
resistance parameter and wall shear stress remains same.
61