International Journal of Scientific & Engineering Research, Volume 5, Issue 2, February-2014 753
ISSN 2229-5518
Slip Effects on Steady Flow Through a Stenosed
Blood Artery
Manish Gaur1 & Manoj Kumar Gupta2
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decrease the flow resistance but the symmetric stenosis
gives maximum resistance to flow. L.M. Srivastava (1985)
t is now a well proved fact that stenosis has become a serious threat to the life which needs an immediate attention. The artery becomes stenosed when its wall becomes fatty due to abnormal development along the lumen of the wall. Because of this stenosis the hemodynamic behaviour of the blood flow is badly affected. The stenosis of the artery gives rise to many medical problems like stroke, heart attack and serious
circulatory disorders.
Many researchers have proved that the blood shows a very interesting behaviour. It behaves like a Newtonian fluid at high shear rate and it behaves like a non
– Newtonian fluid at low shear rate. Y. Nuber (1971) studied the blood flow, slip and viscometry and the study showed that the viscosity indications would exhibit a flow dependent behaviour of much the same pattern as the actual indications supplied by the usual viscometers if the slip function is of plausible form. M.D. Despande et al. (1976) discussed the steady laminar flow through modelled vascular stenoses and compared the theoretical results with available experimental values. J.B. Shukla et. al. (1980) analyzed the effects of stenosis on non – Newtonian flow of the blood in an artery and showed that the increments in the size of the stenosis produce small increments in the flow resistance and wall shear stress as the blood shows a non – Newtonian behaviour. K. Haldar (1985) studied the effects of stenosis shape on blood flow resistance and
proved that the variations in the stenosis shape may
also discussed the flow of couple stress fluid through stenotic blood vessels and showed that the flow resistance and wall shear stress in case of mild stenosis of non – Newtonian blood are increased over those with no stenosis by 60% and 62% respectively in comparison to the Newtonian fluid. J.C. Misra et al. (1993) presented a non – Newtonian model for blood flow through arteries under stenotic conditions and gave a qualitative analysis for the frequency variations of flow rate at various points of the artery, phase velocities and transmission per wavelength. J.C. Misra et al. (2007) discussed the role of slip velocity in blood flow through stenosed arteries considering the blood as a Herschel – Bulkley fluid and investigated the influence of the slip at the wall of the vessel with mild, moderate and severe stenoses. D. Biswas et al. (2011) gave a non – Newtonian model to study the steady blood flow through a stenosed artery taking blood as a Herschel – Bulkley fluid and observed that axial velocity, flow rate increase with slip and decrease with yield stress.
Laminar steady flow of an incompressible Casson fluid through a cylindrical artery having axially symmetric stenosis is considered. The geometry of the artery is described below:
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1 Department of Mathematics, Government College, Kota, Rajasthan, India,
E mail – manishbhartigaur@gmail.com
2 Pursuing Ph.D. at Department of Mathematics, Government College, Kota
Rajasthan, India.
E mail- manoj_ibs@yahoo.co.in
IJSER © 2014
International Journal of Scientific & Engineering Research, Volume 5, Issue 2, February-2014 754
ISSN 2229-5518
Using following non – dimensional quantities:
R(z) = R �(z�)
z�1+ls−z� r�
h� ∂p
∂p�/∂z�
R�0 , z =
, r = , H = ,
ls R�0 R�0
= , τc =
∂z p�0
τ�c
τ�0
v�c β�
p�0R�0/2 , τ0 = p�0R�0/2, vc = p�0R�2/2�k , β = R�
. (2.7)
0 c 0
where p�0 is the absolute typical pressure gradient.
The non – dimensional radius of the stenotic area of the
artery is
1 − H cos2 πz ; 0 ≤ z ≤ 1
�1 ; otherwise (2.8)
The non – dimensional forms of the equations of the motion
(2.2) and (2.3) are
∂p 1 ∂
Let R�(z�) be the radius of the artery in the stenotic region
and R� 0 in the non – stenotic area given as (Young, 1968):
−2 + (rτ ) = 0 (2.9)
∂z r ∂r
∂p
h� 2π
= 0 (2.10)
∂r
R�(z�) = �R� 0 − 2 �1 + cos l
R� 0
�z�1 + ls − z��� ; z�1 ≤ z� ≤ z�1 + ls
; otherwise
The constitutive equations (2.4) and (2.5) of the Casson fluid in the dimensionless forms, can be written as
(2.1)
− ∂vc = (τ1/2 − τ1/2 )2 for τ ≥ τ
(2.11)
where h�, ls and z�1 are the maximum height, length and the
∂r c 0 c 0
location of the stenosis in the artery with whole length l.
∂vc = 0 for τ ≤ τ
(2.12)
Also, let r and z� are the radial and axial coordinates. ∂r c 0
With above considerations, the equations of motion for the blood can be given as
The non – dimensional boundary conditions are
∂vc
v = β
∂r
at r = R(z)
� (2.13)
− ∂p� + 1 ∂ (rτ� ) = 0 (2.2)
τc = Finite value at r = 0
∂z�
r� ∂r� c
∂p� = 0 (2.3)
∂r�
Here p� denotes the pressure at any point and τ�c gives the
Using boundary conditions (2.13) in equation (2.9), we get the expressions for the shear stress τc and wall shear stress τR in the following forms:
shear stress of Casson fluid with the following simplified constitutive equations:
τ = −r
∂z
∂p
(2.14)
∂v�c
1 1/2
1/2 2
τR = −R(z)
(2.15)
F(τ�c ) = −
∂r�
= k�c
�τ�c
− τ�0 �
for τ�c ≥ τ�0 (2.4)
∂z
From equations (2.14) and (2.15),
∂v�c = 0 for τ�
∂r�
≤ τ�0
(2.5)
τc = r τR R
(2.16)
where v�c is the axial velocity of fluid, τ�0 represents the yield
stress and k� c is the fluid viscosity.
The flow is subject to slip boundary conditions as follows:
∂v�c
where R = R(z)
v�c = β�
∂r�
at r = R�(z�)
� (2.6)
Integrating equation (2.11) using equations (2.13) to (2.15),
r�p
τ�c = Finite value at r = 0
where β� represents the slip length in the axial direction
the velocity profile for rp ≤ r ≤ R(z) where rp =
0
non – dimensional radius of the plug flow region, is
is the
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International Journal of Scientific & Engineering Research, Volume 5, Issue 2, February-2014 755
ISSN 2229-5518
R 2
2 8 1/2
3/2
3/2
vc =
2τR
�(τR − τc ) −
3 τ0
�τR
− τc
� + 2τ0 (τR − τc )� −
1/2
1/2 2
β�τR
− τ0 �
(3.1)
Within pug flow region i.e. 0 ≤ r ≤ rp, τc = τ0 at r = rp.
Then from equation (3.1), the plug flow velocity is
The velocity profile for the axial velocity in the non – plug flow region has been obtained in equation (3.1) and results
are analyzed using graphs in figures 1(a) and 1(b).
R 2
1 2 8
1/2
3/2
1/2
1/2 2
vp =
2τR
�τR −
3 τ − 3 τ0 τR
+ 2τ0 τR� − β�τR
− τ0 �
(3.2)
The non – dimensional volumetric flow rate in the form for
the region 0 ≤ r ≤ R(z) is calculated as
R
Q = 4 � rv(r)dr
0
= 4 rp
0
Hence
rv dr + 4 ∫R
p
rvcdr
Q = 2R
�1 τ4 − 4 τ1/2 τ7/2 + 1 τ τ3 − 1 τ4 � − 2R2 β�τ1/2 −
3 R 0 R
R
1/2 2
3 0 R
84 0 R
τ0 �
(3.3)
If τ0 ≪ τR i.e.
3
τ0 ≪ 1, then equation (3.3) becomes
τR
2
Figures 1(a) shows the variations of the axial velocity along the axial distance z for the different values of
Q = R �τ
− 16 τ1/2 τ1/2 + 4 τ � − 2R2 β�τ1/2 − τ1/2 �
(3.4)
the shear stress τc and slip length β with some fixed values
2 R 7 0 R 3 0 R 0
τR = 0.070, τo
= 0.010 and H = 0.1. It is clear that the axial
which can also be used to get the wall shear stress for the stenosed artery given as
τR =
velocity first increases and then decreases after attaining a maximum value along the axial distance z. It also clarifies
that the axial velocity increases whenever the velocity slip β
increase and it decreases for the increasing values of the
�4 �2R−7β� τ1/2 + � 2Q
16 (2R−7β)2
4 �R−3β
1/2 2
shear stress.
7 R−4β 0
R2(R−4β) + 49 (R−4β)2 τ0 − 3
� τ � �
R−4β
(3.5)
For an artery without stenosis i.e. R(z) = R0, the wall shear stress is given as
τ = �4
�2R0−7β
� τ1/2
+ � 2
+ 16 (2R0−7β)2
τ0 −
7 R0−4β
1/2 2
R0(R0−4β)
49 (R0−4β)2
4 �R0−3β� τ � �
(3.6)
3 R0−4β 0
Now using equation (3.5) in equation (2.15), we can
compute the pressure gradient as
∂p = − 1 �4 �2R−7β� τ1/2 + � 2Q
16 (2R−7β)2
∂z R
7 R−4β 0
1/2 2
R2(R−4β) + 49 (R−4β)2 τ0 −
4 �R−3β� τ � �
(3.7)
3 R−4β 0
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ISSN 2229-5518
Figure 1(b) shows the variations of the axial
velocity along the radial distance R(z) for the different
values of the shear stress τc and slip length β with some
fixed values τR = 0.070 and τo = 0.010. Graph shows that
the axial velocity is increasing along the radial distance.
Also that the axial velocity increases when the velocity slip increases and it decreases as the shear stress increases.
The axial velocity for the plug flow region obtained through equation (3.2) has been analyzed in figure 2(a) which shows the variations of the plug flow velocity along the axial distance z taken for the different values of the
yield stress τ0 and slip length β with fixed values τR =
0.070 and H = 0.1. It is observed here that the plug flow
velocity is showing wavy variations along the axial distance
z. Also the plug flow velocity increases as the velocity slip increases and it decreases when the yield stress increases.
Figure 2(b) shows the variations of the axial
velocity along radial distance R(z) for the different values
of the yield stress τ0 and slip length β with other fixed
values τR = 0.070. It shows that the plug flow velocity
increases along the radial distance and it decreases when
the yield stress increases. Also the plug flow velocity increases as the velocity slip increases. It is to be noted here that for the greater values of the yield stress, the plug flow velocity increases slowly as the velocity slip increases as compared to the lower values of the yield stress.
The volumetric flow rate derived through equation (3.4) has been graphically presented in figures 3(a) and 3(b). Figure 3(a) shows the variations of the volumetric flow rate
along the radial distance R(z) for the various values of the yield stress τ0 and slip length β with a fixed value τR =
0.070. Clearly the volumetric flow rate increases along the
radial distance. It is observed that the volumetric flow rate
increases as the velocity slip increases but it decreases when the yield stress increases. Also we see that the volumetric flow rate increases at a little slower rate for the greater values of the yield stress in comparison to the lower yield
stress.
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ISSN 2229-5518
Figure 3(b) presents the variations of the volumetric flow
rate are shown along the height of the stenosis H for the
different values of the yield stress τ0 and wall slip length β
with fixed values τR = 0.070 and z = 0.4. It is obvious that
the volumetric flow rate slows down along the height of the
stenosis with increments in yield stress but it increases when the velocity slip increases.
Figure 4(a) explains the variations of the wall shear stress obtained in equation (3.5) along the radial distance
R(z) for the different values of the slip length β with a fixed value Q = 1. It shows that the wall shear stress decreases
along the radial distance. Also the wall shear stress
decreases when velocity slip increases.
Figure 4(b) shows the variations of the of the wall
shear stress along the height of the stenosis H for the
different values of the yield stress τ0 and wall slip length β
with some fixed values Q = 1 and z = 0.4. It is clear that the
wall shear stress increases along the height of the stenosis.
Also the wall shear stress increases as the yield stress increases and it decreases when the velocity slip increases.
The variations of the pressure gradient obtained in equation (3.7) are shown in figures 5(a) and 5(b). Figure 5(a) shows that the variations of the pressure gradient along the
radial distance R(z) for the different values of the slip length β with a fixed value Q = 1. Obviously the pressure
gradient increases along the radial distance. Also it
increases as the velocity slip increases.
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ISSN 2229-5518
Figure 5(b) gives the variations of the pressure
gradient along the height of the stenosis H for the various
values of the yield stress τ0 and wall slip length β with
some fixed values Q = 1 and z = 0.4. It is observed that the
pressure gradient decreases greatly along the height of the
stenosis. Also the pressure gradient decreases whenever the yield stress increases and it increases as the velocity slip increases.
In the present study authors made an attempt to present the theoretical observations of the different flow features by considering a stenosed artery with blood behaving like a Casson fluid. The results are explained analytically and graphically by choosing some suitable parameters. The graphical analysis of the study reveals that the axial velocity is showing the wave like variations along the axial distance z and for increments in velocity slip it increases in both plug flow and non – plug flow domains. Also the axial velocity increases along the radial distance as the slip length increases in both plug flow and non – plug flow regions. The volumetric flow rate increases along the radial distance as the velocity slip increases. The axial velocity and the volumetric flow rate decrease when the yield stress increase. It is observed that the plug flow velocity and the volumetric flow rate increase gradually for the greater values of the yield stress as compared to the lower yield stress. The wall shear stress decreases and the pressure gradient increases along the radial distance as the velocity slip increases. The analysis regarding the effect of the stenosis over other flow properties like volumetric flow
rate, wall shear stress and pressure gradient has also been
done. The volumetric flow rate and the pressure gradient decrease when the yield stress increases but they increase with increments in velocity slip along the height of the stenosis. Also the wall shear stress increases as the yield stress increases and it decreases when the velocity slip increases along the height of the stenosis.
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[4] Shukla, J.B.; Parihar, R.S. and Rao, B.R.P. (1980):
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– 294.
[5] Haldar, K. (1985): Effects of the shape of stenosis on the resistance to blood flow through an artery, Bulletin of Mathematical Biology, Vol. 47, No. 4, pp. 545 – 550.
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[8] Misra, J.C. and Shit, G.C. (2007): Role of slip velocity in blood flow through stenosed arteries: a non-Newtonian model, Journal of Mechanics in Medicine and Biology, vol. 7, no. 3, pp. 337 – 353.
[9] Biswas, D. and Laskar, R.B. (2011): Steady flow
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