Volume 2010, Article ID 465835,26pages doi:10.1155/2010/465835

*Research Article*

**Pulsatile Flow of a Two-Fluid Model for Blood Flow** **through Arterial Stenosis**

**D. S. Sankar**

*School of Mathematical Sciences, University Science Malaysia, 11800 Penang, Malaysia*

Correspondence should be addressed to D. S. Sankar,sankar ds@yahoo.co.in Received 25 January 2010; Accepted 4 April 2010

Academic Editor: Saad A. Ragab

Copyrightq2010 D. S. Sankar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Pulsatile flow of a two-fluid model for blood flow through stenosed narrow arteries is studied through a mathematical analysis. Blood is treated as two-phase fluid model with the suspension of all the erythrocytes in the as Herschel-Bulkley fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is used to solve the system of nonlinear partial diﬀerential equations. The expressions for velocity, wall shear stress, plug core radius, flow rate and resistance to flow are obtained. The variations of these flow quantities with stenosis size, yield stress, axial distance, pulsatility and amplitude are analyzed. It is found that pressure drop, plug core radius, wall shear stress and resistance to flow increase as the yield stress or stenosis size increases while all other parameters held constant. It is observed that the percentage of increase in the magnitudes of the wall shear stress and resistance to flow over the uniform diameter tube is considerably very low for the present two-fluid model compared with that of the single-fluid model of the Herschel- Bulkley fluid. Thus, the presence of the peripheral layer helps in the functioning of the diseased arterial system.

**1. Introduction**

The analysis of blood flow through stenosed arteries is very important because of the fact that the cause and development of many arterial diseases leading to the malfunction of the cardiovascular system are, to a great extent, related to the flow characteristics of blood together with the geometry of the blood vessels. Among the various arterial diseases, the development of arteriosclerosis in blood vessels is quite common which may be attributed to the accumulation of lipids in the arterial wall or pathological changes in the tissue structure 1. Arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting in stenosed arteries. When an obstruction is developed in an artery, one of the most serious consequences is the increased resistance and the associated reduction of the blood flow to the particular vascular bed supplied by the artery. Also, the continual flow of blood may lead to shearing of the superficial layer of the plaques, parts of which may be

deposited in some other blood vessel forming thrombus. Thus, the presence of a stenosis can lead to the serious circulatory disorder.

Several theoretical and experimental attempts have been made to study the blood flow characteristics due to the presence of a stenosis in the arterial lumen of a blood vessel2–10.

It has been reported that the hydrodynamic factors play an important role in the formation
of stenosis11,12and hence, the study of the blood flow through a stenosed tube is very
important. Many authors have dealt with this problem treating blood as a Newtonian fluid
and assuming the flow to be steady13–16. Since the blood flow through narrow arteries is
highly pulsatile, more attempts have been made to study the pulsatile flow of blood treating
blood as a Newtonian fluid3,6–8,17–19. The Newtonian behavior may be true in larger
arteries, but, blood, being a suspension of cells in plasma, exhibits nonNewtonian behavior
at low-shear rates*γ <*˙ 10/sccin small diameter arteries0.02 mm–0.1 mm; particularly, in
diseased state, the actual flow is distinctly pulsatile2,20–25. Several attempts have been
made to study the nonNewtonian behavior and pulsatile flow of blood through stenosed
tubes2,4,9,10,26–28.

Bugliarello and Sevilla29and Cokelet30have shown experimentally that for blood flowing through narrow blood vessels, there is an outer phaseperipheral layerof plasma Newtonian fluidand an inner phasecore regionof suspension of all the erythrocytes as a nonNewtonian fluid. Their experimentally measured velocity profiles in the tubes confirm the impossibility of representing the velocity distribution by a single-phase fluid model which ignores the presence of the peripheral layer outer layer that plays a crucial role in determining the flow patterns of the system. Thus, for a realistic description of blood flow, perhaps, it is more appropriate to treat blood as a two-phase fluid model consisting of a core regioninner phasecontaining all the erythrocytes as a nonNewtonian fluid and a peripheral layerouter phase of plasma as a Newtonian fluid. Several researchers have studied the two-phase fluid models for blood flow through stenosed arteries treating the fluid in the inner phase as a nonNewtonian fluid and the fluid in the outer phase as a Newtonian fluid25,26,31–33. Srivastava and Saxena25have analyzed a two-phase fluid model for blood flow through stenosed arteries treating the suspension of all the erythrocytes in the core regioninner phaseas a Casson fluid and the plasma in the peripheral layerouter phaseis represented by a Newtonian fluid. In the present model, we study a two-phase fluid model for pulsatile flow of blood through stenosed narrow arteries assuming the fluid in the core region as a Herschel-Bulkley fluid while the fluid in the peripheral region is represented by a Newtonian fluid.

Chaturani and Ponnalagar Samy28and Sankar and Hemalatha2have mentioned
that for tube diameter 0.095 mm blood behaves like Herschel-Bulkley fluid rather than power
law and Bingham fluids. Iida 34 reports “The velocity profile in the arterioles having
diameter less than 0.1 mm are generally explained fairly by the Casson and Herschel-Bulkley
fluid models. However, the velocity profile in the arterioles whose diameters less than
0.0650 mm does not conform to the Casson fluid model, but, can still be explained by the
Herschel-Bulkley model”. Furthermore, the Herschel-Bulkley fluid model can be reduced to
the Newtonian fluid model, power law fluid model and Bingham fluid model for appropriate
values of the power law index nand yield index τ*y*. Since the Herschel-Bulkley fluid
model’s constitutive equation has one more parameter than the Casson fluid model; one can
get more detailed information about the flow characteristics by using the Herschel-Bulkley
fluid model. Moreover, the Herschel-Bulkley fluid model could also be used to study the
blood flow through larger arteries, since the Newtonian fluid model can be obtained as a
particular case of this model. Hence, we felt that it is appropriate to represent the fluid in

*R*

*Rz* *R*1z

*R*0 *βR*_{0} *δ**p*

*R**P*

*z*
Newtonian fluid

Herschel-Bulkley fluid Plug flow

*μ*_{H}*, u*_{H}*δ*_{C}*μ*_{N}*, u**N*

*d* *L*0

*L*

**Figure 1: Flow geometry of an arterial stenosis with peripheral layer.**

the core region of the two-phase fluid model by the Herschel-Bulkley fluid model rather than
the Casson fluid model. Thus, in this paper, we study a two-phase fluid model for blood
flow through mild stenosed narrow arteriesof diameter 0.02 mm–0.1 mmat low-shear rates
*γ <*˙ 10/sectreating the fluid in the core regioninner phaseas a Herschel-Bulkley fluid
and the plasma in the peripheral regionouter phaseas a Newtonian fluid.

In this study, the eﬀects of the pulsatility, stenosis, peripheral layer and the nonNew- tonian behavior of blood are analyzed using an analytical solution. Section 2 formulates the problem mathematically and then nondimensionalises the governing equations and boundary conditions. In Section 3, the resulting nonlinear coupled implicit system of diﬀerential equations is solved using the perturbation method. The expressions for the velocity, flow rate, wall shear stress, plug core radius, and resistance to flow have been obtained.Section 4analyses the variations of these flow quantities with stenosis height, yield stress, amplitude, power law index and pulsatile Reynolds number through graphs. The estimates of wall shear stress increase factor and the increase in resistance to flow factor are calculated for the two-phase Herschel-bulkley fluid model and single-phase fluid model.

**2. Mathematical Formulation**

Consider an axially symmetric, laminar, pulsatile and fully developed flow of blood
assumed to be incompressiblein the*z*direction through a circular artery with an axially
symmetric mild stenosis. It is assumed that the walls of the artery are rigid and the blood is
represented by a two-phase fluid model with an inner phasecore regionof suspension of
all erythrocytes as a Herschel-Bulkley fluid and an outer phaseperipheral layerof plasma
as a Newtonian fluid. The geometry of the stenosis is shown inFigure 1. We have used the
cylindrical polar coordinatesr, φ, zwhose origin is located on the vesselstenosed artery
axis. It can be shown that the radial velocity is negligibly small and can be neglected for a
low Reynolds number flow in a tube with mild stenosis. In this case, the basic momentum
equations governing the flow are

*ρ*_{H}*∂u*_{H}

*∂t* −*∂p*

*∂z*−1
*r*

*∂*

*∂r*r τ*H* in 0≤*r* ≤*R*_{1}z, 2.1

*ρ*_{N}*∂u**N*

*∂t* −*∂p*

*∂z* −1
*r*

*∂*

*∂r*r τ*N* in*R*1z≤*r*≤*Rz,* 2.2

0−*∂p*

*∂r,* 2.3

where the shear stress *τ* |τ*r z*| −τ*r z* since*τ* *τ** _{H}* or

*τ*

*τ*

*. Herschel-Bulkley fluid is a nonNewtonian fluid which is widely used in many areas of fluid dynamics, for example, dam break flows, flow of polymers, blood, and semisolids. Herschel-Bulkley fluid is a nonNewtonian fluid with nonzero yield stress which is generally used in the studies of blood flow through narrow arteries at low-shear rate. Herschel-Bulkley equation is an empirical relation which connects shear stress and shear rate through the viscosity which is given in2.4and 2.5. The relations between the shear stress and the strain rate of the fluids in motion in the core regionfor Herschel-Bulkley fluidand in the peripheral region for Newtonian fluidare given by*

_{N}*τ**H* ^{n}

*μ*_{H}*∂u**H*

*∂r*

*τ**y* if*τ**H*≥*τ**y**, R**p*≤*r* ≤*R*1z, 2.4

*∂u**H*

*∂r* 0 if*τ**H*≤*τ**y**,* 0≤*r*≤*R**p**,* 2.5
*τ**N* *μ*_{N}

−*∂u**N*

*∂r*

if*R*1z≤*r* ≤*Rz,* 2.6

where*u**H*,*u**N*are the axial component of the fluid’s velocity in the core region and peripheral
region; *τ** _{H}*,

*τ*

*are the shear stress of the Herschel-Bulkley fluid and Newtonian fluid;*

_{N}*μ*_{H}*, μ** _{N}*are the viscosities of the Herschel-Bulkley fluid and Newtonian fluid with respective
dimensions ML

^{−1}

*T*

^{−2}

^{n}*T*and

*ML*

^{−1}

*T*

^{−1};

*ρ*

_{H}*, ρ*

*are the densities of the Herschel-Bulkley fluid and Newtonian fluid;*

_{N}*p*is the pressure,

*t; is the time;τ*

*is the yield stress. From2.5, it is clear that the velocity gradient vanishes in the region where the shear stress is less than the yield stress which implies a plug flow whenever*

_{y}*τ*

*≤*

_{H}*τ*

*. However, the fluid behavior is indicated whenever*

_{y}*τ*

*≥*

_{H}*τ*

*. The geometry of the stenosis in the peripheral region as shown inFigure 1is given by*

_{y}*Rz *

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*R*0 in the normal artery region,

*R*_{0}−*δ*_{p}

2 1cos2π
*L*_{0}

*z*−*d*− *L*_{0}
2

in*d*≤*z*≤*dL*_{0}*,*

2.7

where*Rz*is the radius of the stenosed artery with peripheral layer,*R*_{0} is the radius of the
normal artery,*L*0is the length of the stenosis,*d*indicates its location, and*δ**p*is the maximum
depth of the stenosis in the peripheral layer such that δ*P**/R*_{0} 1. The geometry of the

stenosis in the core region as seen inFigure 1is given by

*R*1z

⎧⎪

⎨

⎪⎩

*βR*_{0} in the normal artery region,

*βR*0−*δ**C*

2 1cos2π
*L*0

*z*−*d*− *L*0

2

in*d*≤*z*≤*dL*0*,* 2.8

where*R*1zis the radius of the stenosed core region of the artery,*β*is the ratio of the central
core radius to the normal artery radius, *βR*_{0} is the radius of the core region of the normal
artery, and*δ**C*is the maximum depth of the stenosis in the core region such thatδ*C**/R*01.

The boundary conditions are

i *τ** _{H}* is finite and

*∂u*

_{H}*∂r* 0 at*r*0,
ii *τ*_{H}*τ** _{N}* at

*r*

*R*

_{1}z,

iii *u**H**u**N* at*r* *R*1z,
iv *u**N*0 at*rRz.*

2.9

Since the pressure gradient is a function of*z*and*t, we take*

−*∂p*

*∂z* *qzf*
*t*

*,* 2.10

where*qz *−∂p/∂zz,0,*ft *1*A*sin*ωt,A*is the amplitude of the flow and*ω*is the
angular frequency of the blood flow. Since any periodic function can be expanded in a series
of sines of multiple angles using Fourier series, it is reasonable to choose*ft *1*A*sin*ωt*
as a good approximation. We introduce the following nondimensional variables

*z* *z*

*R*_{0}*,* *Rz * *Rz*

*R*_{0} *,* *R*_{1}z *R*_{1}z

*R*_{0} *,* *r* *r*

*R*_{0}*,* *tωt,* *d* *d*

*R*_{0}*,* *L*_{0} *L*_{0}
*R*_{0}*,*
*qz * *qz*

*q*_{0} *,* *u*_{H}*u*_{H}*q*_{0}*R*^{2}_{0}*/4μ*_{0}

*,* *u*_{N}*u*_{N}*q*_{0}*R*^{2}_{0}*/4μ*_{N}

*,* *τ*_{H}*τ*_{H}

*q*_{0}*R*_{0}*/2,* *τ*_{N}*τ*_{N}*q*_{0}*R*_{0}*/2,*

*θ* *τ**y*

*q*_{0}*R*0*/2,* *α*^{2}_{H}*R*^{2}_{0}*ω ρ*_{H}

*μ*_{0} *,* *α*^{2}_{N}*R*^{2}_{0}*ω ρ*_{N}

*μ*_{N}*,* *R**p* *R**p*

*R*0

*,* *δ**p* *δ**p*

*R*0

*,* *δ**C* *δ**C*

*R*0

*,*

2.11

where *μ*_{0} *μ** _{H}*2/q

_{0}

*R*

_{0}

*, having the dimension as that of the Newtonian fluid’s viscosity,*

^{n−1}*q*

_{0}is the negative of the pressure gradient in the normal artery,

*α*

*H*is the pulsatile Reynolds number or generalized Wormersly frequency parameter and when

*n*1, we get

the Wormersly frequency parameter*α**N* of the Newtonian fluid. Using the nondimensional
variables,2.1,2.2,2.4,2.5, and2.6reduce, respectively, to

*α*^{2}_{H}*∂u**H*

*∂t* 4qzft−2
*r*

*∂*

*∂r*rτ*H* if 0≤*r*≤*R*1z, 2.12
*α*^{2}_{N}*∂u**N*

*∂t* 4qzft−2
*r*

*∂*

*∂r*rτ*N* if *R*1z≤*r*≤*Rz,* 2.13
*τ**H* ^{n}

−1 2

*∂u**H*

*∂r* *θ* if*τ**H*≥*θ, R**p*≤*r* ≤*R*1z, 2.14

*∂u**H*

*∂r* 0 if*τ**H*≤*θ,* 0≤*r*≤*R**p**,* 2.15
*τ**N* −1

2

*∂u**N*

*∂r* if*R*1z≤*r*≤*Rz,* 2.16

where*ft *1*A*sin*t. The boundary conditions*in dimensionless formare
i *τ** _{H}* is finite at

*r*0,

ii *∂u*_{H}

*∂r* 0 at*r* 0,
iii *τ*_{H}*τ** _{N}* at

*r*

*R*

_{1}z, iv

*u*

_{H}*u*

*at*

_{N}*rR*

_{1}z,

v *u**N*0 at*rRz.*

2.17

The geometry of the stenosis in the peripheral regionin dimensionless formis given by

*Rz *

⎧⎪

⎨

⎪⎩

1 in the normal artery region,

1−*δ**p*

2

1cos 2π
*L*0

*z*−*d*− *L*_{0}
2

in*d*≤*z*≤*dL*_{0}*.* 2.18

The geometry of the stenosis in the core regionin dimensionless formis given by

*R*1z

⎧⎨

⎩

*β* in the normal artery region,

*β*−*δ**C*

2

1cos 2π
*L*_{0}

*z*−*d*−*L*0

2

in*d*≤*z*≤*dL*_{0}*.* 2.19

The nondimensional volume flow rate*Q*is given by

*Q*4
_{Rz}

0

*ur, z, tr dr,* 2.20

where*QQ/πR*^{4}_{0}*q*_{0}*/8μ*_{0},*Q*is the volume flow rate.

**3. Method of Solution**

When we nondimensionalize the constitutive equations2.1 and 2.2,*α*^{2}* _{H}* and

*α*

^{2}

*occur naturally and these pulsatile Reynolds numbers are time dependent and hence, it is more appropriate to expand2.12–2.16about*

_{N}*α*

^{2}

*and*

_{H}*α*

^{2}

*. The plug core velocity*

_{N}*u*

*p*, the velocity in the core region

*u*

*H*, the velocity in the peripheral region

*u*

*N*, the plug core shear stress

*τ*

*p*, the shear stress in the core region

*τ*

*, the shear stress in the peripheral region*

_{H}*τ*

*, and the plug core radius*

_{N}*R*

*p*are expanded as follows in terms of

*α*

^{2}

*and*

_{H}*α*

^{2}

*where*

_{N}*α*

^{2}

*1 and*

_{H}*α*

^{2}

*1:*

_{N}*u** _{P}*z, t

*u*

_{0P}z, t

*α*

^{2}

_{H}*u*

_{1P}z, t · · ·

*,*3.1

*u*

*H*r, z, t

*u*0Hr, z, t

*α*

^{2}

_{H}*u*1Hr, z, t · · ·

*,*3.2

*u*

*r, z, t*

_{N}*u*

_{0N}r, z, t

*α*

^{2}

_{N}*u*

_{1N}r, z, t · · ·

*,*3.3

*τ*

*P*z, t

*τ*0Pz, t

*α*

^{2}

_{H}*τ*1Pz, t · · ·

*,*3.4

*τ*

*H*r, z, t

*τ*0Hr, z, t

*α*

^{2}

_{H}*τ*1Hr, z, t · · ·

*,*3.5

*τ*

*r, z, t*

_{N}*τ*

_{0N}r, z, t

*α*

^{2}

_{N}*τ*

_{1N}r, z, t · · ·

*,*3.6

*R*

*P*z, t

*R*0Pz, t

*α*

^{2}

_{H}*R*1Pz, t · · ·

*.*3.7

Substituting3.2, 3.5 in 2.12 and then equating the constant terms and*α*^{2}* _{H}* terms, we
obtain

*∂*

*∂r*rτ0H 2qzftr, 3.8

*∂u*_{0H}

*∂t* −2
*r*

*∂*

*∂r*rτ1H. 3.9

Applying3.2,3.5in2.14and then equating the constant terms and*α*^{2}* _{H}*terms, one can
get

−*∂u*0H

*∂r* 2τ_{0H}* ^{n−1}*τ0H−

*nθ,*3.10

−*∂u*1H

*∂r* 2nτ_{0H}^{n−2}*τ*1Hτ0H−n−1θ*.* 3.11

Using3.3and3.6in2.13and then equating the constant terms and*α*^{2}* _{N}*terms, we get

*∂*

*∂r*rτ0N 2qzftr, 3.12

*∂u*0N

*∂t* −2
*r*

*∂*

*∂r*rτ1N. 3.13

On substituting3.3and3.6in2.16and then equating the constant terms and*α*^{2}* _{N}* terms,
one can obtain

−*∂u*_{0N}

*∂r* 2τ_{0N}*,* 3.14

−*∂u*_{1N}

*∂r* 2τ_{1N}*.* 3.15

Using3.1–3.6in2.17and then equating the constant terms and*α*^{2}* _{H}*and

*α*

^{2}

*terms, the boundary conditions are simplified, respectively, to*

_{N}*τ*0P*, τ*1P are finite at*r* 0, 3.16

*∂u*0P

*∂r* 0, *∂u*1P

*∂r* 0 at*r* 0, 3.17

*τ*0H*τ*0N at*r* *R*1z, 3.18

*τ*_{1H}*τ*_{1N} at*r* *R*_{1}z, 3.19

*u*_{0H}*u*_{0N} at*r* *R*_{1}z, 3.20

*u*1H*u*1N at*r* *R*1z, 3.21

*u*_{0N} 0 at*rRz,* 3.22

*u*_{1N}0 at*rRz.* 3.23

Equations3.8–3.11and3.12–3.15are the system of diﬀerential equations which can be
solved for the unknowns*u*_{0H}*, u*_{1H}*, τ*_{0H}*, τ*_{1H}and*u*_{0N}*, u*_{1N}*, τ*_{0N}*, τ*_{1N}, respectively, with the help
of boundary conditions3.16–3.23. Integrating3.8between 0 and*R*0P and applying the
boundary condition3.16, we get

*τ*_{0P} *qzftR*0P*.* 3.24

Integrating3.8between*R*_{0P} and*r*and then making use of3.24, we get

*τ*0H*qzftr.* 3.25

Integrating3.12between*R*_{1}and*r*and then using3.18, one can get

*τ*0N*qzftr.* 3.26

Integrating3.14between*r*and*R*and then making use of3.22, we obtain

*u*_{0N} *qzftR*^{2}

1−*r*
*R*

2

*.* 3.27

Integrating3.10between*r*and*R*1and using the boundary condition3.20, we get

*u*0H

*qzftR*
*R*

1−

*R*_{1}
*R*

_{2}

2

*qzftR*1

_{n}*R*1

1 n1

1−

*r*
*R*1

_{n1}

− *k*^{2}
*R*1

1−

*r*
*R*1

_{n}*,*

3.28

where*k*^{2} *θ/qzft. The plug core velocityu*0Pcan be obtained from3.28by replacing
*r*by*R*_{0P} as

*u*_{0P}

*qzftR*
*R*

1−

*R*1

*R*
_{2}

2

*qzftR*1

_{n}*R*1

1 n1

1−

*R*0p

*R*1

*n1*

− *k*^{2}
*R*1

1−

*R*0p

*R*1

*n*
*.*

3.29

Neglecting the terms with *α*^{2}* _{H}* and higher powers of

*α*

*H*in 3.7 and using 3.24, the expression for

*R*0P is obtained as

*r|*_{τ}_{0P}_{θ}*R*_{0P}
*θ*

*qzft*

*k*^{2}*.* 3.30

Similarly, solving3.9,3.11, 3.13, and 3.15with the help of 3.24–3.29, and using
3.19,3.21and3.23, the expressions for*τ*_{1P}*, τ*_{1H}*, τ*_{1N}*, u*_{1H}, and*u*_{1P} can be obtained as

*τ*1P −1
4

*qzftR*
*BR*^{2}

*k*^{2}
*R*

1−

*R*1

*R*
2

−

*qzftR*1

*n*

*BR*^{2}_{1}

⎡

⎣ *n*
2n1

*k*^{2}
*R*1

−n−1 2

*k*^{2}
*R*1

_{2}

− *n*

2n1
*k*^{2}

*R*1

* _{n2}*⎤

⎦*,*

3.31

*τ*1H−1
4

*qzf*tR
*BR*^{2}*r*

*R*

1−
*R*1

*R*
2

−

*qzf*tR1

_{n}*BR*^{2}_{1}

× *n*

n1n3

*n*3
2

*r*
*R*1

−
*r*

*R*1

_{n2}

−n−1 n2

*k*^{2}
*R*1

*n*2
2

*r*
*R*1

−
*r*

*R*1

_{n1}

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*_{1}

_{n3}*R*1

*r*

⎤⎦*,*

3.32

*τ*_{1N}−

*qzf*tR

*BRR*_{1} 1
4

*r*
*R*1

−1 8

*R*_{1}
*R*

_{2}
*R*_{1}

*r*

−1 8

*R*_{1}
*R*

_{2}
*r*
*R*1

_{3}

−

*qzftR*1

_{n}

*BR*^{2}_{1} *n*
2n3

*R*_{1}
*r*

−*nn*−1
2n2

*k*^{2}
*R*1

*R*_{1}
*r*

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*_{1}

_{n3}*R*1

*r*

⎤⎦*,*

3.33

*u*_{1N}−2

*qzftR*
*BR*^{2}*R*_{1}

1 8

*R*
*R*1

1−*r*

*R*
_{2}

−1 8

*R*_{1}
*R*

_{3}
log

*R*
*r*

− 1 32

*R*
*R*_{1}

1−*r*

*R*
4

−2

*qzftR*1

_{n}

*BR*^{3}_{1}log
*R*

*r*

*n*

2n3− *nn*−1
2n2

*k*^{2}
*R*1

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*1

* _{n3}*⎤

⎦*,*

3.34

*u*1H−2

*qzftR*

*BR*^{2}*R*1 3
32

*R*
*R*_{1}

−1 8

*R*1

*R*

1 32

*R*1

*R*
3

1 8

*R*1

*R*
3

log
*R*1

*R*

2

*qzftR*1

*n*

*BR*^{3}_{1}log
*R*_{1}

*R*

×

⎡

⎣ *n*

2n3−*nn*−1
2n2

*k*^{2}
*R*_{1}

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*_{1}

* _{n3}*⎤

⎦

−*n*

*qzf*tR1

_{n}*BR*1*R*^{2}

1−

*R*1

*R*
2

× 1 2n1

1−

*r*
*R*_{1}

_{n1}

−n−1 2n

*k*^{2}
*R*_{1}

1−

*r*
*R*_{1}

*n*

−2n

*qzftR*1

_{2n−1}
*BR*^{3}_{1}

× *n*

2n1^{2}

1−
*r*

*R*_{1}
_{n1}

− n−1 2n1

*k*^{2}
*R*_{1}

1−

*r*
*R*_{1}

_{n}

− *n*

2n1^{2}n3

1−
*r*

*R*1

_{2n2}

n−1

2n^{2}6n3
n1n2n32n1

*k*^{2}
*R*_{1}

1−

*r*
*R*_{1}

_{2n1}

− n−1 2n1

*k*^{2}
*R*1

1−

*r*
*R*1

_{n1}

n−1^{2}
2n

*k*^{2}
*R*1

_{2}
1−

*r*
*R*1

*n*

− n−1^{2}
2nn2

*k*^{2}
*R*1

_{2}
1−

*r*
*R*1

_{2n}

− 3

*n*^{2}2n−2
2n−1n2n3

*k*^{2}
*R*_{1}

* _{n3}*
1−

*r*
*R*_{1}

_{n−1}

3n−1

*n*^{2}2n−2
2n−2n2n3

*k*^{2}
*R*_{1}

* _{n4}*
1−

*r*
*R*_{1}

* _{n−2}*⎤

⎦*,*

3.35
*u*1P −2

*qzftR*

*BR*^{2}*R*1 3
32

*R*
*R*_{1}

−1 8

*R*1

*R*

1 32

*R*1

*R*
3

1 8

*R*1

*R*
3

log
*R*1

*R*

2

*qzf*tR1

*n*

*BR*^{3}_{1}log
*R*_{1}

*R*

×

⎡

⎣ *n*

2n3−*nn*−1
2n2

*k*^{2}
*R*_{1}

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*_{1}

* _{n3}*⎤

⎦

−*n*

*qzftR*1

_{n}*BR*1*R*^{2}

1−

*R*1

*R*
2

×

⎡

⎣ 1 2n1

⎧⎨

⎩1−

*k*^{2}
*R*_{1}

* _{n1}*⎫

⎬

⎭−n−1 2n

*k*^{2}
*R*_{1}

1−

*k*^{2}
*R*_{1}

* _{n}*⎤

⎦−2n

*qzftR*1

_{2n−1}
*BR*^{3}_{1}

×

⎡

⎣ *n*
2n1^{2}

⎧⎨

⎩1−
*k*^{2}

*R*1

* _{n1}*⎫

⎬

⎭− n−1 2n1

*k*^{2}
*R*1

1−

*k*^{2}
*R*1

_{n}

− *n*
2n1^{2}n3

⎧⎨

⎩1−
*k*^{2}

*R*1

_{2n2}⎫

⎬

⎭

n−1

2n^{2}6n3
n1n2n32n1

*k*^{2}
*R*_{1}

⎧⎨

⎩1−
*k*^{2}

*R*_{1}

_{2n1}⎫

⎬

⎭

− n−1 2n1

*k*^{2}
*R*1

⎧⎨

⎩1−
*k*^{2}

*R*1

* _{n1}*⎫

⎬

⎭n−1^{2}
2n

*k*^{2}
*R*1

_{2}
1−

*k*^{2}
*R*1

_{n}

− n−1^{2}
2nn2

*k*^{2}
*R*_{1}

2⎧

⎨

⎩1−
*k*^{2}

*R*_{1}
2n⎫

⎬

⎭

− 3

*n*^{2}2n−2
2n−1n2n3

*k*^{2}
*R*1

* _{n3}*⎧

⎨

⎩1−
*k*^{2}

*R*1

* _{n−1}*⎫

⎬

⎭

3n−1

*n*^{2}2n−2
2n−2n2n3

*k*^{2}
*R*_{1}

* _{n4}*⎧

⎨

⎩1−
*k*^{2}

*R*_{1}
* _{n−2}*⎫

⎬

⎭

⎤

⎦*,*

3.36
where*B* 1/ftdft/dt. The expression for velocity*u**H* can be easily obtained from
3.2,3.28and 3.35. Similarly, the expressions for*u*_{N}*, τ** _{H}*, and

*τ*

*can be obtained. The expression for wall shear stress*

_{N}*τ*

*can be obtained by evaluating*

_{w}*τ*

*at*

_{N}*r*

*R*and is given below:

*τ*_{w}

*τ*_{0N}*α*^{2}_{N}*τ*_{1N}

*rR* *τ*_{0w}*α*^{2}_{N}*τ*_{1w}

*qzftR*
*α*^{2}_{N}

−1 8

*qzftR*

*BR*^{2} 1−
*R*_{1}

*R*
_{4}

*α*^{2}_{N}

−

*qzftR*1

*n*

2n2n3*BR*^{2}_{1}
*R*_{1}

*R*

×

⎡

⎣*nn*2−*nn*−1n3
*k*^{2}

*R*_{1}

−3

*n*^{2}2n−2*k*^{2}
*R*_{1}

* _{n3}*⎤

⎦

⎫⎬

⎭*.*

3.37

From 2.20 and 3.27, 3.28, 3.29, 3.34, 3.35, and 3.36, the volume flow rate is calculated and is given by

*Q*4

*R*0P

0

*u*0P*α*^{2}_{H}*u*1P

*r dr*

_{R}_{1}

*R*0P

*u*0H*α*^{2}_{H}*u*1H

*r dr*

_{R}

*R*1

*u*0N*α*^{2}*u*1N

*r dr*

4

*qzf*tR
*R*^{3}

1−

*R*_{1}
*R*

_{2}⎡

⎣
*k*^{2}

*R*1

_{2}
1

4

1−
*R*_{1}

*R*
_{2}⎤

⎦

4

*qzftR*1

_{n}*R*^{3}_{1}
n2n3

⎡

⎣n2−*nn*3
*k*^{2}

*R*_{1}

*n*^{2}2n−2*k*^{2}
*R*_{1}

* _{n3}*⎤

⎦

4α^{2}* _{H}* −

*qzftR*
*BR*^{2}*R*^{3}_{1}

3 32

*R*
*R*_{1}

−1 8

*R*1

*R*

1 32

*R*1

*R*
3

1 8

*R*1

*R*
3

log
*R*1

*R*

*qzftR*1

_{n}

*BR*^{5}_{1}log
*R*_{1}

*R*

×

⎧⎨

⎩
*n*

2n3−*nn*−1
2n2

*k*^{2}
*R*_{1}

− 3

*n*^{2}2n−2

2n2n3

*k*^{2}
*R*_{1}

* _{n3}*⎫

⎬

⎭

−*n*

*qzftR*1

_{n}*BR*^{2}*R*^{3}_{1}

1−

*R*_{1}
*R*

_{2}

×

⎧⎨

⎩ 1

4n3− n−1 4n2

*k*^{2}
*R*_{1}

*n*^{2}*n*−5

4n2n3

*k*^{2}
*R*_{1}

* _{n3}*⎫

⎬

⎭

−*n*

*qzftR*1

_{2n−1}
*BR*^{5}_{1}

×

*n*

2n2n3− *nn*−1

4n^{2}12n5
n2n32n12n3

*k*^{2}
*R*_{1}

*nn*−1^{2}

2n1n2

*k*^{2}
*R*1

_{2}

*n*^{3}−2n^{2}−11n6
2n1n2n3

*k*^{2}
*R*1

_{n3}

−n−1

*n*^{3}−2n^{2}−11n6
2nn2n3

*k*^{2}
*R*1

_{n4}

−

4n^{5}14n^{4}−8n^{3}−45n^{2}−3n18
2nn1n2n32n3

*k*^{2}
*R*_{1}

_{2n4}⎫

⎬

⎭

⎤

⎦
4α^{2}_{N}

−

*qzftR*
*BR*^{4}*R*_{1}

× 1

24
*R*

*R*1

− 3 32

*R*_{1}
*R*

5

96
*R*_{1}

*R*
_{5}

− 1 8

*R*_{1}
*R*

_{3}

log*R*_{1}
1−

*R*_{1}
*R*

_{2}

−

*qzftR*1

*n*

*BR*^{2}*R*^{3}_{1}

1−
*R*_{1}

*R*
_{2}

12 log*R*_{1}

×

⎧⎨

⎩
*n*

4n3−*nn*−1
4n2

*k*^{2}
*R*1

− 3

*n*^{2}2n−2

4n2n3

*k*^{2}
*R*1

* _{n3}*⎫

⎬

⎭

⎤

⎦*.* 3.38

The second approximation to plug core radius*R*1P can be obtained by neglecting the terms
with *α*^{4}* _{H}* and higher powers of

*α*

*in3.7in the following manner. The shear stress*

_{H}*τ*

_{H}*τ*

_{0H}

*α*

^{2}

_{H}*τ*

_{1H}at

*rR*

*is given by*

_{P}!!!τ0H*α*^{2}_{H}*τ*1H!!!

*rR**P* *θ.* 3.39

Equation3.39reflects the fact that on the boundary of the plug core region, the shear stress
is the same as the yield stress. Using the Cityplace Taylor’s series of*τ*_{0H}and*τ*_{1H} about*R*_{0P}
and using*τ*0H|_{rR}_{0P} *θ, we get*

*R*_{1P}
1

*qzft*

−τ1H|_{rR}_{0P}

*.* 3.40

With the help of3.7,3.30,3.32, and3.40, the expression for*R** _{P}* can be obtained as

*R*_{P}*k*^{2}

*Bα*^{2}_{H}*R*^{2}
4

*qzftR*
*k*^{2}

*R*

1−
*R*_{1}

*R*
_{2}

*nBα*^{2}_{H}*R*^{2}_{1}
2n1

*qzftR*1

* _{n}*⎧

⎨

⎩
*k*^{2}

*R*_{1}

−

*n*^{2}−1
*n*

*k*^{2}
*R*_{1}

2

−
*k*^{2}

*R*_{1}
* _{n2}*⎫

⎬

⎭*.*

3.41

The resistance to flow in the artery is given by

Λ

*qzft*

*Q* *.* 3.42

When*R*_{1} *R, the present model reduces to the single fluid model*Herschel-Bulkley fluid
modeland in such case, the expressions obtained in the present model for velocity*u**H*, shear
stress*τ**H**,wall shear stressτ**w**,*flow rate*Q, and plug core radiusR**P* are in good agreement
with those of Sankar and Hemalatha2.

**4. Numerical Simulation of Results and Discussion**

The objective of the present model is to understand and bring out the salient features of the
eﬀects of the pulsatility of the flow, nonNewtonian nature of blood, peripheral layer and
stenosis size on various flow quantities. It is generally observed that the typical value of the
power law index*n*for blood flow models is taken to lie between 0.9 and 1.1 and we have used
the typical value of*n*to be 0.95 for*n <*1 and 1.05 for*n >*12. Since the value of yield stress
is 0.04 dyne/cm^{2} for blood at a haematocrit of 4035, the nonNewtonian eﬀects are more
pronounced as the yield stress value increases, in particular, when it flows through narrow
blood vessels. In diseased state, the value of yield stress is quite highalmost five times 28.

In this study, we have used the range from 0.1 to 0.3 for the nondimensional yield stress*θ.*

To compare the present results with the earlier results, we have used the yield stress value as

0.01 and 0.04. Though the range of the amplitude A varies from 0 to 1, we use the range from 0.1 to 0.5 to pronounce its eﬀect.

The ratio *α* *α*_{N}*/α** _{H}*between the pulsatile Reynolds numbers of the Newtonian
fluid and Herschel-Bulkley fluid is called pulsatile Reynolds number ratio. Though the
pulsatile Reynolds number ratio

*α*ranges from 0 to 1; it is appropriate to assume its value as 0.5 25. Although the pulsatile Reynolds number

*α*

*of the Herschel-Bulkley fluid also ranges from 0 to 1 2, the values 0.5 and 0.25 are used to analyze its eﬀect on the flow quantities. Given the values of*

_{H}*α*and

*α*

*H*, the value of

*α*

*N*can be obtained from

*α*

*α*

*N*

*/α*

*H*. The value of the ratio

*β*of central core radius

*βR*0 to the normal artery radius

*R*0 in the unobstructed artery is generally taken as 0.95 and 0.98525. Following Shukla et al.26, we have used the relations

*R*

_{1}

*βR*and

*δ*

_{C}*βδ*

*to estimate*

_{P}*R*

_{1}and

*δ*

*. The maximum thickness of the stenosis in the peripheral region*

_{C}*δ*

*P*is taken in the range from 0.1 to 0.1525.

To compare the present results with the results of Sankar and Hemalatha2for single fluid
model, we have used the value 0.2 for*δ** _{C}*. To deduce the present model to a single fluid model
Newtonian fluid model or Herschel-Bulkley fluid modeland to compare the results with
earlier results, we have used the value of

*β*as 1.

It is observed that in3.38,*ft, R, andθ*are known and*Q*and*qz*are the unknowns
to be determined. A careful analysis of3.38reveals the fact that*qz*is the pressure gradient
of the steady flow. Thus, if steady flow is assumed, then3.38can be solved for*qz 2,*10.

For steady flow,3.38reduces to

*R*^{2}−*R*^{2}_{1} 4θ^{2}
*R*

*R*_{1}
2

*R*^{2}−*R*^{2}_{1}
*x*^{3}

4 n2n3

×"

n2R^{n3}_{1} *x** ^{n3}*−

*nn*3θR

^{n2}_{1}

*x*

^{n2}*n*^{2}2n−2
*θ** ^{n3}*#

−*Q**S**x*^{3}0,

4.1

where *x* *qz* and *Q** _{S}* is the steady state flow rate. Equation4.1 can be solved for

*x*numerically for a given value of

*n,*

*Q*

*and*

_{S}*θ. Equation*4.1 has been solved numerically for

*x*using Newton-Raphson method with variation in the axial direction and yield stress with

*β*0.95 and

*δ*

*0.1. Throughout the analysis, the steady flow rate*

_{P}*Q*

*value is taken as 1.0. Only that root which gives the realistic value for plug core radius has been considered there are only two real roots in the range from 0 to 20 and the other root gives values of plug core radius that exceeds the tube radius*

_{S}*R.*

**4.1. Pressure Gradient**

The variation of pressure gradient with axial distance for diﬀerent fluid models in the core
region is shown inFigure 2. It has been observed that the pressure gradient for the Newtonian
fluidsingle fluid modelis lower than that of the two fluid models with*n* 1.05 and*θ*
0.1 from*z* 4 to 4.5 and *z* 5.5 to 6, and higher than that of the two fluid models from
*z*4.5 to*z* 5.5 and these ranges are changed with increase in the value of the yield stress
*θ* and a decrease in the value of the power law index*n. The plot for the Newtonian fluid*
modelsingle phase fluid modelis in good agreement with that in Figure 2 of Sankar and
Hemalatha 2. Figure 2 depicts the eﬀects of nonNewtonian nature of blood on pressure
gradient.

0 0.5 1 1.5 2 2.5 3

Pressuregradient*q**z*

4 4.5 5 5.5 6

Axial distance*z*
*n*0.95, θ0.1
*n*0.95, θ0.2

*n*1.05, θ0.1

Power law fluid with*n*0.95
Newtonian fluidsingle

fluid model with*β*1

**Figure 2: Variation of pressure gradient with axial direction for diﬀerent fluids in the core region with**
*δ**P* 0.1.

25 20 15 10 5 0

0 30 60 90 120 150 180 210 240 270 300 330 360
Time*t*^{◦}

Pressuredrop∆*p*

*A*=0.5,*θ*=0.1,*δ**P*=0.1

*A*=0.2,*θ*=0.1,*δ**P*=0.1

*A*=0.5,*θ*=0.15,*δ**P*=0.15
*A*=0.5,*θ*=0.15,*δ**P*=0.1

**Figure 3: Variation of pressure drop in a time cycle for diﬀerent values of***A,θ*and*δ**P*with*nβ*0.95.

**4.2. Pressure Drop**

The variation of pressure dropΔp across the stenosis, i.e., from*z* 4 to*z* 6in a time
cycle for diﬀerent values of*A,θ, andδ**P* with*n* *β*0.95 is depicted inFigure 3. It is clear
that the pressure drop increases as time*t*increases from 0^{◦}to 90^{◦}and then decreases from 90^{◦}
to 270^{◦} and again it increases from 270^{◦} to 360^{◦}. The pressure drop is maximum at 90^{◦}and
minimum at 270^{◦}. It is also observed that for a given value of*A, the pressure drop increases*
with the increase of the stenosis height*δ** _{P}* or yield stress

*θ*when the other parameters held

*constant. Further, it is noticed that as the amplitude A increases, the pressure drop increases*when

*t*lies between 0

^{◦}and 180

^{◦}and decreases when

*t*lies between 180

^{◦}and 360

^{◦}while

*θ*and

*δ*

*are held fixed. Figure 3 shows the simultaneous eﬀects of the stenosis size and nonNewtonian nature of blood on pressure drop.*

_{P}**4.3. Plug Core Radius**

The variation of plug core radiusR*P*with axial distance for diﬀerent values of the amplitude
A and stenosis thickness*δ**P* in the peripheral layerwith*nβ*0.95,*α**H* 0.5,*θ*0.1, and
*t*60^{◦}is shown inFigure 4. It is noted that the plug core radius decreases as the axial variable
*zvaries from 4 to 5 and it increases as z varies from 5 to 6. It is further observed that for a*
given value of*δ**P*, the plug core radius decreases with the increase of the amplitude*A*and the
same behavior is noted as the peripheral layer stenosis thickness increases for a given value

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

4 4.5 5 5.5 6

Axial distance*z*
Plugcoreradius*R**P*

*A*=0.2,*δ**P*=0.1

*A*=0.5,*δ**P*=0.1 *A*=0.5,*δ**P*=0.15

**Figure 4: Variation of plug core radius with axial distance for diﬀerent values of A and***δ**P*with*nβ*0.95,
*α**H*0.5,*θ*0.1 and*t*60^{◦}.

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0 30 60 90 120 150 180 210 240 270 300 330 360

Plugcoreradius*R**P* *α**H*=0.5,*θ*=0.15

*α**H*=0.5,*θ*=0.1

*α**H*=0.1,*θ*=0.1

Time*t*^{◦}

**Figure 5: Variation of plug core radius in a time cycle for diﬀ**erent values of*α**H* and*θ*with*nβ*0.95,
*δ**P* 0.1,*A*0.5,*t*60^{◦}and*z*5.

of the amplitude*A.*Figure 4depicts the eﬀects of stenosis height on the plug core radius of
the blood vessels.

Figure 5sketches the variation of plug core radius in a time cycle for diﬀerent values
of the pulsatile Reynolds number *α** _{H}* of the Herschel-Bulkley fluid and yield stress

*θ*with

*nβ*0.95,

*A*0.5,

*z*5,

*t*60

^{◦}, and

*δ*

*0.1. It is noted that the plug core radius decreases as time*

_{P}*t*increases from 0

^{◦}to 90

^{◦}and then it increases from 90

^{◦}to 270

^{◦}and then again it decreases from 270

^{◦}to 360

^{◦}. The plug core radius is minimum at

*t*90

^{◦}and maximum at

*t*270

^{◦}. It has been observed that for a given value of the pulsatile Reynolds number

*α*

*, the plug core radius increases as the yield stress*

_{H}*θ*increases. Also, it is noticed that for a given value of yield stress

*θ*and with increasing values of the pulsatile Reynolds number

*α*

*, the plug core radius increases when*

_{H}*t*lies between 0

^{◦}and 90

^{◦}and also between 270

^{◦}and 360

^{◦}and decreases when

*t*lies between 90

^{◦}and 270

^{◦}.Figure 5depicts the simultaneous eﬀects of the pulsatility of the flow and the nonNewtonian nature of the blood on the plug core radius of the two-phase model.

**4.4. Wall Shear Stress**

Wall shear stress is an important parameter in the studies of the blood flow through arterial stenosis. Accurate predictions of wall shear stress distributions are particularly useful in the

0
0.5
1
1*.*5
2
2*.*5
3

Wallshearstress*τ**w*

4 4*.*5 5 5*.*5 6

Axial distance*z*
*θ*0.1,*α**N*0.1

*θ*0*.*1,*α**N*0*.*8

*θ*0*.*04,*α**N*0*.*5,*β*1
single fluid model

*θ*0.2,*α**N*0.8

**Figure 6: Variation of wall shear stress with axial distance for diﬀerent values***θ*and*α**N*with*t* 45^{◦},
*nβ*0.95,*A*0.5 and*δ**P*0.1.

3.5 3 2.5 2 1.5 1 0.5 0

0 30 60 90 120 150 180 210 240 270 300 330 360
Wallshearstress*τ**w*

*A*=0.2,*δ**P*=0.1

*A*=0.5,*δ**P*=0.1
*A*=0.5,*δ**P*=0.15

Time*t*^{◦}

**Figure 7: Variation of wall shear stress in a time cycle for diﬀ**erent values of*A*and*δ**P* with*θ* 0.1,
*nβ*0.95,*z*5 and*α**N*0.5.

understanding of the eﬀects of blood flow on the endothelial cells36, 37. The variation
of wall shear stress in the axial direction for diﬀerent values of yield stress*θ* and pulsatile
Reynolds number*α** _{N}*of the Newtonian fluid with

*t*45

^{◦},

*nβ*0.95,

*A*0.5, and

*δ*

*0.1 is plotted inFigure 6. It is found that the wall shear stress increases as the axial variable*

_{P}*z*increases from 4 to 5 and then it decreases symmetrically as

*z*increases further from 5 to 6. For a given value of the pulsatile Reynolds number

*α*

*, the wall shear stress increases considerably with the increase in the values of the yield stress*

_{N}*θ*when the other parameters held constant. Also, it is noticed that for a given value of the yield stress

*θ*and increasing values of the pulsatile Reynolds number

*α*

*, the wall shear stress decreases slightly while the other parameters are kept as invariables. It is of interest to note that the plot for the single fluid Herschel-Bulkley model is in good agreement with that in Figure 8 of Sankar and Hemalatha 2.Figure 6shows the eﬀects of pulsatility of the blood flow and nonNewtonian eﬀects of the blood on the wall shear stress of the two-phase model.*

_{N}Figure 7depicts the variation of wall shear stress in a time cycle for diﬀerent values
of the amplitude*A*and peripheral stenosis height*δ** _{P}* with

*n*

*β*0.95,

*θ*0.1,

*α*

*0.5 and*

_{N}*z*5. It can be easily seen that the wall shear stress increases as time

*t*in degrees increases from 0

^{◦}to 90

^{◦}and then it decreases as

*t*increases from 90

^{◦}to 270

^{◦}and then again it increases as

*t*increases further from 270

^{◦}to 360

^{◦}. The wall shear stress is maximum at 90

^{◦}and