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

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 ratesγ <˙ 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 τy. 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 R1z

R0 βR₀ δp

RP

z Newtonian fluid

Herschel-Bulkley fluid Plug flow

μ_H, u_H δ_C μ_N, uN

d L0

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 thezdirection 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

∂

∂rr τH in 0≤r ≤R₁z, 2.1

ρ_N∂uN

∂t −∂p

∂z −1 r

∂

∂rr τN inR1z≤r≤Rz, 2.2

0−∂p

∂r, 2.3

where the shear stress τ |τr z| −τr z sinceτ τ_H orτ τ_N. 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

τH ⁿ

μ_H ∂uH

∂r

τy ifτH≥τy, Rp≤r ≤R1z, 2.4

∂uH

∂r 0 ifτH≤τy, 0≤r≤Rp, 2.5 τN μ_N

−∂uN

∂r

ifR1z≤r ≤Rz, 2.6

whereuH,uNare the axial component of the fluid’s velocity in the core region and peripheral region; τ_H, τ_N are the shear stress of the Herschel-Bulkley fluid and Newtonian fluid;

μ_H, μ_Nare the viscosities of the Herschel-Bulkley fluid and Newtonian fluid with respective dimensions ML⁻¹T⁻²ⁿT and ML⁻¹T⁻¹;ρ_H, ρ_N are the densities of the Herschel-Bulkley fluid and Newtonian fluid;pis the pressure,t; is the time;τ_y 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 wheneverτ_H≤τ_y. However, the fluid behavior is indicated wheneverτ_H≥τ_y. The geometry of the stenosis in the peripheral region as shown inFigure 1is given by

Rz

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

R0 in the normal artery region,

R₀−δ_p

2 1cos2π L₀

z−d− L₀ 2

ind≤z≤dL₀,

2.7

whereRzis the radius of the stenosed artery with peripheral layer,R₀ is the radius of the normal artery,L0is the length of the stenosis,dindicates its location, andδpis the maximum depth of the stenosis in the peripheral layer such that δP/R₀ 1. The geometry of the

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

R1z

⎧⎪

⎨

⎪⎩

βR₀ in the normal artery region,

βR0−δC

2 1cos2π L0

z−d− L0

2

ind≤z≤dL0, 2.8

whereR1zis the radius of the stenosed core region of the artery,βis the ratio of the central core radius to the normal artery radius, βR₀ is the radius of the core region of the normal artery, andδCis the maximum depth of the stenosis in the core region such thatδC/R01.

The boundary conditions are

i τ_H is finite and ∂u_H

∂r 0 atr0, ii τ_Hτ_N atr R₁z,

iii uHuN atr R1z, iv uN0 atrRz.

2.9

Since the pressure gradient is a function ofzandt, we take

−∂p

∂z qzf t

, 2.10

whereqz −∂p/∂zz,0,ft 1Asinωt,Ais the amplitude of the flow andωis 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 chooseft 1Asinωt as a good approximation. We introduce the following nondimensional variables

z z

R₀, Rz Rz

R₀ , R₁z R₁z

R₀ , r r

R₀, tωt, d d

R₀, L₀ L₀ R₀, qz qz

q₀ , u_H u_H q₀R²₀/4μ₀

, u_N u_N q₀R²₀/4μ_N

, τ_H τ_H

q₀R₀/2, τ_N τ_N q₀R₀/2,

θ τy

q₀R0/2, α²_H R²₀ω ρ_H

μ₀ , α²_N R²₀ω ρ_N

μ_N , Rp Rp

R0

, δp δp

R0

, δC δC

R0

,

2.11

where μ₀ μ_H2/q₀R₀ⁿ⁻¹, having the dimension as that of the Newtonian fluid’s viscosity,q₀is the negative of the pressure gradient in the normal artery,αH is the pulsatile Reynolds number or generalized Wormersly frequency parameter and whenn 1, we get

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

α²_H∂uH

∂t 4qzft−2 r

∂

∂rrτH if 0≤r≤R1z, 2.12 α²_N∂uN

∂t 4qzft−2 r

∂

∂rrτN if R1z≤r≤Rz, 2.13 τH ⁿ

−1 2

∂uH

∂r θ ifτH≥θ, Rp≤r ≤R1z, 2.14

∂uH

∂r 0 ifτH≤θ, 0≤r≤Rp, 2.15 τN −1

2

∂uN

∂r ifR1z≤r≤Rz, 2.16

whereft 1Asint. The boundary conditionsin dimensionless formare i τ_H is finite atr0,

ii ∂u_H

∂r 0 atr 0, iii τ_Hτ_N atr R₁z, iv u_Hu_N atrR₁z,

v uN0 atrRz.

2.17

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

Rz

⎧⎪

⎨

⎪⎩

1 in the normal artery region,

1−δp

2

1cos 2π L0

z−d− L₀ 2

ind≤z≤dL₀. 2.18

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

R1z

⎧⎨

⎩

β in the normal artery region,

β−δC

2

1cos 2π L₀

z−d−L0

2

ind≤z≤dL₀. 2.19

The nondimensional volume flow rateQis given by

Q4 _Rz

0

ur, z, tr dr, 2.20

whereQQ/πR⁴₀q₀/8μ₀,Qis the volume flow rate.

3. Method of Solution

When we nondimensionalize the constitutive equations2.1 and 2.2,α²_H and α²_N occur naturally and these pulsatile Reynolds numbers are time dependent and hence, it is more appropriate to expand2.12–2.16aboutα²_Handα²_N. The plug core velocityup, the velocity in the core regionuH, the velocity in the peripheral regionuN, the plug core shear stressτp, the shear stress in the core regionτ_H, the shear stress in the peripheral regionτ_N, and the plug core radiusRpare expanded as follows in terms of α²_H andα²_N whereα²_H 1 and α²_N1:

u_Pz, t u_0Pz, t α²_Hu_1Pz, t · · · , 3.1 uHr, z, t u0Hr, z, t α²_Hu1Hr, z, t · · · , 3.2 u_Nr, z, t u_0Nr, z, t α²_Nu_1Nr, z, t · · ·, 3.3 τPz, t τ0Pz, t α²_Hτ1Pz, t · · · , 3.4 τHr, z, t τ0Hr, z, t α²_Hτ1Hr, z, t · · ·, 3.5 τ_Nr, z, t τ_0Nr, z, t α²_Nτ_1Nr, z, t · · ·, 3.6 RPz, t R0Pz, t α²_HR1Pz, t · · ·. 3.7

Substituting3.2, 3.5 in 2.12 and then equating the constant terms andα²_H terms, we obtain

∂

∂rrτ0H 2qzftr, 3.8

∂u_0H

∂t −2 r

∂

∂rrτ1H. 3.9

Applying3.2,3.5in2.14and then equating the constant terms andα²_Hterms, one can get

−∂u0H

∂r 2τ_0Hⁿ⁻¹τ0H−nθ, 3.10

−∂u1H

∂r 2nτ_0Hⁿ⁻²τ1Hτ0H−n−1θ. 3.11

Using3.3and3.6in2.13and then equating the constant terms andα²_Nterms, we get

∂

∂rrτ0N 2qzftr, 3.12

∂u0N

∂t −2 r

∂

∂rrτ1N. 3.13

On substituting3.3and3.6in2.16and then equating the constant terms andα²_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 andα²_Handα²_N terms, the boundary conditions are simplified, respectively, to

τ0P, τ1P are finite atr 0, 3.16

∂u0P

∂r 0, ∂u1P

∂r 0 atr 0, 3.17

τ0Hτ0N atr R1z, 3.18

τ_1Hτ_1N atr R₁z, 3.19

u_0Hu_0N atr R₁z, 3.20

u1Hu1N atr R1z, 3.21

u_0N 0 atrRz, 3.22

u_1N0 atrRz. 3.23

Equations3.8–3.11and3.12–3.15are the system of differential equations which can be solved for the unknownsu_0H, u_1H, τ_0H, τ_1Handu_0N, u_1N, τ_0N, τ_1N, respectively, with the help of boundary conditions3.16–3.23. Integrating3.8between 0 andR0P and applying the boundary condition3.16, we get

τ_0P qzftR0P. 3.24

Integrating3.8betweenR_0P andrand then making use of3.24, we get

τ0Hqzftr. 3.25

Integrating3.12betweenR₁andrand then using3.18, one can get

τ0Nqzftr. 3.26

Integrating3.14betweenrandRand then making use of3.22, we obtain

u_0N qzftR²

1−r R

2

. 3.27

Integrating3.10betweenrandR1and using the boundary condition3.20, we get

u0H

qzftR R

1−

R₁ R

₂

2

qzftR1

_n R1

1 n1

1−

r R1

_n1

− k² R1

1−

r R1

_n ,

3.28

wherek² θ/qzft. The plug core velocityu0Pcan be obtained from3.28by replacing rbyR_0P as

u_0P

qzftR R

1−

R1

R ₂

2

qzftR1

_n R1

1 n1

1−

R0p

R1

n1

− k² R1

1−

R0p

R1

n .

3.29

Neglecting the terms with α²_H and higher powers of αH in 3.7 and using 3.24, the expression forR0P is obtained as

r|_τ0PθR_0P θ

qzft

k². 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 forτ_1P, τ_1H, τ_1N, u_1H, andu_1P can be obtained as

τ1P −1 4

qzftR BR²

k² R

1−

R1

R 2

−

qzftR1

n

BR²₁

⎡

⎣ n 2n1

k² R1

−n−1 2

k² R1

₂

− n

2n1 k²

R1

_n2⎤

⎦,

3.31

τ1H−1 4

qzftR BR²r

R

1− R1

R 2

−

qzftR1

_n BR²₁

× n

n1n3

n3 2

r R1

− r

R1

_n2

−n−1 n2

k² R1

n2 2

r R1

− r

R1

_n1

− 3

n²2n−2

2n2n3

k² R₁

_n3 R1

r

⎤⎦,

3.32

τ_1N−

qzftR

BRR₁ 1 4

r R1

−1 8

R₁ R

₂ R₁

r

−1 8

R₁ R

₂ r R1

₃

−

qzftR1

_n

BR²₁ n 2n3

R₁ r

−nn−1 2n2

k² R1

R₁ r

− 3

n²2n−2

2n2n3

k² R₁

_n3 R1

r

⎤⎦,

3.33

u_1N−2

qzftR BR²R₁

1 8

R R1

1−r

R ₂

−1 8

R₁ R

₃ log

R r

− 1 32

R R₁

1−r

R 4

−2

qzftR1

_n

BR³₁log R

r

n

2n3− nn−1 2n2

k² R1

− 3

n²2n−2

2n2n3

k² R1

_n3⎤

⎦,

3.34

u1H−2

qzftR

BR²R1 3 32

R R₁

−1 8

R1

R

1 32

R1

R 3

1 8

R1

R 3

log R1

R

2

qzftR1

n

BR³₁log R₁

R

×

⎡

⎣ n

2n3−nn−1 2n2

k² R₁

− 3

n²2n−2

2n2n3

k² R₁

_n3⎤

⎦

−n

qzftR1

_n BR1R²

1−

R1

R 2

× 1 2n1

1−

r R₁

_n1

−n−1 2n

k² R₁

1−

r R₁

n

−2n

qzftR1

_2n−1 BR³₁

× n

2n1²

1− r

R_{1 n1}

− n−1 2n1

k² R₁

1−

r R₁

_n

− n

2n1²n3

1− r

R1

_2n2

n−1

2n²6n3 n1n2n32n1

k² R₁

1−

r R₁

_2n1

− n−1 2n1

k² R1

1−

r R1

_n1

n−1² 2n

k² R1

₂ 1−

r R1

n

− n−1² 2nn2

k² R1

₂ 1−

r R1

_2n

− 3

n²2n−2 2n−1n2n3

k² R₁

_n3 1−

r R₁

_n−1

3n−1

n²2n−2 2n−2n2n3

k² R₁

_n4 1−

r R₁

_n−2⎤

⎦,

3.35 u1P −2

qzftR

BR²R1 3 32

R R₁

−1 8

R1

R

1 32

R1

R 3

1 8

R1

R 3

log R1

R

2

qzftR1

n

BR³₁log R₁

R

×

⎡

⎣ n

2n3−nn−1 2n2

k² R₁

− 3

n²2n−2

2n2n3

k² R₁

_n3⎤

⎦

−n

qzftR1

_n BR1R²

1−

R1

R 2

×

⎡

⎣ 1 2n1

⎧⎨

⎩1−

k² R₁

_n1⎫

⎬

⎭−n−1 2n

k² R₁

1−

k² R₁

_n⎤

⎦−2n

qzftR1

_2n−1 BR³₁

×

⎡

⎣ n 2n1²

⎧⎨

⎩1− k²

R1

_n1⎫

⎬

⎭− n−1 2n1

k² R1

1−

k² R1

_n

− n 2n1²n3

⎧⎨

⎩1− k²

R1

_2n2⎫

⎬

⎭

n−1

2n²6n3 n1n2n32n1

k² R₁

⎧⎨

⎩1− k²

R₁

_2n1⎫

⎬

⎭

− n−1 2n1

k² R1

⎧⎨

⎩1− k²

R1

_n1⎫

⎬

⎭n−1² 2n

k² R1

₂ 1−

k² R1

_n

− n−1² 2nn2

k² R₁

2⎧

⎨

⎩1− k²

R₁ 2n⎫

⎬

⎭

− 3

n²2n−2 2n−1n2n3

k² R1

_n3⎧

⎨

⎩1− k²

R1

_n−1⎫

⎬

⎭

3n−1

n²2n−2 2n−2n2n3

k² R₁

_n4⎧

⎨

⎩1− k²

R_{1 n−2}⎫

⎬

⎭

⎤

⎦,

3.36 whereB 1/ftdft/dt. The expression for velocityuH can be easily obtained from 3.2,3.28and 3.35. Similarly, the expressions foru_N, τ_H, and τ_N can be obtained. The expression for wall shear stressτ_wcan be obtained by evaluatingτ_N atr Rand is given below:

τ_w

τ_0Nα²_Nτ_1N

rR τ_0wα²_Nτ_1w

qzftR α²_N

−1 8

qzftR

BR² 1− R₁

R ₄

α²_N

−

qzftR1

n

2n2n3BR²₁ R₁

R

×

⎡

⎣nn2−nn−1n3 k²

R₁

−3

n²2n−2k² R₁

_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

Q4

R0P

0

u0Pα²_Hu1P

r dr

_R1

R0P

u0Hα²_Hu1H

r dr

_R

R1

u0Nα²u1N

r dr

4

qzftR R³

1−

R₁ R

₂⎡

⎣ k²

R1

₂ 1

4

1− R₁

R ₂⎤

⎦

4

qzftR1

_n R³₁ n2n3

⎡

⎣n2−nn3 k²

R₁

n²2n−2k² R₁

_n3⎤

⎦

4α²_H −

qzftR BR²R³₁

3 32

R R₁

−1 8

R1

R

1 32

R1

R 3

1 8

R1

R 3

log R1

R

qzftR1

_n

BR⁵₁log R₁

R

×

⎧⎨

⎩ n

2n3−nn−1 2n2

k² R₁

− 3

n²2n−2

2n2n3

k² R₁

_n3⎫

⎬

⎭

−n

qzftR1

_n BR²R³₁

1−

R₁ R

₂

×

⎧⎨

⎩ 1

4n3− n−1 4n2

k² R₁

n²n−5

4n2n3

k² R₁

_n3⎫

⎬

⎭

−n

qzftR1

_2n−1 BR⁵₁

×

n

2n2n3− nn−1

4n²12n5 n2n32n12n3

k² R₁

nn−1²

2n1n2

k² R1

₂

n³−2n²−11n6 2n1n2n3

k² R1

_n3

−n−1

n³−2n²−11n6 2nn2n3

k² R1

_n4

−

4n⁵14n⁴−8n³−45n²−3n18 2nn1n2n32n3

k² R₁

_2n4⎫

⎬

⎭

⎤

⎦ 4α²_N

−

qzftR BR⁴R₁

× 1

24 R

R1

− 3 32

R₁ R

5

96 R₁

R ₅

− 1 8

R₁ R

₃

logR₁ 1−

R₁ R

₂

−

qzftR1

n

BR²R³₁

1− R₁

R ₂

12 logR₁

×

⎧⎨

⎩ n

4n3−nn−1 4n2

k² R1

− 3

n²2n−2

4n2n3

k² R1

_n3⎫

⎬

⎭

⎤

⎦. 3.38

The second approximation to plug core radiusR1P can be obtained by neglecting the terms with α⁴_H and higher powers of α_H in3.7in the following manner. The shear stressτ_H τ_0Hα²_Hτ_1HatrR_Pis given by

!!!τ0Hα²_Hτ1H!!!

rRP 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 ofτ_0Handτ_1H aboutR_0P and usingτ0H|_rR0P θ, we get

R_1P 1

qzft

−τ1H|_rR0P

. 3.40

With the help of3.7,3.30,3.32, and3.40, the expression forR_P can be obtained as

R_P k²

Bα²_HR² 4

qzftR k²

R

1− R₁

R ₂

nBα²_HR²₁ 2n1

qzftR1

_n⎧

⎨

⎩ k²

R₁

−

n²−1 n

k² R₁

2

− k²

R_{1 n2}⎫

⎬

⎭.

3.41

The resistance to flow in the artery is given by

Λ

qzft

Q . 3.42

WhenR₁ R, the present model reduces to the single fluid modelHerschel-Bulkley fluid modeland in such case, the expressions obtained in the present model for velocityuH, shear stressτH,wall shear stressτw,flow rateQ, and plug core radiusRP 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 indexnfor blood flow models is taken to lie between 0.9 and 1.1 and we have used the typical value ofnto be 0.95 forn <1 and 1.05 forn >12. Since the value of yield stress is 0.04 dyne/cm² 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 stressθ.

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_Hbetween the pulsatile Reynolds numbers of the Newtonian fluid and Herschel-Bulkley fluid is called pulsatile Reynolds number ratio. Though the pulsatile Reynolds number ratioαranges from 0 to 1; it is appropriate to assume its value as 0.5 25. Although the pulsatile Reynolds numberα_H 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 ofαandαH, the value ofαNcan be obtained fromα αN/αH. The value of the ratio β of central core radius βR0 to the normal artery radiusR0 in the unobstructed artery is generally taken as 0.95 and 0.98525. Following Shukla et al.26, we have used the relationsR₁ βRand δ_C βδ_P to estimate R₁ andδ_C. The maximum thickness of the stenosis in the peripheral regionδP 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 forδ_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 ofβas 1.

It is observed that in3.38,ft, R, andθare known andQandqzare the unknowns to be determined. A careful analysis of3.38reveals the fact thatqzis the pressure gradient of the steady flow. Thus, if steady flow is assumed, then3.38can be solved forqz 2,10.

For steady flow,3.38reduces to

R²−R²₁ 4θ² R

R₁ 2

R²−R²₁ x³

4 n2n3

×"

n2Rⁿ³₁ xⁿ³−nn3θRⁿ²₁ xⁿ²

n²2n−2 θⁿ³#

−QSx³0,

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_Sand θ. Equation 4.1 has been solved numerically forxusing Newton-Raphson method with variation in the axial direction and yield stress withβ 0.95 andδ_P 0.1. Throughout the analysis, the steady flow rateQ_Svalue 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 radiusR.

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 withn 1.05 andθ 0.1 fromz 4 to 4.5 and z 5.5 to 6, and higher than that of the two fluid models from z4.5 toz 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 indexn. 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

Pressuregradientqz

4 4.5 5 5.5 6

Axial distancez n0.95, θ0.1 n0.95, θ0.2

n1.05, θ0.1

Power law fluid withn0.95 Newtonian fluidsingle

fluid model withβ1

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

25 20 15 10 5 0

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

Pressuredrop∆p

A=0.5,θ=0.1,δP=0.1

A=0.2,θ=0.1,δP=0.1

A=0.5,θ=0.15,δP=0.15 A=0.5,θ=0.15,δP=0.1

Figure 3: Variation of pressure drop in a time cycle for different values ofA,θandδPwithnβ0.95.

4.2. Pressure Drop

The variation of pressure dropΔp across the stenosis, i.e., fromz 4 toz 6in a time cycle for different values ofA,θ, andδP withn β0.95 is depicted inFigure 3. It is clear that the pressure drop increases as timetincreases 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 ofA, the pressure drop increases with the increase of the stenosis heightδ_P or yield stressθwhen 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_P are held fixed. Figure 3 shows the simultaneous effects of the stenosis size and nonNewtonian nature of blood on pressure drop.

4.3. Plug Core Radius

The variation of plug core radiusRPwith axial distance for different values of the amplitude A and stenosis thicknessδP in the peripheral layerwithnβ0.95,αH 0.5,θ0.1, and t60^◦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 ofδP, the plug core radius decreases with the increase of the amplitudeAand 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 distancez PlugcoreradiusRP

A=0.2,δP=0.1

A=0.5,δP=0.1 A=0.5,δP=0.15

Figure 4: Variation of plug core radius with axial distance for different values of A andδPwithnβ0.95, αH0.5,θ0.1 andt60^◦.

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

PlugcoreradiusRP αH=0.5,θ=0.15

αH=0.5,θ=0.1

αH=0.1,θ=0.1

Timet^◦

Figure 5: Variation of plug core radius in a time cycle for different values ofαH andθwithnβ0.95, δP 0.1,A0.5,t60^◦andz5.

of the amplitudeA.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 θwith nβ0.95,A0.5,z5,t60^◦, andδ_P 0.1. It is noted that the plug core radius decreases as timet 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 t270^◦. It has been observed that for a given value of the pulsatile Reynolds numberα_H, the plug core radius increases as the yield stressθincreases. Also, it is noticed that for a given value of yield stressθand with increasing values of the pulsatile Reynolds numberα_H, the plug core radius increases whentlies between 0^◦and 90^◦and also between 270^◦and 360^◦and decreases whentlies 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

Wallshearstressτw

4 4.5 5 5.5 6

Axial distancez θ0.1,αN0.1

θ0.1,αN0.8

θ0.04,αN0.5,β1 single fluid model

θ0.2,αN0.8

Figure 6: Variation of wall shear stress with axial distance for different valuesθandαNwitht 45^◦, nβ0.95,A0.5 andδP0.1.

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 Wallshearstressτw

A=0.2,δP=0.1

A=0.5,δP=0.1 A=0.5,δP=0.15

Timet^◦

Figure 7: Variation of wall shear stress in a time cycle for different values ofAandδP withθ 0.1, nβ0.95,z5 andαN0.5.

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 stressθ and pulsatile Reynolds numberα_Nof the Newtonian fluid witht45^◦,nβ0.95,A0.5, andδ_P 0.1 is plotted inFigure 6. It is found that the wall shear stress increases as the axial variable z increases from 4 to 5 and then it decreases symmetrically asz increases further from 5 to 6. For a given value of the pulsatile Reynolds numberα_N, the wall shear stress increases considerably with the increase in the values of the yield stressθwhen the other parameters held constant. Also, it is noticed that for a given value of the yield stressθ and increasing values of the pulsatile Reynolds numberα_N, 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.

Figure 7depicts the variation of wall shear stress in a time cycle for different values of the amplitudeAand peripheral stenosis heightδ_P withn β 0.95,θ 0.1,α_N 0.5 and z 5. It can be easily seen that the wall shear stress increases as time tin degrees increases from 0^◦to 90^◦and then it decreases astincreases from 90^◦to 270^◦and then again it increases astincreases further from 270^◦to 360^◦. The wall shear stress is maximum at 90^◦and