In this paper we assume that the blood is to be a Non-Newtonian and incompressible and Homogeneous fluid. An investigation has been done for the resistance to flow across mild stenosis situated symmetrically on steady blood flow through arteries with uniform or non-uniform cross section. An analytical solution for Power law fluid has been obtained. For the physiological insight of the problem various parameters systemic and pulmonary artery are taken and the study reveals that as the height of the stenosis increases in uniform or non-uniform portion of the artery, the resistance parameter and shear stress also steadily increases, whereas, flow rate decreases steadily and we analyze some cases between flux, pressure gradient and radius and give some significant results.

1. Journal for Research| Volume 01| Issue 09 | November 2015
ISSN: 2395-7549
All rights reserved by www.journalforresearch.org 23
Effects of Stenosis on Power Law Fluid Flow of
Blood in Blood Vessels
Subhash Chandra Pragya Singh
Department of Mathematical Science Department of Mathematical Science
Indian Institute of Technology (Baranas Hindu University),
Varanasi 221002, India
Indian Institute of Technology (Baranas Hindu University),
Varanasi 221002, India
Rana Waleed Bhnam Hndoosh
Department of Software Engineering
College of Computers Sciences & Mathematics, Mosul University, Iraq
Abstract
In this paper we assume that the blood is to be a Non-Newtonian and incompressible and Homogeneous fluid. An investigation
has been done for the resistance to flow across mild stenosis situated symmetrically on steady blood flow through arteries with
uniform or non-uniform cross section. An analytical solution for Power law fluid has been obtained. For the physiological insight
of the problem various parameters systemic and pulmonary artery are taken and the study reveals that as the height of the
stenosis increases in uniform or non-uniform portion of the artery, the resistance parameter and shear stress also steadily
increases, whereas, flow rate decreases steadily and we analyze some cases between flux, pressure gradient and radius and give
some significant results.
Keywords: Resistance Parameter, Arterial Stenosis, Power Law Fluid, Wall Shear Stress, Flux, Pressure Gradient,
Pulmonary Artery, Flow Rate
_______________________________________________________________________________________________________
I. INTRODUCTION
Diseases in the blood vessels and in the heart, such as heart attack and stroke, are the major causes of mortality worldwide. The
underlying cause for these events is the formation of lesions, knows as atherosclerosis, in the large and medium sized arteries in
the human circulation.
The term stenosis or we can say that the deposit of cholesterol is the development of arteriosclerotic or other types of
abnormal tissue development. When stenosis is constructed in artery then blood flow is obstructed. Stenosis could affect one or
more segments of the human cardiovascular system studies on initiation and growth of stenosis (arteriosclerotic plaques) in the
human cardiovascular system have been carried out from several view-points. Arteriosclerosis is a common disease, which
severely influences human health. Our body is made up of miles and miles of hollow tubes. Many small and large hollow space
and even well establish and reinforce canals. It has been found that the initiation and localization of arteriosclerosis is closely
related to local heamodynamic factors. Due to these serious consequences, attention has been given in studies of blood flow in
stenosis region under different conditions. Different mathematical models have been studied by some researchers to explore the
various aspects of blood flow in stenosed artery (Smith et al. 2002 and Shukla et al. (1980a, b). Srivastava (2002) investigated
the effects of stenosis shape and red cell concentration (hematocrit) on blood flow characteristics due to the presence of stenosis.
Ponalagusamy (2007) considered a mathematical model for blood flow through stenosed arteries with axially variable peripheral
layer thickness and variable slip at the wall. Mishra et al. (2010) studied that as the height of the stenosis increases in blood
vessels, the shear stress and resistance parameter steadily increases whereas, flow rate decreases steadily.
Fig. 1: Blood Artery Fig. 2: Stenosis in Artery

2. Effects of Stenosis on Power Law Fluid Flow of Blood in Blood Vessels
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Fig. 3: Bone in our Spines May Harden and Become Overgrown
As we age, the bone in our spines may harden and become overgrown. This can lead to a narrowing of the spinal canal, called
stenosis
Fig. 4: blocked in the right coronary artery Fig. 5: Development of atherosclerotic plaques
Fig. 6: Geometry of Tube with Multi Stenosis
In this paper, blood is assumed to be Non-Newtonian, incompressible and homogeneous fluid; cylindrical polar co-ordinate is
used, with the axis of symmetry of artery taken as Z -axis. The stenosis is mild and the motion of the fluid is laminar and steady.
The inertia term is neglected, as the motion is slow. No body force acts on the fluid and there is no slip at the wall.
II. NOMENCLATURE
  Density of blood,   Viscosity of blood, p  Pressure, 1R  Radius of uniform portion of tube,  R z  Radius of
obstructed portion of tube,  R ns z  Radius of obstructed portion of due to the nth
stenosis of tube, ns  Amplitude of nth
stenosis, nL = Length of nth
stenosis, nd  Location of nth stenosis, nl  Length of uniform portion of tube, l  Length of tube,
K  Wall exponent parameter,   Wall shear stress, 0  Measure of yield stress, e  Strain late du
dr
 
  
 
, u  Velocity of

3. Effects of Stenosis on Power Law Fluid Flow of Blood in Blood Vessels
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fluid, cR  Radius of the core region of the tube,   Resistance to flow at the wall for the flow of blood, 0  Resistance to
flow at the wall for the flow of blood in uniform portion of tube, '
  Resistance parameter
We assume the following non- dimensional quantities:
 
 ' ' ' ' ' ' ' '1 1
1 1
1 0
, , , , , , 'n n
n n n
R zd L l Rz z
Z d L l S R z R
l l l l l R l

 

             
                   
            
III.BASIC EQUATIONS
In the present analysis, it is assumed that the stenosis develops in the arterial wall in an axially symmetric manner and depends
upon the axial distance z, and the height of its growth (figure 9). In such a case the radius of the artery,  ,R z (by Young and
Tsai (1973a, b)) can be written as follows:
   1 1 1 1 1 1;0 &sR Z R z R z d d L z l      
   1
1 1
1 1 1 1 1
1
2
1 cos ;
2 2
s
s L
R Z R z R z d d z d L
L
    
           
   
     2 2 1 2 2 2; &sR Z R z R z l z d d L z l       …… (1)
     2
2 2
2 2 2 2 2
2
2
1 cos ;
2 2
s
s L
R Z R z R z z d d z d L
L
    
           
   
   2
1
1 1;K z l
R Z R e l z l
   ………. (2)
For the steady flow through circular artery, the wall shear stress is given by,
22
rGdz
dp
r
 … (3)
Where,
dp
G
dz
 
  
 
is the pressure gradient … (4)
The flow rate Q through the artery, is the sum of the flow through the core region and that in the peripheral region, i.e.,
core peripheralQ Q Q  … (5)
Where the flow rate through the core and peripheral region respectively is given by
2
core c cQ u R
… (6)
2
c
R
peripheral
R
Q urdr 
… (7)
Tne resistance to flow at the wall for the flow of blood can also be expressed as
Q
dp

Development of the Model:A.
The constitutive relationship for the power fluid is given by the relationship
 n
e ; 1n   … (8)
The velocity of the fluid through the tube thus can be expressed in terms of
2
n
rG du
dr

 
  
 
 11
2 1
n
nnG nr
u C
n

 
   
        
 
… (9)
Where C is constant of integration.
Applying the boundary conditions 0; ,u r R  we have
1
( 1)/
2 1
n nn
G nR
C
n

  
   
    … (10)
Thus the velocity of the fluid in the tube is given by equation (11)

4. Effects of Stenosis on Power Law Fluid Flow of Blood in Blood Vessels
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1
(n 1)/n ( 1)/
R
2 1
n
n nG n
u r
n
    
         … (11)
The flow through the artery can be obtained from the basic equation
0
2
R
Q urdr 
… (12)
For 1
,
2
n  we get the expression of flow through the blood vessel as
    
2
5 2 ' '
1 1 1 1 1 2 22
5 5
20
P
Q R l Kl l l s L L

 

      
… (13)
Also the expression of wall shear stress through the blood vessel is given in equation (14)
2 2 2
3
' '1 1 22
1 1 1 1 1 2 2 22 2
1 2
3 3
{ } 1 1 ( )
2 2 2 2
kl L L
R l l l s L d s L d
L L
 
  
          
                
         
… (14)
For 1
,
3
n  resistance to flow at the wall for the flow of blood is given by
3
16 6
0
48 2 2
l
cR R dz




   
The resistance to flow at the wall for the flow of blood in uniform portion of blood vessel is,
3
16 6
0 1
0
48 2 2
l
cR R dz




   
Thus the resistance parameter for the flow of blood in the blood vessel is expressed as
          
3' 6 ' ' 6 6 ' ' '
1 1 1 2 2 1 2 11 6 1 6 1 2c c cl L L s R L s R R l l L K l l                … (15)
Effect of Wall Shear Stress:B.
Power law model
'
1S 
K = 0.001 K = 0 K = - 0.001
.027 35.324 20.174 5.024
.034 38.354 23.20 8.050
.040 41.384 26.234 11.084
.046 44.414 29.264 14.114
.053 47.444 32.294 17.144
.060 50.474 35.324 20.174
Table 1: Variation of  against '
1S for K = -0.001, 0, 0.001.
Effect of Flow Rate:C.
Power law model
'
1S Q
K = 0.001 K = 0 K = -0.001
.027 40.48 39.89 39.24
.034 40.41 39.77 39.21
.040 40.37 39.74 39.17
.046 40.33 39.72 39.12
.053 40.30 39.69 39.07
.060 40.25 39.64 39.04
Table 2: Variation of Q against '
1S for K = -0.001, 0, 0.001
Effect of Resistance Parameter:D.
Power law modal
'
2S '

K = -0.001 K = 0 K=0.001
.027 5.150996 5.0028200003 4.854644
.034 5.151416 5.0032400003 4.855064
.040 5.151776003 5.003600003 4.855424003
.046 5.152136 5.003960003 4.855784
.053 5.152556003 5.0043800003 4.856204003

5. Effects of Stenosis on Power Law Fluid Flow of Blood in Blood Vessels
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.060 5.152976003 5.0048000003 4.856624003
.067 5.153126001 5.005200003 4.856927003
Table 3: Variation of '
 against '
2S for K = -0.001, 0, 0.001
We assume one stenosis each in uniform and non- uniform portion of the artery (Figure 6). To observe explicitly the effect of
various parameters on resistance, wall shear stress and viscosity to the flow, the following function has been assumed for the
artery radius, which is non- uniform.
Fig. 7: Variation of  against '
1s for various value of K
Fig. 8: Variation of Q against
'
1s for various value of K
Fig. 9: Variation of
'
 against
'
2s for various value of K
Effect of Various Parameters on the Flow of Blood in Stented Blood Vessels:E.
In order to get a physiological insight into the effect of stenosis on the wall shear stress, flow rate and resistance parameter
against '
1S or '
2S or both, for different values of wall exponent parameter K, i.e. K>0(divergence of artery), K=0(uniform
portion of capillary) and K<0(convergence of veins), computations are made for power law model are shown in the below
sections.
Analysis:1)
In the Power law model developed, we observe that as the height of stenosis increases in the blood vessels, wall shear stress also
steadily increases for different values of wall exponent parameter, i.e. K>0(divergence of artery), K=0(uniform portion of
capillary) and K<0(convergence of veins). The mean arterial blood pressure (MAP) in arteries is around 100mm Hg, in the
capillaries the MAP is 25mm Hg and in the veins and venae-cavae its mean pressure falls progressively to about 0 mm Hg in the
systemic circulation. Similarly in the pulmonary circulation the MAP is 16 mm Hg, whereas, in the pulmonary capillary it is 7

6. Effects of Stenosis on Power Law Fluid Flow of Blood in Blood Vessels
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mm Hg and in the pulmonary veins its mean pressure falls progressively to about 0 mm Hg like in systemic circulation. The
above model depict the physiological conditions like K>0, K=0 and K<0.
Analysis:2)
In the Power law model developed, we observe that as the height of the stenosis increases in the blood vessels, flow rate steadily
decreases for different values of wall exponent parameter, i.e., K<0(convergence of artery), K=0(uniform portion of artery) and
K>0(divergence of artery).
In the above model developed, we observe that as the height of stenosis increases in the blood vessels, resistance parameter
steadily increases for different values of wall exponent parameter, i.e. K<0(convergence of artery), K=0(uniform portion of
artery) and K>0(divergence of artery).
IV.DISCUSSION
Wall shear stress is an important factor in the study of blood flow. Accurate predictions of the distribution of the wall shear stress
are particularly useful for the understanding of the effect of blood flow on endothelial cells. However, the flow rate in the arteries
is affected much compared to veins, as arteries are resistance vessels, whereas veins are capacitance vessels. In hypertensive
patients, the sustained increased pressure in arteries will lead to remodeling of the blood vessels and heart, especially in the
resistance vessels where the pressure is very high. Arteries tend to become less elastic and stiff. In the model discussed, the
trends observed show that as the stenosis increases there is an increase in the MAP in the resistance vessels which may lead to
remodeling of the arteries. The remodeling is not prominent in capillaries and veins, where the resistance to flow is least
parameter compared to arteries. In the model developed above, we observe that Power law fluid model well suit for the
physiological data.
V. CONCLUSION
The study reveals that as the height of the stenosis increases in the uniform or non- uniform or both portions of the artery the
resistance to the flow also increases.
Physical Significance:A.
Model can help the user understand the spread of a disease and then specify the question that needs to be answered, and identify
important data needs, including regular surveillance and outbreak investigations. Which can help health officials focus on critical
factors.
ACKNOWLEDGEMENT
I am also thankful to my brother Mr. Sanjay Kumar for his moral support and belief in my abilities!
REFERENCES
[1] Shukla, J. B., Parihar, R. S. and Gupta, S. P. (1980a) “Effect of peripheral layer viscosity on blood flow through the artery with mild stenosis.” Bull. Math.
Biol., 42:797-805.
[2] Shukla, J. B., Parihar, R. S.; Rao, B. R. P. and Gupta, S.P. (1980b) “Effect of peripheral layer viscosity on peristaltic transport of a biofluid.” J. Fluid
Mech., 97: 225-235.
[3] Smith, N. P.; Pullan, A. J. and Hunter, P. J. (2002) “An anatomically based model of transient coronary blood flow in the heart.” SIAM J. Applied
Mathematics, 62, 990-1018.
[4] Srivastava, V.P. (2002) “Particular suspension blood flow through stenotic arteries: effect of hematocrit and stenosis shape.” Indian J. Pure Appl. Math.
33(9): 1353-1360.
[5] Ponalagusamy, R. (2007) “Blood flow through an artery with mild stenosis: A two layered model, different shapes of stenosis and slip velocity at the wall.”
Journal of Applied Sciences, 7(7): 1071-1077.
[6] Mishra, B. K. and Verma, N. (2010) “Effects of stenosis on Non- Newtonian flow of blood in blood vessels.” Australian Journal of Basic and Applied
Sciences, 4(4): 588-601.
[7] Young, D. F. and Tsai, F. Y. (1973a) “Flow characteristic in model of arterial stenosis-I.” J. Biomechanics, 6: 395-410.
[8] Young, D. F. and Tsai, F. Y. (1973b) “Flow characteristic in model of arterial stenosis-II.” J. Biomechanics, 6: 547-599.
[9] Chandra, S. and Kumar, S. (2009) “Heat transfer and fluid flow characteristic of blood flow in multi-stenosis artery with effect of magnetic field.” Indian
Journal of Biomechanics, Special Issue NCBM (2009), ISSN 0974-0783, PP 186-190.