Blood Flow through an Inclined Stenosed Artery

Applied Mathematical Sciences, Vol. 10, 2016, no. 5, 235 - 254

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511701

Blood Flow through an Inclined Stenosed Artery

Ahmed Bakheet1,2, Esam A. Alnussaiyri1,

Zuhaila Ismail1, and Norsarahaida Amin1*

1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi

Malaysia, UTM, 81310 Johor Bahru, Johor, Malaysia

2Department of Mathematics, Faculty of Science, Al-Mustansiriya University

Baghdad, Iraq *Corresponding author

Copyright © 2015 Ahmed Bakheet et al. This article is distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

The present work is to investigate the effect of the inclination angle on blood

flow through the flexible stenosed artery. The mathematical model of blood

considered laminar, incompressible, unsteady and fully developed, is represented

by the generalized power law model. The numerical Marker and Cell method

(MAC) has been utilized to solve the continuity and momentum equations in

cylindrical coordinate. The Poisson equation for the pressure has been solved

using Successive-Over-Relaxation method (SOR) and the pressure–velocity

correction formula has been derived. The effects of inclination angle ( ) and

power law index (n) on the flow characteristics of blood such as velocity, pressure

drop, and wall shear stress are presented by their representation graphs.

Keywords: Generalized power law, inclined artery, Marker and Cell

1 Introduction

The effect of the inclination angle of the blood flow has been done by many

authors such as Chaturani and Upadhya [1] who studied the gravity flow of fluid

with couple stress along an inclined plane with application to blood flow.

Vajravelu et al. [2] investigated the peristaltic transport of Herschel-Bulkley fluid

through an inclined tube. Rathod et al. [3] studied the pulsatile flow of blood

through rigid inclined circular tubes under the influence of periodic body

acceleration. Sanyal et al. [4] studied the characteristics of blood flow in a rigid

inclined circular tube with periodic body acceleration under the influence of a uni-

236 Ahmed Bakheet et al.

form magnetic field. Prasad and Radhakrishnamacharya [5] considered the steady

blood flow through an inclined non-uniform tube with multiple stenoses. Rathod

and Habeeb [6] studied pulsatile inclined two layered flow under periodic body

acceleration. Krishna et al. [7] investigated the peristaltic pumping of a Casson

model in an inclined channel under the effect of a magnetic field. Krishna et al. [8]

proposed a mathematical modelling and numerical solution for the flow of a

micropolar fluid under the effect of magnetic field in an inclined channel of half

width under the considerations of low Reynolds number. Kavitha et al. [9]

investigated peristaltic flow of a micropolar fluid in a vertical channel with long

wavelength approximation. Sreenadh et al. [10] presented a mathematical model

to study the flow of Casson fluid through an inclined tube of non-uniform cross

section with multiple stenosis. Chakraborty et al. [11] showed blood flow through

an inclined tube to account for the slip at stenotic wall, hematocrit and inclination

of the artery has been represented by a particle-fluid suspension. It is assumed that

the wall of the tube is rigid and the blood flow is represented by a two-fluid model.

Biswas and Paul [12] observed the steady flow of blood through an inclined

tapered constricted the blood as Newtonian fluid and the artery with an axial slip

in velocity at the vessel wall. Also, their analysis includes Poiseuille flow of blood

with one-dimensional blood flow models through tapered vessels with inclined

geometries. Verma and Srivastava [13] studied the blood flow considered to be

Newtonian and incompressible viscous in a rigid inclined circular tube in the

presence of transverse magnetic field. Sharma et al. [14] studied pulsatile blood

flow in a catheterized inclined artery with a velocity slip at the stenosed arterial

wall under the influence of magnetic field.

Many investigators, including Haynes and Burton [15], Merrill et al. [16],

Hershey et al. [17], Charm and Kurland [18], and Lih [19] have shown that blood

being a suspension of corpuscles behaves like a non-Newtonian fluid at low shear

rates. In particular, Hershey et al. [20] and Huckaback and Hahn [21] presented

that blood flowing through a tube of diameter less than 0.2 mm and at low shear

rate less than 120 s . The experimental finding of Perktold et al. [22] indicated

that the power law fluid exhibits a non-Newtonian influence. Tu and Deville [23]

observed that the blood of patients in some disease conditions, for instance,

patients with severe myocardial infarction, cerebrovascular diseases and

hypertension, exhibits power law behaviour. Furthermore, Johnston et al. [24]

investigated the blood flow in human right coronary arteries in steady state. They

studied Newtonian and non-Newtonian blood models such as Carreau model,

Walburn-Schneck model, power law, Casson model and generalized power law

model in order to judge the significance of the blood models. Their research

showed that the Power Law model over-estimates the wall shear stress at low inlet

velocities and under-estimates the wall shear stress at high inlet velocities. In

addition, Walburn-Schneck model under-estimates wall shear stress in high inlet

velocities while Newtonian model under-estimates wall shear stress at low inlet

velocities. Thus, they concluded that the Newtonian model of blood viscosity is a

good approximation in the regions of mid-range to high shear; generalized power

Blood flow through an inclined stenosed artery 237

law model is advisable to use to achieve a better approximation of wall shear

stress at low shear. Again, Johnston et al. [25] studied five non-Newtonian models

of blood flow in human right coronary arteries, this time for steady and unsteady

state simulation, respectively, and they recommended that the generalized power

law model approximates wall shear stress better than the Newtonian model for

low inlet velocities and in regions of low shear.

In the present study a mathematical model is developed to investigate the

influence of inclination angle on unsteady two-dimensional blood flow through an

elastic stenosed artery in which the streaming blood is considered as a generalized

power law model including both the shear-thickening and shear-thinning rheology

of blood. The MAC method based on staggered grids has been used to solve the

governing equations in cylindrical polar coordinate system numerically and

achieved the computed results with the desired degree of accuracy. The pressure

was calculated using the pressure correction method. The velocity profile and wall

shear stress for various values of inclination angle have been shown graphically.

2 Mathematical Formulation

We assumed that blood is a non-Newtonian fluid characterized by

generalized power law model having both shear-thickening and shear-thinning

rheology of blood. In addition the flow is assumed to be unsteady, incompressible

and axisymmetric.

2.1. Governing Equation

The blood flow through an axisymmetric stenosed artery is simulated in two

dimensions by making use of a cylindrical coordinate system. The dimensionless

equations of motion in the unsteady state can be written in conservative form as

( )0

w urr

z r

(1)

2( ) ( ) 1 1 sin

( ) ( )Re Fr

rz zz

w wu w wu pr

t r z r z r r z

(2)

2 2( ) 1 1 cos

( ) ( )Re Fr

rr rz

u u wu u pr

t r z r r r r z

(3)

with 1

12 2 2 2 2

2 2 2 2

n

zz

u u w u w w

r r z z r z

(4)


1

12 2 2 2 2

2 2 2

n

rz

u u w u w u w

r r z z r z r

(5)

11

2 2 2 2 2

2 2 2 2

n

rr

u u w u w u

r r z z r r

(6)

where r and z are the dimensionless coordinates scaled with respect to r0, with the

z-axis located along the axis of the artery. As there is no secondary or rotational

flow the total velocity is defined by the dimensionless radial and axial

components, u and w are scaled with respect to the cross-sectional average

velocity U. The generalized Reynolds number Re, the dimensionless pressure p

and Froude number Fr are defined as

0

2Re

n

n

r

mU

2

pp

U

and

2

0

Fr .U

gR

where is the density of the blood, p the pressure, g is a gravitation parameter

and, m and n being the consistency and power indices of the generalized

power-law model, respectively.

The geometry of the stenosis considered in this study is the cosine curve

11 0

0 0

1

( )1 1 cos ( ) , 2

( , ) 2

( ) otherwise

z za t d z d z

R z t r z

a t

(7)

where ( , )R z t represents the radius of the artery in constricted region and 0 ,r

in the normal region, is the severity of the stenosis, 0z is the half length of

stenosis and 1z is the center of the stenosis (see Fig.1). The time-variant

parameter 1( )a t is given by 1( ) 1 cos( )Ra t k t where Rk represents the

amplitude parameter and is the phase angle. r

z

R0

z=Lg

sinzg

cosrg

0

( , )R z t

Figure1. Geometry of artery with stenosis


2.2. Boundary and Initial Condition

For the boundary conditions, there is no radial flow along the axis of the

artery and the axial velocity gradient of the streaming blood may be assumed to be

equal to zero, i.e., there is no shear rate of fluid along the axis.

( , , )0, ( , , ) 0 on 0

w r z tu r z t r

r

(8)

( , , ) 0, ( , , ) on ( , )R

w r z t u r z t r R z tt

(9)

2

( , ,0) 2 1 for 0r

w r z U zR

(10)

and

( , ,0) 0 for 0u r z z (11)

( , , ) ( , , )0 at

w r z t u r z tz L

z z

(12)

( , ,0) 0 ( , ,0)w r z u r z and ( , ,0) 0 for 0p r z z (13)

3 Radial Coordinate Transformation

To avoid interpolation errors while discretizing the governing equations, we

use the following transformation ( , )

rx

R z t , then the flow equations (1)-(3) are

transformed as follows:

10

w u u x R w

z xR R x R z x

(14)

1w R R w w px w u w

t R z t x z z

1 1 1 s i n

R e F r

x z z z z zxz

x R

xR R x R z x z

(15)


2 21 ( ) 1u u u x wu R u p

wt R x z R x z xR R x

1 1 1 cos.

Re Fr

xz zz zzxz

x R

xR R x R z x z

(16)

with

2 2 21

2 2 2 2zz

u u w x R w

R x xR z R z x

11

2 21

n

u x R u w w x R w

z R z x R x z R z x

(17)

2 2 21

2 2 2 2xx

u u w x R w

R x xR z R z x

11

2 21 1

n

u x R u w u

z R z x R x R x

,

(18)

2 2 21

2 2 2xz

u u w x R w

R x xR z R z x

11

2 21 1

n

u x R u w u x R u w

z R z x R x z R z x R x

. (19)

The boundary and initial conditions (8)–(13) become

( , , )0, ( , , ) 0 on 0

w x z tu x z t x

r

(20)

( , , ) 0, ( , , ) on 1R

w x z t u x z t xt

(21)

2( , ,0) 2(1 ) at 0w x z x z (22)


and

( , ,0) 0 at 0u x z z (23)

( , , ) ( , , )0 at

w x z t u x z tz L

z z

(24)

( , ,0) 0 ( , ,0)w x z u x z and ( , ,0) 0 for 0p x z z (25)

4 Numerical Solution

The governing equations with the set of initial and boundary conditions are

discretized using a finite difference approximation which is borne out in staggered

grid, known as the MAC method proposed firstly by Harlow and Welch [26]. In

this type of grid alignment, the pressure and velocities are calculated at different

locations as indicated in Fig.2. The advantage in using MAC is that the pressure

boundary conditions at the inlet and outlet are not required. Where the vector of

velocity is specified the discretization involves the combination of central and

second order upwind differencing, from which the pressure Poisson equation and

pressure-velocity correction formulae are derived.

jip ,

),( 1

2

1 ji

xz , 1i ju

),(2

1

2

1ji

xz

,i jw

),(2

1j

i xz ),(2

11j

i xz

1,i jw

),(2

1 ji

xz ,i ju

Figure 2. Typical MAC

Using the central differencing scheme, the discretized form of the continuity

equation at the (i, j) cell takes the form

, 1, , ,1 12

1 1 11

2 2 22

( ) 0i j i j i j i j

n n n nnj jn b a

j i ji

w w x u x uw w Rx R x

z x z x

(26)

where

, 1, 1, 1 , 1 , 1, , 1 , 10.25 ( ), 0.25 ( ),n n n n n n n nb i j i j i j i j a i j i j i j i jw w w w w w w w w w


1 1

2 2

,2 2

j ij i

x zx x z z

and 1 1

2 2

( )n

i iR R z (27)

For the radial momentum equation, the first order upwind for the time derivatives

with a forward difference is approximated to obtain the finite difference quotient,

the time derivative in the radial momentum equation is approximated to obtain the

difference quotient. Then the finite difference equation of the radial momentum

equation at ( , )i j cell can be written as

1

, , , , 1

, ,

1

2

1 1 coscon diff .

Re Fr

n n n

i j i j i j i j n n

i j i jn

i

u u p pu u

t R x

(28)

where ,con n

i ju and ,diff n

i ju are the finite difference representation of

convective and diffusive terms of the radial momentum for the nth time level at

the cell ( , )i j and they are discretized as follows:

n

ji

n

i

n

i

jn

jix

u

t

R

R

xu

,2

1

2

1

,con

x

uu

x

uu

R

uBBuTTBT

n

i

22

2

1

)1(1

x

uu

x

uwuw uBBuTTLLRR )1(

n

ij

n

jiuBBuTTBBTT

n

i

n

i

j

Rx

u

x

ww

x

uwuw

z

R

R

x

2

1

2

,

2

1

2

1

)()1(

(29)

with

1, 1 1, , , 1 , 1, 1, 1 , 1, , , ,

2 2 2 2

n n n n n n n ni j i j i j i j i j i j i j i j

L R B T

w w w w w w w ww w w w

1, , , 1, , , 1 , , 1, , ,

2 2 2 2

n n n n n n n ni j i j i j i j i j i j i j i j

L R B T

u u u u u u u uu u u u

is a combination factor calculated from the numerical stability criteria.

When =0 the scheme converts to central differencing and =1 it is a

second-order upwind differencing. L, R, B and T represent the corresponding

velocities calculated at the left, right bottom and top position of the cell faces,

respectively. In the second-order upwind differencing scheme the choice of taking

the momentum flux passing through the interface of the control volume

depends on the sign of the velocities at that interface [27].

The expression for uL , uR , uB and uT are defined as follows:


n

jiuTT

n

jiuTT

n

jiuBB

n

jiuBB

n

jiuRR

n

jiuRR

n

jiuLL

n

jiuLL

uwuw

uwuw

uwuw

uwuw

,1,

1,,

,,1

,1,

,0if,,0if

,,0if,,0if

,,0if,,0if

,,0if,,0if

Similarly,

n

jiuTT

n

jiuTT

n

jiuLB

n

jiuBB

uuuu

uuuu

,1,

1,,

,0if,,0if

,,0if,,0if

n

i

n

ji

zz

n

i

j

n

ji

zz

n

ij

n

jixz

n

ji

xz

n

i

n

jiz

R

xR

x

zRxxRu

2

1,

2

1,

2

1

,

,

2

1

,

)(1diff

(30)

where

n

ji

n

i

n

i

j

n

ji

n

ij

n

ji

n

ji

n

i

n

jixzx

u

z

R

R

x

z

w

Rx

u

x

u

R ,2

1

2

1,

2

2

1

,

2

,

2

1

, 221

22)(

11

2 2

1, , , ,1 1 1

22 2 2

1 1,

n

n n n n n

j

n n n

i j i j i i j i ji i i

xw u R u w

R x z R z x R x

(31)

n

ji

n

i

n

i

j

n

ji

n

ij

n

ji

n

ji

n

i

n

jizzx

u

z

R

R

x

z

w

Rx

u

x

u

R ,2

1

2

1,

2

2

1

,

2

,

2

1

, 221

22)(

n

ji

n

i

n

i

j

n

ji

n

n

ji

n

ix

w

z

R

R

x

z

w

x

w

R ,2

1

2

1,

1

2

12

,

2

1

1. (32)

The axial momentum equation in the finite difference form at ( , )i j cell can be

written as


11, , , 1, , 1 1, 1 , 1 1, 12

1

2

4

n n n n n n n nnj

i j i j i j i j i j i j i j i j

nii

xw w p p p p p pR

t z z xR

(33)

where n

jiw ,con and ,diff ni jw are the finite difference representation of

convective and diffusive terms of the axial momentum for the nth time level at the

cell ( , )i j and they are discretized as follows:

n

ji

n

i

n

i

jn

jix

w

t

R

R

x

w,

2

1

,con

x

uw

x

uwuw

R

wBBwTTBBTT

n

i

)1(

1

z

ww

z

ww wLLwRRLR

22

)1(

1 2 2,2

1

2

(1 ) ,

n nj

i j mT wT B wBT B

n n

ii ij

xw uw ww wR

R z x x x R

(34)

with

, 1, , 1,,

2 2

n n n ni j i j i j i j

L R

w w w ww w

, , 1 , , 1,

2 2


B T

w w w ww w

, 1 1, 1 , 1,,

2 2


B T

u u u uu u

and

.

2

T Bm

u uu

, 1,

1, ,

, , 1

, 1 ,

if 0, , if 0, ,

if 0, , if 0, ,

if 0, , if 0, ,

if 0, , if 0,

n nL wL i j L wL i j

n nR wR i j R wR i j

n nB wB i j B wB i j

n nT wT i j T wT i j

w w w w

w w w w

w w w w

w w w w

Similarly,

, , 1

, 1 ,

if 0, , if 0, ,

if 0, , if 0,

n nB wB i j B wL i j

n nT wT i j T wT i j

u w u w

u w u w


n

i

n

ji

zz

n

i

jn

ji

zz

n

ij

n

jixz

n

ji

xz

n

i

n

jiz

R

xR

x

zRxxRw

,

2

1

,

,

,

,

)(1diff

(35)

The Poisson equation for the pressure is found by combining the discretized

form of the momentum and continuity equations. The final form of the Poisson

equation for the pressure is

1

, ,

, , , 1, , 1, , , 1 , , 1

Div Divn n

i j i j n n n n n

i j i j i j i j i j i j i j i j i j i jA p B p C p D p D pt

n

jiji

n

jiji

n

jiji

n

jiji pIpHpGpF 1,1,1,1,1,1,1,1,

1 1

2 2, , 1 , , 1

1 1con con diff diff

Re Re

n

j jn n n ni j i j i j i j

x R

w w w wz

1

, 1 , 1 , , 1

1con con diff diff

Re Re

j jn n n nj i j j i j i j i j

x xx u x u u u

x

(36)

where

1 1 1 1, 1 , 11 1 12 2

, , 1,Div ( )

n

n nj j

j i j j i jn n ni j i j i j

x Rx u x u

w wz x

and

1 1

, 1 , 12 2, , 1,Div ( )

n

n nj j

j i j j i jn n ni j i j i j

x Rx u x u

w wz x

The expressions of jijijijijijijijiji IHGFEDCBA ,,,,,,,,, and,,,,,,, are given by

n

i

j

n

i

j

n

ij

jiRx

x

Rx

x

z

Rx

A

2

1

2

1

2

1

22

2

1

2

1

,)()()(

2

,

2

2

1

2

1

,)(

2

z

Rx

B

n

ij

ji

,

2

2

1

2

1

,)(

2

z

Rx

C

n

ij

ji

n

i

j

n

ijn

i

n

i

n

i

n

i

jiRx

x

zx

Rx

z

R

Rz

R

RD

2

1

2

2

1

2

2

1

11

,)(4

11

,


n

i

j

n

ijn

i

n

i

n

i

n

i

jiRx

x

zx

Rx

z

R

Rz

R

RE

2

1

2

12

1

2

2

1

11

,)(4

11

,

n

i

n

i

n

ij

jiz

R

zxR

Rx

F

4

2

1

2

2

1

, ,

jiji FG ,, ,

n

i

n

i

n

ij

jiz

R

zxR

Rx

H11

2

1

2

2

1

,4

,

jiji HI ,, .

The Poisson equation for the pressure (36) has been solved iteratively by the

successive over-relaxation method (SOR) with a confident iterations number to

get the pseudo pressure field at the nth time step. The over-relaxation parameter is

taken as 1.2 in order to get the maximized rate of convergence.

4.1 Pressure and Velocity Corrections

The velocity calculated by solving the momentum equations using a pseudo

pressure-field may not satisfy the continuity equation. Thus, the correction is

necessary to get more accurate velocity-field that satisfies the continuity equation.

The pressure correction formula is *

, , 0 ,

n

i j i j i jp p p (37)

where *

, jip is found by solving the Poisson equation, 0( 0.3) is an under

relaxation parameter and

ji

ji

jitA

p,

*

,

,

Div

(38)

*

,Div ji is the value of the divergence of the velocity field at the cell (i, j) and it is

obtained by solving the Poisson equation. The velocity correction formulae are

1 *, , , ,n

i j i j i j

tw w p

z

(39)

1 *1, 1, , ,n

i j i j i j

tw w p

z

(40)

1 *, , ,

1

2

,ni j i j i j

i

tu u p

R x

(41)

1 *, 1 , 1 ,

1

2

,ni j i j i j

i

tu u p

R x

(42)


where *

1,

*

,

*

,1

*

, ,,, jijijiji uuww represent the updated velocity-field obtained by

solving the Poisson equation.

4.2 Stability Restriction

To make the finite-difference scheme numerically stable, certain restrictions

are imposed on the mesh sizes ,x z and also on t . Markham and Proctor [28]

suggested that the restriction is related to the convection of the fluid, requiring

that the fluid cannot move through more than one cell boundary in a given time

step. So the time step must satisfy the inequality

1

,

Min , .

i j

z xt

w u

(43)

A more appropriate treatment, following Welch et al. [29], the momentum

must not diffuse more than one cell in one time step. In this condition, which is

related to the viscous effects, stability analysis implies 2 2

2 2 2

,

ReMin .

2i j

x zt

x z

(44)

The time step to be used at a given point in the calculation satisfies

1 2 ,Min , .

i jt t t (45)

Hence, in our computation, we take

ji

ttct,21,Min (46)

where c is a constant lying between 0.2 and 0.5 [28].

Moreover, the upwind combination factor β appearing in (29) is chosen

according to the inequality

,

1 Max , .i j

w t u t

z x

(47)

This inequality yields a very small value of the parameter.

5. Results and Discussion

The numerical results in this work are presented after the steady state is

achieved, when the non-dimensional time is 50. The expression for velocity

profile obtained at the throat of the stenosis (z=15) and downstream the stenosis

(z=23), and has been depicted in figures by plotting for different values of

inclination angle . For purpose of the numerical computations of the wanted

quantities of main physiological significance, we employed the following

parameter values from [30] as follows:

3 3

07, 0.02, Re 300, 0.276 and 1.05 10 .d r kgm


Pressure drop across a stenosis is due to large changes of energy in the blood

resulting from increasing the blood flow through the stenosis [31]. Table.1 shows

comparisons of this study with predicted dimensionless normalized pressure drops

across a cosine stenosis in [32] and across single irregular stenoses in [31] for

different Reynolds number.

Table1. Comparison of predicted dimensionless pressure drops across the stenosis

Re 20 100 200 500 1000

Mustapha et al. (2010) 53.6008 11.7804 - 4.5152 3.0864

Andersson et al. (2000) - - 8.122 - 3.239

Present study 56.113 13.6772 8.5069 4.9669 3.5828

Fig.1 shows agreement of the axial velocity result with Ikbal et al. [30]. The

numerical computations have been carried out with the following parameter

values: (z = 14; Re = 300; n = 0.639). Fig.2 shows the variation in axial velocity

for different n at different z. It appears from the results that although the nature of

the velocity profile corresponding to different n is analogous to some extent the

velocity corresponding to shear-thickening predict lower values than those of

shear-thinning rheology and Newtonian. Further, higher velocity occur at the

throat of the stenosis z=15. Fig.3 shows the variation in radial velocity for

different n at different z. It can be observed from the figure that the velocity

profile corresponding to the downstream of the stenosis reversed considerably to

that at the throat. The variation of the axial velocity profile at the throat of the

stenosis (z=15) and downstream of the stenosis (z=23) for 0.639n and different

value of is shown in Fig.4. From the figure it is clear that for o90 the

maximum velocity occurs and then gradually decreases as decreases. The

variations of the radial velocity are shown for 0.639n and different

inclination angle in Fig.5 and Fig.6. From the figure it is seen that the radial

velocity starts from zero on the axis, and then increases as one moves away from

it to approach some finite value on the wall of the artery. This finite value clearly

reflects the radial movement of the wall under consideration. One can observe

from the figure that the increasing of increase the radial velocity. In addition,

increasing is causing accelerated the flow making a pressure drop that is larger

than that due to wall shear stress alone. The velocity has a steeper slope shape

which results in an increase in the wall shear stress. Fig.7 reveals the variation

wall shear stress for 0.639n and different values of . It has been noticed that

wall shear stress increases as increases and the increase is higher in the throat

of the stenosis as compared to upstream and downstream of the stenosis. These

results agree with those of [2] and [11].


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Dimensionless radial position

Dim

ensi

onle

ss a

xial

vel

ocity

Ikbal et al.(2009)

present study

Figure 1. Comparison of axial velocity profile at z =14 with [30]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Dim

en

sio

nle

ss a

xia

l ve

loci

ty

n=0.639, z=15

n=0.639, z=23

n=1.2, z=15

n=1.2, z=23

Newtonian, z=15

Newtonian, z=23

Figure 2. Variation in axial velocity for different n and different z for o15 .

0 0.2 0.4 0.6 0.8 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Dim

ensi

onle

ss ra

dial

vel

ocity

n=0.639,z=15

n=0.639,z=23

n=1.2,z=15

n=1.2,z=23

Newtonian,z=15

Newtonian,z=23

Figure3. Variation in radial velocity for different n for o15 .


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5


Dim

ensio

nle

ss a

xia

l velo

city

=30,z=15

=60,z=15

=90,z=15

=30,z=23

=60,z=23

=90,z=23

Figure 4. Variation in axial velocity for different and different z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012


Dim

ensio

nle

ss r

adia

l velo

city

= 30

= 60

= 90

Figure 5. Variation in radial velocity for different at z=23

0 0.2 0.4 0.6 0.8 1-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0


Dim

ensio

nle

ss r

adia

l velo

city

= 30

= 60

= 90

Figure 6. Variation in radial velocity for different at z=15


0 5 10 15 20 25 300

0.5

1

1.5

Dimensionless axial position

Dim

en

sio

nle

ss w

all

sh

ea

r str

ess

= 30o

= 60o

= 90o

Figure 7. Variation in dimensionless wall shear stress for different

6 Conclusion

A calculation of how far the inclination angle in an artery of abnormal

cross-section affects the velocity of blood flow and wall shear stress has been

presented. Here we have considered the influence of inclination angle, index of

the generalized power low model and stenosis on the flow of blood through an

artery containing abnormal segments. The study reveals that as the angle

increases, the velocity and wall shear stress increase.

Acknowledgements. Financial supports provided by Vot 00M25, Research

University Grant Scheme, Universiti Teknologi Malaysia are gratefully

acknowledged. A. Bakheet and Esam are grateful to Ministry of Higher Education,

Iraq for providing study leave and a fellowship to continue doctoral studies.

References

[1] P. Chaturani and V.S. Upadhya, Gravity flow of fluid with couple stress

along an inclined plane with application to blood flow, Journal of

Biorheology, 14 (1977), 237-246.

[2] K. Vajravelu, S. Sreenadh and V. R. Babu, Peristaltic transport of a

Herschel–Bulkley fluid in an inclined tube, International Journal of

Non-Linear Mechanics, 40 (2005), 83-90.

http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.001

[3] V.P. Rathod, N. Begum and R. I. Sheeba, Inclined pulsatile flow of blood

with periodic body acceleration, Acta Ciencia Indica, 32 (2006), 1509-1516.

http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.001


[4] D.C. Sanyal, K. Das and S. Debnath, Effect of magnetic field on pulsatile

blood flow through an inclined circular tube with periodic body acceleration,

Journal of Physical Sciences, 11 (2007), 43-56.

[5] M.K. Prasad and G. Radhakrishnamacharya, Flow of Herschel-Bulkley fluid

through an inclined tube of non-uniform cross-section with multiple stenosis,

Archives of Mechanics, 60 (2008), 161-172.

[6] V.P. Rathod and A. Habeeb, Pulsatile inclined two layered flow under

periodic body acceleration, Journal of Indian Academy of Mathematics, 30

(2008), 137-151.

[7] K.S. Krishna, M.M. Ramana, K.R. Chenna and K.Y. Ravi, Peristaltic

pumping of a magnetohydrodynamic Casson fluid in an inclined channel,

Advances in Applied Science Research, 2 (2011), 428-436.

[8] K.S. Krishna, V.D. Saroj and R.Y. Ravi, Peristaltic motion of a micropolar

fluid under the effect of a magnetic field in an inclined channel, The

International Journal of Engineering and Science, 2 (2013), 31-40.

[9] A. Kavitha, R.R. Hemadri, S. Sreenadh, R. Saravana and A.N.S. Srinivas,

Peristaltic flow of a micropolar fluid in a vertical channel with longwave

length approximation, Advances in Applied Science Research, 2 (2011), 269

- 279.

[10] S. Sreenadh, A.R. Pallavi and B.H. Satyanarayana, Flow of a Casson fluid

through an inclined tube of non-uniform cross section with multiple stenoses,

Advances in Applied Science Research, 2 (2011), 340-349.

[11] S. Chakraborty, D. Biswas and M. Paul, Suspension model blood flow

through an inclined tube with an axially non-symmetrical stenosis,

Korea-Australia Rheology Journal, 23 (2011), 25-32.

http://dx.doi.org/10.1007/s13367-011-0004-8

[12] D. Biswas and M. Paul, Study of blood flow inside an inclined non-uniform

stenosed artery, International Journal of Mathematical Archive, 4 (2013),

33-42.

[13] S.R. Verma and A. Srivastava, Effect of magnetic field on steady blood flow

through an inclined circular tube, International Journal of Engineering

Research and Applications, 3 (2013), 428-432.

[14] M.K. Sharma, K. Singh and S. Bansal, Pulsatile MHD flow in an inclined

catheterized stenosed artery with slip on the wall, J. Biomedical Science and

Engineering, 7 (2014), 194-207. http://dx.doi.org/10.4236/jbise.2014.74023

http://dx.doi.org/10.1007/s13367-011-0004-8

http://dx.doi.org/10.4236/jbise.2014.74023


[15] R.H. Haynes and A.C. Burton, Role of Non-Newtonian behavior of blood in

homodynamic, American Journal of Physiology, 197 (1959), 943-952.

[16] E.W. Merril, G.R. Cokelet, A. Britten and R.E. Wells, Non-Newtonian

rheology of human blood - effect of fibrinogen deduced by subtraction,

Circulation Research, 13 (1963), 48-56.

http://dx.doi.org/10.1161/01.res.13.1.48

[17] D. Hershey, R.E. Byrnes, R.L. Deadens and A.M. Roam, Blood Theology:

Temperature Dependent of the Power Law Model, Boston: Presented at the

A.I.CH.E. Meeting, 1964.

[18] S.E. Charm and G.S. Kurland, Viscometry of human blood for shear rate of

0 – 100,000 sec-1, Nature, 206 (1965). 617-618.

http://dx.doi.org/10.1038/206617a0

[19] M.M. Lih, Transport Phenomena in Medicine and Biology, John Wiley and

Sons, New York, 1975.

[20] D. Hershey and S.J. Chow, Blood flow in rigid tubes: Thickness and slip

velocity of plasma film at the wall, Journal of Applied Physiology, 21 (1966),

27-36.

[21] C.E. Huckaback and A.W. Hahn, A generalized approach to the modeling of

arterial blood flow, Journal of Applied Physiology, 27 (1968), 27-34.

[22] K. Perktold, R. Peter and M. Resh, Pulsatile Non-Newtonian blood flow

simulation through a bifurcation with an aneurism, Biorheology, 26 (1989),

1011-1030.

[23] C. Tu and M. Deville, Pulsatile flow of non-Newtonian fluids through

arterial stenoses, Journal of Biomechanics, 29 (1996), 899-908.

http://dx.doi.org/10.1016/0021-9290(95)00151-4

[24] B.M. Johnston, P. Johnston, S. Corney and D. Kilpatrick, Non-Newtonian

blood flow in human right coronary arteries: Steady state simulations,

Journal of Biomechanics, 37 (2004), 709-720.

http://dx.doi.org/10.1016/j.jbiomech.2003.09.016

[25] B.M. Johnston, P. Johnston, S. Corney and D. Kilpatrick, Non-Newtonian

blood flow in human right coronary arteries: Transient simulations, Journal

of Biomechanics, 39 (2006), 1116-1128.


http://dx.doi.org/10.1161/01.res.13.1.48

http://dx.doi.org/10.1038/206617a0

http://dx.doi.org/10.1016/0021-9290%2895%2900151-4




[26] F. H. Harlow, and J. E. Welch, Numerical calculation of time-dependent

viscous incompressible flow of fluid with free surface, Physics of Fluids, 8

(1965), 2182-2189. http://dx.doi.org/10.1063/1.1761178

[27] C. Midya, G.C. Layek, A.S. Gupta and T.R. Mahapatra, Magneto-

hydrodynamic viscous flow separated in a channel with constrictions,

Journal of Fluid Engineering, 125 (2003), 952-962.

http://dx.doi.org/10.1115/1.1627834

[28] G. Markham and M.V. Proctor, Modifications to the two-dimensional

incompressible fluid flow code ZUNI to provide enhanced performance,

Central Electricity Generating Board Report TPRD/L/0063/M82. 1983.

[29] J.E. Welch, F.H. Harlow, J. P. Shannon and B.J. Daly, The MAC method,

Los Alamos Scientific Lab. Report LA-3425. Los Alamos, 1996.

[30] Md.A. Ikbal, S. Chakravarty, K.L. Wong, J. Mazumdar and P.K. Mandal,

Unsteady response of non-Newtonian blood flow through a stenosed artery in

magnetic field, Journal of Computational and Applied Mathematics, 230

(2009), 243-259. http://dx.doi.org/10.1016/j.cam.2008.11.010

[31] N. Mustapha, P.K. Mandal, P.R. Johnston and N. Amin, A Numerical

simulation of unsteady blood flow through multi-irregular arterial stenoses,

Applied Mathematical Modelling, 34 (2010), 1559-1573.

http://dx.doi.org/10.1016/j.apm.2009.09.008

[32] H.I. Andersson, R. Halden and T. Glomsaker, Effects of surface irregularities

on flow resistance in differently shaped arterial stenoses, J. Biomech., 33

(2000), 1257–1262. http://dx.doi.org/10.1016/s0021-9290(00)00088-9

Received: November 29, 2015; Published: January 19, 2016

http://dx.doi.org/10.1063/1.1761178

http://dx.doi.org/10.1115/1.1627834

http://dx.doi.org/10.1016/j.cam.2008.11.010

http://dx.doi.org/10.1016/j.apm.2009.09.008

http://dx.doi.org/10.1016/s0021-9290%2800%2900088-9

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Blood Flow through an Inclined Stenosed Artery