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1. CHAPTER 3
Thermal Effects of Non-Newtonian
Fluid Flow Through a Permeable
Non-Uniform Tube with Multiple
Constrictions
16

2. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 17
3.1 Flow of Nanofluid Through a Permeable Non-
Uniform Tube having Multiple Constrictions
3.1.1 Introduction
Stenosis is the term used for the abnormal narrowing in the blood vessel or structure.
The world mortality rate is increasing due to problems related with cardiac. This
is because of high grade of stenosis in arteries. Due to stenosis, the effected arteries
become hard. Stenosis is the main reason for clotting of blood, cerebral stokes and
failure of the heart. Stenosis increases the resistance of the flow in arteries. Due to
this, the pressure of the flow increases. This leads to substantial changes in pressure
distribution and wall shear stress. A detailed knowledge of the flow in the artery
with stenosis gives better understanding of blood flow in the physiological systems.
In the past, many researchers studied blood flow in the artery systems by treating
blood as Newtonian or non-Newtonian. (Young [1968], Padmanabhan [1980] and
Shukla et al. [1980]). Most of these theoretical models studied the blood flow
in a circular tube or channel having single stenosis. But in the reality, there is a
possibility of forming multiple stenoses or overlapping stenoses in the arteries. The
researchers like Awgichew and Radhakrishnamacharya [2013a], Agarwal
and Varshney [2016] investigated blood flow in arteries with multiple stenosis. In
all these studies they considered the wall of the tube is rigid.
Due to high thermal conductivity of nanofluids and its many applications in biomed-
ical field, more number of researchers are making significant research in this field.
(Choi and Eastman [1995], Nadeem et al. [2013], El-dabe et al. [2015],
Bali et al. [2016])
In this chapter, nanofluid in the inclined permeable channel with non-uniform cross-
section and two stenoses is considered. The impact of parameters on λ̄ and τh have
been studied.

3. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 18
3.1.2 Mathematical Formulation
Consider a nanofluid flow in an inclined tube having non-uniform channel with
multiple stenosis. Considering cylindrical polar co-ordinate system (r, θ, z) and tube
makes an angle α to the z-axis. The stenosis is assumed to be mild and developed
in an axially symmetric way.
Figure 3.1.1: Geometry of Inclined Stenosed artery having Multiple Stenosis
The tube radius of Figure (3.1.1) is given in Equation (3.1.1) (Prasad et al.
[2015])
h = R (z) =





























R0 ; 0 ≤ z ≤ d1
R0 − δ1
2
h
1 + cos2π
L1
z − d1 − L1
2
i
; d1 ≤ z ≤ d1 + L1
R0 ; d1 + L1 ≤ z ≤ B1 − L2
2
R0 − δ2
2
h
1 + cos2π
L2
(z − B1)
i
; B1 − L2
2
≤ z ≤ B1
R∗
(z) − δ2
2
h
1 + cos2π
L2
(z − B1)
i
; B1 ≤ z ≤ B1 + L2
2
R∗
(z) ; B1 + L2
2
≤ z ≤ B
(3.1.1)
where, R∗(z)
R0
= exp
βB2
(z − B1)2
.
The following restrictions for mild stenosis are supposed to satisfy:
δi ≪ min (R0, Rout), δi ≪ Li, where Rout = R (z) at z = B
Here δi and Li(i = 1, 2) are the maximum heights and lengths of first and second
stenosis respectively.

4. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 19
The equations for an incompressible fluid flow are (Awgichew and Radhakrish-
namacharya [2013a])
1
r̄
∂ (r̄w̄r̄)
∂r̄
+
∂w̄
∂z̄
= 0 (3.1.2)
ρ
w̄r̄
∂w̄r
∂r̄
+ w̄z̄
∂w̄z
∂z̄
= −
∂P̄
∂r̄
+ µc
∂2
w̄r̄
∂r̄2
+
1
r̄
∂w̄r̄
∂r̄
+
∂2
w̄r̄
∂z̄2
−
w̄r̄
r̄2
−
cosα
F
(3.1.3)
ρ
w̄z̄
∂w̄z̄
∂r̄
+ w̄r̄
∂w̄z̄
∂z̄
= −
∂P̄
∂z̄
+ µc
∂2
w̄z̄
∂r̄2
+
1
r̄
∂w̄z̄
∂r̄
+
∂2
w̄z̄
∂z̄2
+ ρgβ T̄ − T̄0
+ ρgβ C̄ − C̄0
+
sinα
F
(3.1.4)
w̄r̄
∂T̄
∂r̄
+ w̄z̄
∂T̄
∂z̄
= β
∂2
T̄
∂r̄2
+
1
r̄
∂T̄
∂r̄
+
∂2
T̄
∂z̄2
+
τ
(
DB
∂C̄
∂r̄
∂T̄
∂r̄
+
∂C̄
∂z̄
∂T̄
∂z̄
+
DT̄
T̄0
∂T̄
∂r̄
2
+
∂T̄
∂z̄
2
#)
(3.1.5)
w̄r̄
∂C̄
∂r̄
+ w̄z̄
∂C
∂z̄
= DB
∂2
C̄
∂r̄2
+
1
r̄
∂C̄
∂r̄
+
∂2
C̄
∂z̄2
+
DT̄
T̄0
∂2
T̄
∂r̄2
+
1
r̄
∂T̄
∂r̄
+
∂2
T̄
∂z̄2
(3.1.6)
where, τ =
(ρC)P
(ρC)f
is ratio of effective heat capacity of the nanoparticle material to
the heat capacity of fluid.
The boundary conditions are
∂w̄
∂r̄
= 0,
∂T̄
∂r̄
= 0,
∂C̄
∂r̄
= 0 at r̄ = 0 (3.1.7)
w̄ = −k
∂w̄
∂r
, T̄ = T̄0 , C̄ = C̄0 at r̄ = R(z) (3.1.8)
Using following non-dimensional quantities
z̄ =
z
B
, ¯
d1 =
d1
B
, L̄1 =
L1
B
, L̄2 =
L2
B
, B̄1 =
B1
B
, v̄ =
B
δW
v , w̄ =
w
W
, δ̄i =
δi
R0
,
R̄ (z) =
R (z)
R0
, P̄ =
P
µWL
R2
0
, q̄ =
q
πR2
0W
, Re =
2ρc1R0
µ
, F̄ =
F
µWλ
, σ =
C − C̄0
C̄0
,
Nt =
(ρC)P DT T̄0
(ρC)f β
, Gr =
gβT̄0R3
0
γ2
, Br =
gβC̄0R3
0
γ2
, θt =
T − T̄0
T̄0
, Nb =
(ρC)P DB̄C̄0
(ρC)f
.

5. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 20
After applying the non-dimensional quantities, the equations becomes
∂w
∂r
+
w
r
+
∂w
∂z
= 0 (3.1.9)
∂P
∂r
= −
cosα
F
(3.1.10)
∂P
∂z
=
1
r
∂
∂r
r
∂w
∂r
+ Grθt + Brσ (3.1.11)
1
r
∂
∂r
r
∂θt
∂r
+ Nb
∂σ
∂r
∂θt
∂r
+ Nt
∂θt
∂r
2
= 0 (3.1.12)
1
r
∂
∂r
r
∂σ
∂r
+
Nt
Nb
1
r
∂
∂r
r
∂θt
∂r
= 0 (3.1.13)
The non-dimensional boundary conditions are
∂w
∂r
= 0 , ∂θt
∂r
= 0 ∂σ
∂r
= 0 at r = 0,
w = −k∂w
∂r
, θt = 0, σ = 0 at r = h(z)
)
(3.1.14)
3.1.3 Solution
The coupled equations (3.1.12) and (3.1.13) are solved by HPM, with the folllowing
initial guesses
θ10 (r, z) =
r2
− h2
4
(3.1.15)
σ10 (r, z) = −
r2
− h2
4
(3.1.16)
The expressions for θt and σ are
θt (r, z) =
(Nb − Nt)
64
r2
− h2
−
Nb
18
r3
− h3
−
Nt Nb
2
+ Nt
2
36864
r4
− h4
r6
− h6
(3.1.17)

6. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 21
σ (r, z) = −
1
4
r2
− h2
Nt
Nb
+
Nt
Nb
1
18
Nb r3
− h3
+
1
36864
Nb
2
+ Nt
2
r6
− h6
(3.1.18)
Using Equations (3.1.17) and (3.1.18) in (3.1.11) and applying boundary conditions,
the solution for the velocity can be written as
w (r, z) =
r2
− h2
4
−
kr
2
−
sin α
F
+
dP
dz
−
r4
16
−
r2
h2
4
+
3h4
16
−
kr3
4
+
krh2
2
Gr
64
(Nb − Nt) +
GrNb
18
r5
25
−
r2
h3
4
+
21h5
100
−
kr4
5
+
krh3
2
+ GrNt
r12
144
−
r6
h6
36
−
r8
h8
64
+
r2
h10
4
−
41h12
192
−
kr11
12
+
kr5
h6
6
+
kr7
h4
8
−
krh10
2
(N2
b + N2
t )
36864
+
Br
4
Nt
Nb
r4
16
−
r2
h2
4
+
3h4
16
−
kr3
4
+
krh2
2
−
BrNt
18
r5
25
−
r2
h3
4
+
21h5
100
−
kr4
5
+
krh3
2
− Br
Nt
Nb
(N2
b + N2
t )
36864
r8
64
−
r2
h6
4
+
15
64
h8
−
kr7
8
+
krh6
2
(3.1.19)
The dimension less flux q is
q =
Z h
0
2rwdr (3.1.20)
By substituting the Equation (3.1.19) in (3.1.20), the flux is given by
q =
h4
8
+
kh3
3
sin α
F
−
dP
dz
−
Gr (Nb − Nt)
64
h6
12
+
7kh5
30
+
27h7
280
+
4kh6
15
GrNb
18
−
GrNt (N2
b + N2
t )
36864
41h14
420
+
887kh13
3276
+
Br
4
h6
12
+
7kh5
30
Nt
Nb
−
BrNt
18
27h7
280
+
4kh6
15
− Br
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.1.21)

7. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 22
From Equation (3.1.21), dP
dz
can be obtained as
dP
dz
=
sin α
F
+
1
h4
8
+ kh3
3
−q−Gr (Nb − Nt)
h6
768
+
7kh5
1920
+GrNb
3h7
560
+
2kh6
135
−
GrNt (N2
b + N2
t )
36864
41h14
420
+
887kh13
3276
+ Br
h6
48
+
7kh5
120
Nt
Nb
− BrNt
3h7
560
+
2kh6
135
− Br
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.1.22)
The pressure drop per wave length ∆p=p(0)-p(λ) is ∆p=−
R 1
0
dP
dz
dz
∆p =
Z 1
0
sin α
F
+
1
h4
8
+ kh3
3
q+Gr (Nb − Nt)
h6
768
+
7kh5
1920
−GrNb
3h7
560
+
2kh6
135
+
GrNt (N2
b + N2
t )
36864
41h14
420
+
887kh13
3276
− Br
h6
48
+
7kh5
120
Nt
Nb
+ BrNt
3h7
560
+
2kh6
135
+ Br
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
dz (3.1.23)
The resistance (or)Impedance to the flow (λ) is λ=∆p
q
λ =
1
q
Z 1
0
sin α
F
+
1
h4
8
+ kh3
3
q+Gr (Nb − Nt)
h6
768
+
7kh5
1920
−GrNb
3h7
560
+
2kh6
135
+
GrNt (N2
b + N2
t )
36864
41h14
420
+
887kh13
3276
− Br
h6
48
+
7kh5
120
Nt
Nb
+ BrNt
3h7
560
+
2kh6
135
+ Br
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
dz (3.1.24)
∆pn is the pressure drop in the absence of stenosis (h = 1) and is attained from
Equation (3.1.23) as
∆pn =
Z 1
0
sin α
F
+
1
1
8
+ k
3
q+Gr (Nb − Nt)
1
768
+
7k
1920
−GrNb
3
560
+
2k
135
+
GrNt (N2
b + N2
t )
36864
41
420
+
887k
3276
− Br
1
48
+
7k
120
Nt
Nb
+ BrNt
3
560
+
2k
135
+ Br
Nt
Nb
(N2
b + N2
t )
36864
9
80
+
11k
36
dz (3.1.25)

8. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 23
The impedance to the flow in the normal artery is (λn), given by λn = ∆pn
q
=
1
q
Z 1
0
sin α
F
+
1
1
8
+ k
3
q + Gr (Nb − Nt)
1
768
+
7k
1920
− GrNb
3
560
+
2k
135
+
GrNt (N2
b + N2
t )
36864
41
420
+
887k
3276
− Br
1
48
+
7k
120
Nt
Nb
+ BrNt
3
560
+
2k
135
+ Br
Nt
Nb
(N2
b + N2
t )
36864
9
80
+
11k
36
dz (3.1.26)
The normalized impedance to the flow is
λ̄ =
λ
λn
(3.1.27)
The wall shear stress is τh = −h
2
dP
dz
= −
h
2
sin α
F
+
1
h4
8
+ kh3
3
−q−Gr (Nb − Nt)
h6
768
+
7kh5
1920
+GrNb
3h7
560
+
2kh6
135
−
GrNt (N2
b + N2
t )
36864
41h14
420
+
887kh13
3276
+ Br
h6
48
+
7kh5
120
Nt
Nb
− BrNt
3h7
560
+
2kh6
135
− Br
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.1.28)
3.1.4 Results and Analysis
The influence of fluid parameters on resistance to the flow (λ̄) and wall shear stress
(τh) are studied.
Figures (3.1.2 - 3.1.9) shows the effects of various parameters on the resistance to the
flow(λ̄) for different parameters. It is observed that, the resistance to the flow (λ̄)
increases with the heights of the stenosis, inclination (α), Thermophoresis parameter
(Nt), local temperature Grashof number (Gr), local nanoparticle Grashof number
(Br) and permeability constant (k), but decreases with Brownian motion number
(Nb) and volumetric flow rate (q).
Figures (3.1.10 - 3.1.17) shows the shear stress acting on the wall (τh) over the height
of stenosis. It can be shown that the shear stress at the wall increases with height
of the stenosis, Nb and q. But it decreases with the increase of Br, Gr, Nt, k and α.

9. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 24
Figures (3.1.18 - 3.1.19) displays the nature of nanoparticle phenomena. It is shown
that σ decreases with increase of the Nb, but increases with the increase of Nt.
Effects of θt are shown in Figures (3.1.20 - 3.1.21). It is seen that, θt decreases with
the increase of Nb, but increases with the increase of Nt.

10. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 25
d2 = 0.01
d2 = 0.02
d2 = 0.03
d2 = 0.04
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
1.30
1.35
d1
l
–
Figure 3.1.2: Variation of δ1 on λ̄ with δ2 varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.3,
Nt = 0.8 , α = π
6 , k = 0.05, F = 0.3
a = p/12
a = p/6
a = p/4
a = p/3
0.02 0.04 0.06 0.08 0.10
1.02
1.03
1.04
1.05
1.06
d1
l
–
Figure 3.1.3: Variation of δ1 on λ̄ with α varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.3,
Nt = 0.8 , δ2 = 0.01, k = 0.05, F = 0.3
Nt = 0.4
Nt = 0.6
Nt = 0.8
Nt = 1.0
0.02 0.04 0.06 0.08 0.10
1.20
1.25
1.30
1.35
1.40
d1
l
–
Figure 3.1.4: Variation of δ1 on λ̄ with Nt varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.3,
δ2 = 0.01 , α = π
6 , k = 0.05, F = 0.3

11. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 26
Nb = 2
Nb = 5
Nb = 8
Nb = 11
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
1.30
d1
l
–
Figure 3.1.5: Variation of δ1 on λ̄ with Nb varying
q = 0.3, Br = 0.3, Gr = 0.2, Nt = 0.8,
δ2 = 0.01 , α = π
6 , k = 0.05, F = 0.3
Gr = 1
Gr = 4
Gr = 7
Gr = 10
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
1.30
1.35
d1
l
–
Figure 3.1.6: Variation of δ1 on λ̄ with Gr varying
q = 0.3, Br = 0.3, Nb = 0.3, Nt = 0.8,
δ2 = 0.01 , α = π
6 , k = 0.05, F = 0.3
Br = 0.20
Br = 0.25
Br = 0.30
Br = 0.35
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
1.8
1.9
d1
l
–
Figure 3.1.7: Variation of δ1 on λ̄ with Br varying
q = 0.3, Gr = 0.2, Nb = 0.3, Nt = 0.8,
δ2 = 0.01 , α = π
6 , k = 0.05, F = 0.3

12. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 27
k = 0.02
k = 0.03
k = 0.04
k = 0.05
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
d1
l
–
Figure 3.1.8: Variation of δ1 on λ̄ with k varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.3,
Nt = 0.8, δ2 = 0.01 , α = π
6 , F = 0.3
q = 0.30
q = 0.31
q = 0.32
q = 0.33
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
1.30
1.35
d1
l
–
Figure 3.1.9: Variation of δ1 on λ̄ with q varying
Br = 0.3, Gr = 0.2, Nb = 0.3, Nt = 0.8,
δ2 = 0.01 , α = π
6 , k = 0.05, F = 0.3
d2 = 0.010
d2 = 0.015
d2 = 0.020
d2 = 0.025
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.10: Variation of δ1 on τh with δ2 varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.05, z = 0.7, F = 0.3

13. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 28
Br = 0.30
Br = 0.32
Br = 0.34
Br = 0.36
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.11: Variation of δ1 on τh with Br varying
q = 0.3, δ2 = 0.01, Gr = 0.2, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.05, z = 0.7, F = 0.3
Gr = 01
Gr = 06
Gr = 11
Gr = 17
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.12: Variation of δ1 on τh with Gr varying
q = 0.3, δ2 = 0.01, Br = 0.3, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.05, z = 0.7, F = 0.3
Nb = 0.10
Nb = 0.11
Nb = 0.12
Nb = 0.13
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.13: Variation of δ1 on τh with Nb varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nt = 0.3, α = π
6 , k = 0.05, z = 0.7, F = 0.3

14. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 29
Nt = 0.40
Nt = 0.43
Nt = 0.46
Nt = 0.49
0.02 0.04 0.06 0.08 0.10
1.2
1.3
1.4
1.5
d1
t
h
Figure 3.1.14: Variation of δ1 on τh with Nt varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, α = π
6 , k = 0.05, z = 0.7, F = 0.3
k = 0.050
k = 0.052
k = 0.054
k = 0.056
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.15: Variation of δ1 on τh with k varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, α = π
6 , z = 0.7, F = 0.3
a = p/6.04
a = p/6.00
a = p/5.96
a = p/5.92
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
1.8
d1
t
h
Figure 3.1.16: Variation of δ1 on τh with α varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, k = 0.05, z = 0.7, F = 0.3

15. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 30
q = 0.300
q = 0.302
q = 0.304
q = 0.306
0.02 0.04 0.06 0.08 0.10
1.4
1.5
1.6
1.7
d1
t
h
Figure 3.1.17: Variation of δ1 on τh with q varying
F = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, α = π
6 , k = 0.05, z = 0.7
Nb = 0.4
Nb = 0.5
Nb = 0.3
-1.0 -0.5 0.0 0.5 1.0
0
1
2
3
4
r
s
Figure 3.1.18: Variation in Nanoparticle σ with Nb varying
Nt = 0.9
Nt = 0.7
Nt = 0.5
-1.0 -0.5 0.0 0.5 1.0
0
1
2
3
4
r
s
Figure 3.1.19: Variation in Nanoparticle σ with Nt varying

16. Chapter 3.1: Flow of Nanofluid Through a Permeable Non-Uniform Tube having
Multiple Constrictions 31
Nb = 0.3
Nb = 0.4
Nb = 0.5
-1.0 -0.5 0.0 0.5 1.0
0.00
0.01
0.02
0.03
0.04
r
q
t
Figure 3.1.20: Variation in Temperature profile θt with Nb varying
Nt = 0.8
Nt = 0.7
Nt = 0.6
-1.0 -0.5 0.0 0.5 1.0
0.00
0.01
0.02
0.03
0.04
r
q
t
Figure 3.1.21: Variation in Temperature profile θt with Nt varying
Resistance to the flow (λ̄)
Straight Tube Inclined Tube
δ1 (α = 0o
) (α = 5o
) (α = 10o
) (α = 15o
) (α = 20o
)
0.02 1.02316 1.02411 1.02514 1.02624 1.02742
0.04 1.03363 1.03502 1.03651 1.03811 1.03981
0.06 1.04495 1.04681 1.0488 1.05094 1.05322
0.08 1.05723 1.05959 1.06213 1.06485 1.06775
0.10 1.01345 1.01401 1.01461 1.01525 1.01593
Table 3.1.1: A comparative analysis of λ̄ with δ1 between the straight tube
and Inclined tube for Nanofluid having multiple stenosis
The comparative study of λ̄ with δ1 between the straight tube and inclined tube for
Nanofluid having multiple stenoses has been made, which is shown in Table (3.1.1).
It is seen that, λ̄ is more in the inclined tube when compared to the straight tube.

17. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 32
3.2 Thermal Effects of Micropolar Fluid Through
a Permeable Non-Uniform Artery having Mul-
tiple Stenosis
3.2.1 Introduction
Stenosis is the most common valvular heart diseases in the developed countries of
the world. Vascular fluid dynamics play an important role in the development of
arterial stenosis, which is one of the most extensive diseases in human being resulting
to failure of the cardiovascular system. The circulation of blood gets interrupted to
an extent depending upon the severity of the stenosis.
Subadra et al. [2015] have studied the peristaltic transport of nanoparticles of
micropolar fluid in an inclined tube with heat and mass transport effect. Many
researchers have done their work using no-slip boundary condition at the walls of
the vessels. But, the walls are permeable in many physiological systems.
In the past, many researchers assumed the blood is a Newtonian fluid. This as-
sumption of Newtonian behaviour of blood is acceptable for high shear rate flow.
But, in some conditions, blood exhibits Non-Newtonian properties. Most of these
theoretical models studied the blood flow in a circular tube or channel having single
stenosis. But in the reality, there is a possibility of forming multiple stenoses or over
lapping Stenoses in the arteries. Notable researchers like Maruthi Prasad and
Radhakrishnamacharya [2008], Vajravelu et al. [2015] investigated blood
flow in arteries with multiple stenosis. In all these studies they considered the wall
of the tube is not flexible.
In this chapter, micropolar fluid in an inclined permeable tube having non-uniform
cross-section with two stenosis is considered. The effects of different parameters on
pressure drop, resistance to the flow and wall shear stress have been studied.

18. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 33
3.2.2 Problem Formulation
Consider an incompressible and steady micropolar fluid flow in an inclined tube
having non-uniform channel with multiple stenosis. Considering cylindrical polar
co-ordinate system (r, θ, z) and tube makes an angle α with z−axis. The stenosis is
assumed to be mild and developed in an axially symmetric manner.
The Geometry of an Inclined stenosed artery with multiple stenosis is given in the
Figure (3.1.1) and radius of the tube is given by the Equation (3.1.1).
The equations for the steady micropolar fluid are (Mekheimer and El Kot [2008])
(∇·W) = 0 (3.2.1)
ρ(W·∇W) = −(∇P) + (K∇ × W) + (µ + K)∇2
W (3.2.2)
ρj(W·∇W) = −(2KV ) + (K∇ × W) − γ(∇ × ∇ × V ) + (α + β + γ)∇(∇ · V )
(3.2.3)
Here P is fluid pressure, j is microgyration parameter, µ,K are respectively the
coefficients of shear stress and vortex viscosities. V and W are respectively the micro
rotation and velocity vectors. α, β, γ are the material constants which satisfies the
in-equalities given below:
2µ + K ≥ 0, 3α + β ≥ 0, γ ≥ |β|
Thus, the equations for the fluid flow are
∂wr
∂r
+
wr
r
+
∂wz
∂z
= 0 (3.2.4)
ρ
wr
∂wz
∂r
+ wz
∂wz
∂z
= −
∂P
∂z
+ (µ + K)
∂2
wz
∂r2
+
1
r
∂wz
∂r
+
∂2
wz
∂z2
+
K
r
∂(rvθ)
∂r
(3.2.5)
ρ
wr
∂wr
∂r
+ wz
∂wr
∂z
= −
∂P
∂r
+ (µ + K)
∂2
wr
∂r2
+
1
r
∂wr
∂r
−
wr
r2
− K
∂vθ
∂z
(3.2.6)
ρj
wr
∂vθ
∂r
+ wz
∂vθ
∂z
= −2Kvθ − K
∂wz
∂r
−
∂wr
∂z
+ γ
∂
∂r
1
r
∂(rvθ)
∂r
+
∂2
vθ
∂z2
(3.2.7)

19. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 34
Here, W=(wr, 0, wz) and V =(0, vθ, 0) are velocity and microrotation vectors respec-
tively.
Introducing the non - dimensional variables
z̄= z
L
, δ̄= δ
R0
, r̄= r
R0
, w̄z=wz
w0
, w̄r=Lwr
w0δ
, w̄θ=R0vθ
w0
, ¯
J= j
R2
0
, p̄= P
µw0L
R2
0
into Equations (3.2.4 - 3.2.7), and by applying the mild stenosis approximations, the
equations becomes
∂P
∂r
= −
cos α
F
(3.2.8)
N
r
∂
∂r
(rvθ) +
∂2
w
∂r2
+
1
r
∂w
∂r
+ (1 − N)
sin α
F
+ (1 − N)(Grθt + Brσ) = (1 − N)
∂P
∂z
(3.2.9)
2vθ +
∂w
∂r
−
2 − N
m2
∂
∂r
1
r
∂
∂r
(rvθ)
= 0 (3.2.10)
1
r
∂
∂r
r
∂θt
∂r
+ Nb
∂σ
∂r
∂θt
∂r
+ Nt
∂θt
∂r
2
= 0 (3.2.11)
1
r
∂
∂r
r
∂σ
∂r
+
Nt
Nb
1
r
∂
∂r
r
∂θt
∂r
= 0 (3.2.12)
Here, w=wz is the velocity in the axial direction, N= K
µ+K
;(0 ≤ N 1), m2
=
R2
0K(2µ+K)
γ(µ+K)
,
where N and m are respectively coupling number and micropolar parameter. θt, σ,
Nt, Nb, Br and Gr are temperature profile, nanoparticle phenomena, thermophore-
sis parameter, Brownian motion parameter, local nanoparticle Grashof number and
local temperature Grashof number.
The Non-dimensional boundary conditions are
∂w
∂r
= 0 , ∂θt
∂r
= 0 , ∂σ
∂r
= 0 at r = 0
w = −k∂w
∂r
, θt = 0 , σ = 0 at r = h(z)





(3.2.13)

20. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 35
3.2.3 Solution
By applying the Homotopy Perturbation Method to solve the equations (3.2.11) and
(3.2.12), we get
θt (r, z) =
(Nb − Nt)
64
r2
− h2
−
Nb
18
r3
− h3
−
Nt Nb
2
+ Nt
2
36864
r4
− h4
r6
− h6
(3.2.14)
σ (r, z) = −
1
4
r2
− h2
Nt
Nb
+
Nt
Nb
1
18
Nb r3
− h3
+
1
36864
Nb
2
+ Nt
2
r6
− h6
(3.2.15)
Using the equations (3.2.14) and (3.2.15) in (3.2.9) and applying boundary condi-
tions, the solution for the velocity can be written as
w (r, z) = (1 − N)
r2
− h2
4
−
kr
2
−
sin α
F
+
dP
dz
− Nvθ (r − h − k) − (1 − N)
Gr
Nb − Nt
64
r4
16
−
r2
h2
4
+
3h4
16
−
kr3
4
+
krh2
2
−
Nb
18
r5
25
−
r2
h3
4
+
21h5
100
−
kr4
5
+
krh3
2
−
Nt (N2
b + N2
t )
36864
r12
144
−
r6
h6
36
−
r8
h8
64
+
r2
h10
4
−
41h12
192
−
kr11
12
+
kr5
h6
6
+
kr7
h4
8
−
krh10
2
− (1 − N) Br
−
1
4
Nt
Nb
r4
16
−
r2
h2
4
+
3h4
16
−
kr3
4
+
krh2
2
+
Nt
18
r5
25
−
r2
h3
4
+
21h5
100
−
kr4
5
+
krh3
2
+
Nt
Nb
(N2
b + N2
t )
36864
r8
64
−
r2
h6
4
+
15
64
h8
−
kr7
8
+
krh6
2
(3.2.16)
The dimension less flux q is
q =
Z h
0
2rwdr (3.2.17)
Substituting the Equation (3.2.16) in Equation (3.2.17), the flux is given by

21. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 36
q = (1 − N)
h4
8
+
kh3
3
sin α
F
−
dP
dz
+ N
h3
3
+ kh2
vθ − (1 − N) Gr
Nb − Nt
64
h6
12
+
7kh5
30
−
Nb
9
−453h7
2800
+
2kh6
15
−
Nt (N2
b + N2
t )
36864
369h14
1120
−
149kh13
468
− (1 − N) Br
7h6
96
−
7kh5
120
Nt
Nb
+
Nt
9
27h7
560
+
2kh6
15
+
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.2.18)
From Equation (3.2.18), dP
dz
can be obtained as
dP
dz
=
sin α
F
+
1
(1 − N) h4
8
+ kh3
3
−q + N
h3
3
+ kh2
vθ − (1 − N) Gr
Nb − Nt
64
h6
12
+
7kh5
30
−
Nb
9
−453h7
2800
+
2kh6
15
−
Nt (N2
b + N2
t )
36864
369h14
1120
−
149kh13
468
− (1 − N) Br
7h6
96
−
7kh5
120
Nt
Nb
+
Nt
9
27h7
560
+
2kh6
15
+
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.2.19)
The pressure drop per wave length ∆p=p(0)-p(λ) is ∆p=−
R 1
0
dP
dz
dz
∆p =
Z 1
0
−
sin α
F
+
1
(1 − N) h4
8
+ kh3
3
q − N
h3
3
+ kh2
vθ + (1 − N) Gr
Nb − Nt
64
h6
12
+
7kh5
30
−
Nb
9
−453h7
2800
+
2kh6
15
−
Nt (N2
b + N2
t )
36864
369h14
1120
−
149kh13
468
+ (1 − N) Br
7h6
96
−
7kh5
120
Nt
Nb
+
Nt
9
27h7
560
+
2kh6
15
+
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
dz
(3.2.20)
The resistance (or)Impedance to the flow (λ) is λ=∆p
q

22. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 37
λ =
1
q
Z 1
0
−
sin α
F
+
1
(1 − N) h4
8
+ kh3
3
q − N
h3
3
+ kh2
vθ + (1 − N) Gr
Nb − Nt
64
h6
12
+
7kh5
30
−
Nb
9
−453h7
2800
+
2kh6
15
−
Nt (N2
b + N2
t )
36864
369h14
1120
−
149kh13
468
+ (1 − N) Br
7h6
96
−
7kh5
120
Nt
Nb
+
Nt
9
27h7
560
+
2kh6
15
+
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
dz
(3.2.21)
∆pn is the pressure drop in the absence of stenosis (h = 1) and is attained from
Equation (3.2.20) as
∆pn =
Z 1
0
−
sin α
F
+
1
(1 − N) 1
8
+ k
3
q − N
1
3
+ k
vθ + (1 − N) Gr
Nb − Nt
64
1
12
+
7k
30
−
Nb
9
−453
2800
+
2k
15
−
Nt (N2
b + N2
t )
36864
369
1120
−
149k
468
+ (1 − N) Br
7
96
−
7k
120
Nt
Nb
+
Nt
9
27
560
+
2k
15
+
Nt
Nb
(N2
b + N2
t )
36864
9
80
+
11k
36
dz
(3.2.22)
The impedance to the flow in the normal artery is (λn), given by
λn =
∆pn
q
=
1
q
Z 1
0
−
sin α
F
+
1
(1 − N) 1
8
+ k
3
q − N
1
3
+ k
vθ + (1 − N) Gr
Nb − Nt
64
1
12
+
7k
30
−
Nb
9
−453
2800
+
2k
15
−
Nt (N2
b + N2
t )
36864
369
1120
−
149k
468
+ (1 − N) Br
7
96
−
7k
120
Nt
Nb
+
Nt
9
27
560
+
2k
15
+
Nt
Nb
(N2
b + N2
t )
36864
9
80
+
11k
36
dz
(3.2.23)
The normalized impedance to the flow is
λ̄ =
λ
λn
(3.2.24)
The wall shear stress is τh = −h
2
dP
dz
and is given by

23. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 38
τh =
h
2
−
sin α
F
−
1
(1 − N) h4
8
+ kh3
3
−q + N
h3
3
+ kh2
vθ − (1 − N) Gr
Nb − Nt
64
h6
12
+
7kh5
30
−
Nb
9
−453h7
2800
+
2kh6
15
−
Nt (N2
b + N2
t )
36864
369h14
1120
−
149kh13
468
− (1 − N) Br
7h6
96
−
7kh5
120
Nt
Nb
+
Nt
9
27h7
560
+
2kh6
15
+
Nt
Nb
(N2
b + N2
t )
36864
9h10
80
+
11kh9
36
(3.2.25)
3.2.4 Result Analysis
The effects of various parameters on λ̄ are shown in figures (3.2.1 - 3.2.8).
It is observed that, the resistance to the flow λ̄
increases with the heights of
the stenosis, local nanoparticle Grashof number (Br), local temperature Grashof
number (Gr) and inclination (α). It is noted that, the velocity of the particles
with the surrounding molecules (Nt) increases with the increases of heights of the
stenosis. It is remarkable to note that, the permeability (k) of the walls of the
artery increases with the increase of resistance to the flow. It is interesting to note
that, the resistance to the flow decreases with the increase of the collision between
the molecules. I.e., Brownian motion parameter (Nb). Also, resistance to the flow
decreases with the increase of volumetric flow rate (q).
The shear stress acting on the wall (τh) over the height of stenosis has shown in
the figures (3.2.9 - 3.2.16). It is shown that, the shear stress at the wall increases
with height of the stenosis. Also, it is observed that when the collision between the
molecules (Nb) increases, the shear stress at the wall also increases. The wall shear
stress increases with the increase of flux (q). It is also noted that, the shear stress at
the wall decreases with Br, Gr, heat and mass transfer coefficient (Nt), inclination
(α) and permeability constant (k).

24. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 39
d2 = 0.01
d2 = 0.02
d2 = 0.03
d2 = 0.04
0.02 0.04 0.06 0.08 0.10
1.05
1.10
1.15
1.20
d1
l
–
Figure 3.2.1: Variation of δ1 on λ̄ with δ2 varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.3,
Nt = 0.8 , α = π
6 , k = 0.05, F = 0.3
!
Br = 0.2
Br = 0.3
Br = 0.4
Br = 0.5
0.02 0.04 0.06 0.08 0.10
1.05
1.10
1.15
1.20
d1
l
–
Figure 3.2.2: Variation of δ1 on λ̄ with Br varying
q = 0.3, δ2 = 0.01, Gr = 0.2, Nb = 0.3,
Nt = 0.8 , α = π
6 , k = 0.05, F = 0.3
!
Gr = 1
Gr = 6
Gr = 11
Gr = 16
0.02 0.04 0.06 0.08 0.10
1.05
1.10
1.15
1.20
d1
l
–
Figure 3.2.3: Variation of δ1 on λ̄ with Gr varying
q = 0.3, δ2 = 0.01, Br = 0.3, Nb = 0.3,
Nt = 0.8 , α = π
6 , k = 0.05, F = 0.3
!

25. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 40
Nb = 0.10
Nb = 0.11
Nb = 0.12
Nb = 0.13
0.02 0.04 0.06 0.08 0.10
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
d1
l
–
Figure 3.2.4: Variation of δ1 on λ̄ with Nb varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nt = 0.8 , α = π
6 , k = 0.05, F = 0.3
!
Nt = 0.4
Nt = 0.8
Nt = 1.2
Nt = 1.6
0.02 0.04 0.06 0.08 0.10
1.06
1.08
1.10
1.12
1.14
1.16
1.18
d1
l
–
Figure 3.2.5: Variation of δ1 on λ̄ with Nt varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.3 , α = π
6 , k = 0.05, F = 0.3
!
a = p/36
a = p/18
a = p/9
a = p/12
0.02 0.04 0.06 0.08 0.10
1.02
1.03
1.04
1.05
1.06
d1
l
–
Figure 3.2.6: Variation of δ1 on λ̄ with α varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.3 , Nt = 0.8, k = 0.05, F = 0.3
!

26. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 41
k = 0.02
k = 0.04
k = 0.06
k = 0.08
0.02 0.04 0.06 0.08 0.10
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
d1
l
–
Figure 3.2.7: Variation of δ1 on λ̄ with k varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.3, Nt = 0.8 , α = π
6 , F = 0.3
!
q = 0.30
q = 0.32
q = 0.34
q = 0.36
0.02 0.04 0.06 0.08 0.10
1.05
1.10
1.15
1.20
d1
l
–
Figure 3.2.8: Variation of δ1 on λ̄ with q varying
δ2 = 0.01, k = 0.05, Br = 0.3, Gr = 0.2,
Nb = 0.3, Nt = 0.8 , α = π
6 , F = 0.3
!
d2 = 0.010
d2 = 0.025
d2 = 0.020
d2 = 0.015
0.02 0.04 0.06 0.08 0.10
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
d1
t
h
Figure 3.2.9: Variation of δ1 on τh with δ2 varying
q = 0.3, Br = 0.3, Gr = 0.2, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.1, z = 0.7, F = 0.3
!

27. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 42
Br = 0.20
Br = 0.23
Br = 0.26
Br = 0.29
0.02 0.04 0.06 0.08 0.10
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
d1
t
h
Figure 3.2.10: Variation of δ1 on τh with Br varying
q = 0.3, δ2 = 0.01, Gr = 0.2, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.1, z = 0.7, F = 0.3
!
Gr = 1
Gr = 3
Gr = 5
Gr = 7
0.02 0.04 0.06 0.08 0.10
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
d1
t
h
Figure 3.2.11: Variation of δ1 on τh with Gr varying
q = 0.3, δ2 = 0.01, Br = 0.3, Nb = 0.1,
Nt = 0.3, α = π
6 , k = 0.1, z = 0.7, F = 0.3
!
Nt = 0.5
Nt = 0.6
Nt = 0.7
Nt = 0.8
0.02 0.04 0.06 0.08 0.10
1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.05
d1
t
h
Figure 3.2.12: Variation of δ1 on τh with Nt varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, α = π
6 , k = 0.1, z = 0.7, F = 0.3
!

28. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 43
Nb = 0.10
Nb = 0.13
Nb = 0.12
Nb = 0.11
0.02 0.04 0.06 0.08 0.10
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
d1
t
h
Figure 3.2.13: Variation of δ1 on τh with Nb varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nt = 0.3, α = π
6 , k = 0.1, z = 0.7, F = 0.3
!
a = p/6.06
a = p/6.00
a = p/5.94
a = p/5.88
0.02 0.04 0.06 0.08 0.10
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
d1
t
h
Figure 3.2.14: Variation of δ1 on τh with α varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, k = 0.1, z = 0.7, F = 0.3
!
k = 0.100
k = 0.103
k = 0.106
k = 0.109
0.02 0.04 0.06 0.08 0.10
1.75
1.80
1.85
1.90
1.95
2.00
2.05
2.10
d1
t
h
Figure 3.2.15: Variation of δ1 on τh with k varying
q = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, α = π
6 , z = 0.7, F = 0.3
!

29. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 44
q = 0.300
q = 0.302
q = 0.304
q = 0.306
0.02 0.04 0.06 0.08 0.10
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
d1
t
h
Figure 3.2.16: Variation of δ1 on τh with q varying
F = 0.3, δ2 = 0.01, Br = 0.3, Gr = 0.2,
Nb = 0.1, Nt = 0.3, α = π
6 , k = 0.1, z = 0.7
!
Streamlines: The streamlines for k are shown in Figure (3.2.17). It is seen that,
streamlines are getting closure with the increase of k.
k = 0.01
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
k = 0.02
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
k = 0.03
0.0 0.2 0.4 0.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Figure 3.2.17: Streamlines for k = 0.01, 0.02 and 0.03

30. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 45
Resistance to the flow (λ̄)
Straight Tube Inclined Tube
δ1 (α = 0o
) (α = 5o
) (α = 10o
) (α = 15o
) (α = 20o
)
0.02 1.01273 1.01465 1.01723 1.02084 1.02620
0.04 1.02218 1.02553 1.03002 1.03631 1.04565
0.06 1.03236 1.03724 1.04379 1.05297 1.06661
0.08 1.04335 1.04989 1.05866 1.07096 1.08922
0.10 1.05524 1.06357 1.07475 1.09042 1.11369
Table 3.2.1: A Comparative analysis between straight tube and Inclined
tube for nanoparticles in Micropolar fluid
A comparative analysis have been made between straight tube and inclined tube for
nanoparticles in micropolar fluid having multiple stenoses is shown in Table (3.2.1).
It is noted that, λ̄ is more in the inclined tube when compared to the straight tube.
δ1, δ2 Nanofluid Micropolar fluid with Nanoparticles
Resistance to the flow (λ̄) Resistance to the flow (λ̄)
δ1 δ2=0.01 δ2=0.02 δ2=0.03 δ2=0.04 δ2=0.01 δ2=0.02 δ2=0.03 δ2=0.04
0.02 1.09506 1.12381 1.15361 1.18452 1.05124 1.06926 1.08793 1.10731
0.04 1.15583 1.18458 1.21438 1.24529 1.08929 1.10731 1.12598 1.14536
0.06 1.22126 1.25001 1.27981 1.31073 1.13028 1.14829 1.16697 1.18634
0.08 1.29187 1.32062 1.35042 1.38134 1.17452 1.19253 1.21121 1.23058
0.10 1.36823 1.39698 1.42678 1.4577 1.22237 1.24039 1.25906 1.27843
Table 3.2.2: Comparitive study of Resistance to the flow (λ̄) with heights
of the stenosis (δ1δ2) between the Nanofluid and Micropolar fluid with
Nanoparticles having multiple stenosis
A comparative study of resistance to the flow (λ̄) with heights of the stenosis (δ1δ2)
between the Nanofluid and Micropolar fluid with nanoparticles in a tube of non-
uniform cross-section having multiple stenosis has been considered and is given in
Table (3.2.2). It is seen that the flow resistance is more in Nanofluid when compared
to Micropolar fluid with nanoparticles.

31. Chapter 3.2: Thermal Effects of Micropolar Fluid Through a Permeable Non-
Uniform Artery having Multiple Stenosis 46
δ1, k Nanofluid Micropolar fluid with Nanoparticles
Resistance to the flow (λ̄) Resistance to the flow (λ̄)
δ1 k=0.02 k=0.03 k=0.04 k=0.05 k=0.02 k=0.03 k=0.04 k=0.05
0.02 1.06232 1.07084 1.08146 1.09506 1.04224 1.04483 1.0478 1.051240
0.04 1.10590 1.11888 1.13508 1.15583 1.07366 1.07815 1.08331 1.08929
0.06 1.15289 1.17067 1.19285 1.22126 1.10754 1.11407 1.12158 1.13028
0.08 1.20368 1.22660 1.25521 1.29187 1.14418 1.15289 1.16290 1.17452
0.10 1.25870 1.28716 1.32269 1.36823 1.18388 1.19492 1.20763 1.22237
Table 3.2.3: Comparitive study of Resistance to the flow (λ̄) with heights
of the stenosis (δ1) and (k) between Nanofluid and Micropolar fluid with
Nanoparticles having multiple stenosis
A comparative study of resistance to the flow (λ̄) with height of the stenosis (δ1)
and permeability constant (k) between the Nanofluid and Micropolar fluid with
nanoparticles in a tube of non-uniform cross-section having multiple stenosis has
been considered and is given in Table (3.2.3). It is shown that the flow resistance is
more in Nanofluid when compared to Micropolar fluid with nanoparticles.