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

Authors: D. S. Sankar
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

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.


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## Body

## 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 vessel [2–10]. It has been reported that the hydrodynamic factors play an important role in the formation of stenosis [11, 12] and 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 steady [13–16]. 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 fluid [3, 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 (γ̇<10/scc) in small diameter arteries (0.02 mm–0.1 mm); particularly, in diseased state, the actual flow is distinctly pulsatile [2, 20–25]. Several attempts have been made to study the nonNewtonian behavior and pulsatile flow of blood through stenosed tubes [2, 4, 9, 10, 26–28].Bugliarello and Sevilla [29] and Cokelet [30] have shown experimentally that for blood flowing through narrow blood vessels, there is an outer phase (peripheral layer) of plasma (Newtonian fluid) and an inner phase (core region) of 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 region (inner phase) containing all the erythrocytes as a nonNewtonian fluid and a peripheral layer (outer 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 fluid [25, 26, 31–33]. Srivastava and Saxena [25] have analyzed a two-phase fluid model for blood flow through stenosed arteries treating the suspension of all the erythrocytes in the core region (inner phase) as a Casson fluid and the plasma in the peripheral layer (outer phase) is 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 Samy [28] and Sankar and Hemalatha [2] have 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 (n) and yield index (τ̅y). 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 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 arteries (of diameter 0.02 mm–0.1 mm) at low-shear rates (γ̇<10/sec) treating the fluid in the core region (inner phase) as a Herschel-Bulkley fluid and the plasma in the peripheral region (outer phase) as a Newtonian fluid.In this study, the effects of the pulsatility, stenosis, peripheral layer and the nonNewtonian behavior of blood are analyzed using an analytical solution. Section2 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 4 analyses 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 incompressible) in thez̅ 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 phase (core region) of suspension of all erythrocytes as a Herschel-Bulkley fluid and an outer phase (peripheral layer) of plasma as a Newtonian fluid. The geometry of the stenosis is shown in Figure 1. We have used the cylindrical polar coordinates (r̅,ϕ̅,z̅) whose origin is located on the vessel (stenosed 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(2.1)ρ̅H∂u̅H∂t̅=-∂p̅∂z̅-1r̅∂∂r̅(r̅τ̅H)in0≤r̅≤R̅1(z̅),(2.2)ρ̅N∂u̅N∂t̅=-∂p̅∂z̅-1r̅∂∂r̅(r̅τ̅N)inR̅1(z̅)≤r̅≤R̅(z̅),(2.3)0=-∂p̅∂r̅,
where the shear stress τ̅=|τ̅r̅z̅|=-τ̅r̅z̅(sinceτ̅=τ̅Horτ̅=τ̅N). 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 in (2.4) and (2.5). The relations between the shear stress and the strain rate of the fluids in motion in the core region (for Herschel-Bulkley fluid) and in the peripheral region (for Newtonian fluid) are given by(2.4)τ̅H=μ̅H(∂u̅H∂r̅)n+τ̅yifτ̅H≥τ̅y,R̅p≤r̅≤R̅1(z̅),(2.5)∂u̅H∂r̅=0ifτ̅H≤τ̅y,0≤r̅≤R̅p,(2.6)τ̅N=μ̅N(-∂u̅N∂r̅)ifR̅1(z̅)≤r̅≤R̅(z̅),
where u̅H, u̅N are the axial component of the fluid's velocity in the core region and peripheral region; τ̅H, τ̅N are the shear stress of the Herschel-Bulkley fluid and Newtonian fluid; μ̅H,μ̅N are the viscosities of the Herschel-Bulkley fluid and Newtonian fluid with respective dimensions [ML-1T-2]nT and ML-1T-1;ρ̅H,ρ̅N are the densities of the Herschel-Bulkley fluid and Newtonian fluid; p̅ is the pressure, t̅; is the time; τ̅y is the yield stress. From (2.5), 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 τ̅H≤τ̅y. However, the fluid behavior is indicated whenever τ̅H≥τ̅y. The geometry of the stenosis in the peripheral region as shown in Figure 1 is given by(2.7)R̅(z̅)={R̅0inthenormalarteryregion,R̅0-δ̅p2[1+cos2πL̅0(z̅-d̅-L̅02)]ind̅≤z̅≤d̅+L̅0,
where R̅(z̅) is the radius of the stenosed artery with peripheral layer, R̅0is the radius of the normal artery, L̅0 is the length of the stenosis, d̅ indicates its location, and δ̅p is the maximum depth of the stenosis in the peripheral layer such that [δ̅P/R̅0]≪1. The geometry of the stenosis in the core region as seen in Figure 1 is given by
(2.8)R̅1(z̅)={βR̅0inthenormalarteryregion,βR̅0-δ̅C2[1+cos2πL̅0(z̅-d̅-L̅02)]ind̅≤z̅≤d̅+L̅0,
where R̅1(z̅) is the radius of the stenosed core region of the artery, β is the ratio of the central core radius to the normal artery radius, βR̅0 is the radius of the core region of the normal artery, and δ̅C is the maximum depth of the stenosis in the core region such that [δ̅C/R̅0]≪1. The boundary conditions are(2.9)(i)τ̅Hisfiniteand∂u̅H∂r̅=0atr̅=0,(ii)τ̅H=τ̅Natr̅=R̅1(z̅),(iii)u̅H=u̅Natr̅=R̅1(z̅),(iv)u̅N=0atr̅=R̅(z̅).Since the pressure gradient is a function of z̅ and t̅, we take(2.10)-∂p̅∂z̅=q̅(z̅)f(t̅),
whereq̅(z̅)=-(∂p̅/∂z̅)(z̅,0), f(t̅)=1+Asinω̅t̅, A is the amplitude of the flow and ω̅ 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 f(t̅)=1+Asinω̅t̅ as a good approximation. We introduce the following nondimensional variables (2.11)z=z̅R̅0,R(z)=R̅(z̅)R̅0,R1(z)=R̅1(z̅)R̅0,r=r̅R̅0,t=ω̅t̅,d=d̅R̅0,L0=L̅0R̅0,q(z)=q̅(z̅)q̅0,uH=u̅Hq̅0R̅02/4μ̅0,uN=u̅Nq̅0R̅02/4μ̅N,τH=τ̅Hq̅0R̅0/2,τN=τ̅Nq̅0R̅0/2,θ=τ̅yq̅0R̅0/2,αH2=R̅02ω̅ρ̅Hμ̅0,αN2=R̅02ω̅ρ̅Nμ̅N,Rp=R̅pR̅0,δp=δ̅pR̅0,δC=δ̅CR̅0,
where μ̅0=μ̅H(2/q̅0R̅0)n-1, having the dimension as that of the Newtonian fluid's viscosity,q̅0 is the negative of the pressure gradient in the normal artery, αH is the pulsatile Reynolds number or generalized Wormersly frequency parameter and when n=1, we get the Wormersly frequency parameter αN of the Newtonian fluid. Using the nondimensional variables, (2.1), (2.2), (2.4), (2.5), and (2.6) reduce, respectively, to(2.12)αH2∂uH∂t=4q(z)f(t)-2r∂∂r(rτH)if0≤r≤R1(z),(2.13)αN2∂uN∂t=4q(z)f(t)-2r∂∂r(rτN)ifR1(z)≤r≤R(z),(2.14)τH=-12∂uH∂rn+θifτH≥θ,Rp≤r≤R1(z),(2.15)∂uH∂r=0ifτH≤θ,0≤r≤Rp,(2.16)τN=-12∂uN∂rifR1(z)≤r≤R(z),
where f(t)=1+Asint. The boundary conditions (in dimensionless form) are
(2.17)(i)τHisfiniteatr=0,(ii)∂uH∂r=0atr=0,(iii)τH=τNatr=R1(z),(iv)uH=uNatr=R1(z),(v)uN=0atr=R(z).
The geometry of the stenosis in the peripheral region (in dimensionless form) is given by(2.18)R(z)={1inthenormalarteryregion,1-δp2[1+cos2πL0(z-d-L02)]ind≤z≤d+L0.
The geometry of the stenosis in the core region (in dimensionless form) is given by(2.19)R1(z)={βinthenormalarteryregion,β-δC2[1+cos2πL0(z-d-L02)]ind≤z≤d+L0.
The nondimensional volume flow rate Q is given by(2.20)Q=4∫0R(z)u(r,z,t)rdr,
where Q=Q̅/[πR̅04q̅0/8μ̅0], Q̅ is the volume flow rate.Figure 1
Flow geometry of an arterial stenosis with peripheral layer.


## 3. Method of Solution

When we nondimensionalize the constitutive equations (2.1) and (2.2), αH2 and αN2 occur naturally and these pulsatile Reynolds numbers are time dependent and hence, it is more appropriate to expand (2.12)–(2.16) about αH2 and αN2. The plug core velocity up, the velocity in the core region uH, the velocity in the peripheral region uN, the plug core shear stress τp, the shear stress in the core region τH, the shear stress in the peripheral region τN, and the plug core radius Rp are expanded as follows in terms of αH2 and αN2 (where αH2≪1 and αN2≪1):(3.1)uP(z,t)=u0P(z,t)+αH2u1P(z,t)+⋯,(3.2)uH(r,z,t)=u0H(r,z,t)+αH2u1H(r,z,t)+⋯,(3.3)uN(r,z,t)=u0N(r,z,t)+αN2u1N(r,z,t)+⋯,(3.4)τP(z,t)=τ0P(z,t)+αH2τ1P(z,t)+⋯,(3.5)τH(r,z,t)=τ0H(r,z,t)+αH2τ1H(r,z,t)+⋯,(3.6)τN(r,z,t)=τ0N(r,z,t)+αN2τ1N(r,z,t)+⋯,(3.7)RP(z,t)=R0P(z,t)+αH2R1P(z,t)+⋯.
Substituting (3.2), (3.5) in (2.12) and then equating the constant terms and αH2 terms, we obtain(3.8)∂∂r(rτ0H)=2q(z)f(t)r,(3.9)∂u0H∂t=-2r∂∂r(rτ1H).
Applying (3.2), (3.5) in (2.14) and then equating the constant terms and αH2 terms, one can get(3.10)-∂u0H∂r=2τ0Hn-1[τ0H-nθ],(3.11)-∂u1H∂r=2nτ0Hn-2τ1H[τ0H-(n-1)θ].
Using (3.3) and (3.6) in (2.13) and then equating the constant terms and αN2 terms, we get(3.12)∂∂r(rτ0N)=2q(z)f(t)r,(3.13)∂u0N∂t=-2r∂∂r(rτ1N).
On substituting (3.3) and (3.6) in (2.16) and then equating the constant terms and αN2 terms, one can obtain(3.14)-∂u0N∂r=2τ0N,(3.15)-∂u1N∂r=2τ1N.
Using (3.1)–(3.6) in (2.17) and then equating the constant terms and αH2 and αN2 terms, the boundary conditions are simplified, respectively, to(3.16)τ0P,τ1Parefiniteatr=0,(3.17)∂u0P∂r=0,∂u1P∂r=0atr=0,(3.18)τ0H=τ0Natr=R1(z),(3.19)τ1H=τ1Natr=R1(z),(3.20)u0H=u0Natr=R1(z),(3.21)u1H=u1Natr=R1(z),(3.22)u0N=0atr=R(z),(3.23)u1N=0atr=R(z).
Equations (3.8)–(3.11) and (3.12)–(3.15) are the system of differential equations which can be solved for the unknowns u0H,u1H,τ0H,τ1H and u0N,u1N,τ0N,τ1N, respectively, with the help of boundary conditions (3.16)–(3.23). Integrating (3.8) between 0 and R0P and applying the boundary condition (3.16), we get(3.24)τ0P=q(z)f(t)R0P.
Integrating (3.8) between R0P and r and then making use of (3.24), we get(3.25)τ0H=q(z)f(t)r.
Integrating (3.12) between R1 and r and then using (3.18), one can get(3.26)τ0N=q(z)f(t)r.
Integrating (3.14) between r and R and then making use of (3.22), we obtain(3.27)u0N=q(z)f(t)R2[1-(rR)2].
Integrating (3.10) between r and R1 and using the boundary condition (3.20), we get(3.28)u0H=[q(z)f(t)R]R{1-(R1R)2}+2[q(z)f(t)R1]nR1[1(n+1){1-(rR1)n+1}-k2R1{1-(rR1)n}],
where k2=θ/[q(z)f(t)]. The plug core velocity u0P can be obtained from (3.28) by replacing r by R0P as(3.29)u0P=[q(z)f(t)R]R{1-(R1R)2}+2[q(z)f(t)R1]nR1[1(n+1){1-(R0pR1)n+1}-k2R1{1-(R0pR1)n}].
Neglecting the terms with αH2 and higher powers of αH in (3.7) and using (3.24), the expression for R0P is obtained as
(3.30)r|τ0P=θ=R0P=(θq(z)f(t))=k2.
Similarly, solving (3.9), (3.11), (3.13), and (3.15) with the help of (3.24)–(3.29), and using (3.19), (3.21) and (3.23), the expressions for τ1P,τ1H,τ1N,u1H, andu1P can be obtained as(3.31)τ1P=-14[q(z)f(t)R]BR2(k2R){1-(R1R)2}-[q(z)f(t)R1]nBR12[n2(n+1)(k2R1)-(n-1)2(k2R1)2-n2(n+1)(k2R1)n+2],(3.32)τ1H=-14[q(z)f(t)R]BR2(rR){1-(R1R)2}-[q(z)f(t)R1]nBR12×[n(n+1)(n+3){(n+32)(rR1)-(rR1)n+2}-(n-1)(n+2)(k2R1){(n+22)(rR1)-(rR1)n+1}-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3(R1r)],(3.33)τ1N=-[q(z)f(t)R]BRR1[14(rR1)-18(R1R)2(R1r)-18(R1R)2(rR1)3]-[q(z)f(t)R1]nBR12[n2(n+3)(R1r)-n(n-1)2(n+2)(k2R1)(R1r)-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3(R1r)],(3.34)u1N=-2[q(z)f(t)R]BR2R1[18(RR1){1-(rR)2}-18(R1R)3log(Rr)-132(RR1){1-(rR)4}]-2[q(z)f(t)R1]nBR13log(Rr)[n2(n+3)-n(n-1)2(n+2)(k2R1)-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3],(3.35)u1H=-2[q(z)f(t)R]BR2R1[332(RR1)-18(R1R)+132(R1R)3+18(R1R)3log(R1R)]+2[q(z)f(t)R1]nBR13log(R1R)×[n2(n+3)-n(n-1)2(n+2)(k2R1)-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3]-n[q(z)f(t)R1]nBR1R2{1-(R1R)2}×[12(n+1){1-(rR1)n+1}-(n-1)2n(k2R1){1-(rR1)n}]-2n[q(z)f(t)R1]2n-1BR13×[n2(n+1)2{1-(rR1)n+1}-(n-1)2(n+1)(k2R1){1-(rR1)n}-n2(n+1)2(n+3){1-(rR1)2n+2}+(n-1)(2n2+6n+3)(n+1)(n+2)(n+3)(2n+1)(k2R1){1-(rR1)2n+1}-(n-1)2(n+1)(k2R1){1-(rR1)n+1}+(n-1)22n(k2R1)2{1-(rR1)n}-(n-1)22n(n+2)(k2R1)2{1-(rR1)2n}-3(n2+2n-2)2(n-1)(n+2)(n+3)(k2R1)n+3{1-(rR1)n-1}+3(n-1)(n2+2n-2)2(n-2)(n+2)(n+3)(k2R1)n+4{1-(rR1)n-2}],(3.36)u1P=-2[q(z)f(t)R]BR2R1[332(RR1)-18(R1R)+132(R1R)3+18(R1R)3log(R1R)]+2[q(z)f(t)R1]nBR13log(R1R)×[n2(n+3)-n(n-1)2(n+2)(k2R1)-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3]-n[q(z)f(t)R1]nBR1R2{1-(R1R)2}×[12(n+1){1-(k2R1)n+1}-(n-1)2n(k2R1){1-(k2R1)n}]-2n[q(z)f(t)R1]2n-1BR13×[n2(n+1)2{1-(k2R1)n+1}-(n-1)2(n+1)(k2R1){1-(k2R1)n}-n2(n+1)2(n+3){1-(k2R1)2n+2}+(n-1)(2n2+6n+3)(n+1)(n+2)(n+3)(2n+1)(k2R1){1-(k2R1)2n+1}-(n-1)2(n+1)(k2R1){1-(k2R1)n+1}+(n-1)22n(k2R1)2{1-(k2R1)n}-(n-1)22n(n+2)(k2R1)2{1-(k2R1)2n}-3(n2+2n-2)2(n-1)(n+2)(n+3)(k2R1)n+3{1-(k2R1)n-1}+3(n-1)(n2+2n-2)2(n-2)(n+2)(n+3)(k2R1)n+4{1-(k2R1)n-2}],
where B=[1/f(t)](df(t)/dt). The expression for velocity uH can be easily obtained from (3.2), (3.28) and (3.35). Similarly, the expressions for uN,τH, and τN can be obtained. The expression for wall shear stress τw can be obtained by evaluating τN at r=R and is given below:(3.37)τw=(τ0N+αN2τ1N)r=R=τ0w+αN2τ1w=[q(z)f(t)R]+αN2{-18[q(z)f(t)R]BR2[1-(R1R)4]}+αN2{-[q(z)f(t)R1]n2(n+2)(n+3)BR12(R1R)×[n(n+2)-n(n-1)(n+3)(k2R1)-3(n2+2n-2)(k2R1)n+3]}.
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
(3.38)Q=4[∫0R0P(u0P+αH2u1P)rdr+∫R0PR1(u0H+αH2u1H)rdr+∫R1R(u0N+α2u1N)rdr]=4[q(z)f(t)R]R3{1-(R1R)2}[(k2R1)2+14{1-(R1R)2}]+4[q(z)f(t)R1]nR13(n+2)(n+3)[(n+2)-n(n+3)(k2R1)+(n2+2n-2)(k2R1)n+3]+4αH2[332(RR1)-[q(z)f(t)R]BR2R13{332(RR1)-18(R1R)+132(R1R)3+18(R1R)3log(R1R)}+[q(z)f(t)R1]nBR15log(R1R)×{n2(n+3)-n(n-1)2(n+2)(k2R1)-3(n2+2n-2)2(n+2)(n+3)(k2R1)n+3}-n[q(z)f(t)R1]nBR2R13{1-(R1R)2}×{14(n+3)-(n-1)4(n+2)(k2R1)+(n2+n-5)4(n+2)(n+3)(k2R1)n+3}-n[q(z)f(t)R1]2n-1BR15×{n2(n+2)(n+3)-n(n-1)(4n2+12n+5)(n+2)(n+3)(2n+1)(2n+3)(k2R1)+n(n-1)22(n+1)(n+2)(k2R1)2+(n3-2n2-11n+6)2(n+1)(n+2)(n+3)(k2R1)n+3-(n-1)(n3-2n2-11n+6)2n(n+2)(n+3)(k2R1)n+4-(4n5+14n4-8n3-45n2-3n+18)2n(n+1)(n+2)(n+3)(2n+3)(k2R1)2n+4}]+4αN2[(RR1)-[q(z)f(t)R]BR4R1×{124(RR1)-332(R1R)+596(R1R)5-18(R1R)3(logR1){1-(R1R)2}}-[q(z)f(t)R1]nBR2R13{1-(R1R)2}(1+2logR1)×{n4(n+3)-n(n-1)4(n+2)(k2R1)-3(n2+2n-2)4(n+2)(n+3)(k2R1)n+3}].
The second approximation to plug core radius R1P can be obtained by neglecting the terms with αH4 and higher powers of αH in (3.7) in the following manner. The shear stress τH=τ0H+αH2τ1H at r=RP is given by
(3.39)|τ0H+αH2τ1H|r=RP=θ.
Equation (3.39) reflects 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 τ0H and τ1H about R0P and using τ0H|r=R0P=θ, we get(3.40)R1P=[1q(z)f(t)][-τ1H|r=R0P].
With the help of (3.7), (3.30), (3.32), and (3.40), the expression for RP can be obtained as(3.41)RP=k2+(BαH2R24)[q(z)f(t)R](k2R){1-(R1R)2}+nBαH2R122(n+1)[q(z)f(t)R1]n{(k2R1)-(n2-1)n(k2R1)2-(k2R1)n+2}.
The resistance to flow in the artery is given by(3.42)Λ=[q(z)f(t)]Q.
When R1=R, the present model reduces to the single fluid model (Herschel-Bulkley fluid model) and in such case, the expressions obtained in the present model for velocity uH, shear stress τH,wall shear stress τw, flow rate Q, and plug core radius RP are in good agreement with those of Sankar and Hemalatha [2].


## 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 indexn 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>1 [2]. Since the value of yield stress is 0.04dyne/cm2 for blood at a haematocrit of 40 [35], 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 high (almost 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 ratioα(=αN/αH) 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 αH 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 α and αH, the value of αN can be obtained from α=αN/αH. The value of the ratio β of central core radius βR̅0 to the normal artery radius R̅0 in the unobstructed artery is generally taken as 0.95 and 0.985 [25]. Following Shukla et al. [26], we have used the relations R1=βR and δC=βδP to estimate R1 and δC. The maximum thickness of the stenosis in the peripheral region δP is taken in the range from 0.1 to 0.15 [25]. To compare the present results with the results of Sankar and Hemalatha [2] for single fluid model, we have used the value 0.2 for δC. To deduce the present model to a single fluid model (Newtonian fluid model or Herschel-Bulkley fluid model) and to compare the results with earlier results, we have used the value of β as 1.It is observed that in (3.38), f(t),R, and θ are known and Q and q(z) are the unknowns to be determined. A careful analysis of (3.38) reveals the fact that q(z) is the pressure gradient of the steady flow. Thus, if steady flow is assumed, then (3.38) can be solved for q(z) [2, 10]. For steady flow, (3.38) reduces to(4.1)(R2-R12)[4θ2(RR1)2+(R2-R12)]x3+[4(n+2)(n+3)]×⌊(n+2)R1n+3xn+3-n(n+3)θR1n+2xn+2+(n2+2n-2)θn+3⌋-QSx3=0,
where x=q(z) and QS is the steady state flow rate. Equation (4.1) can be solved for x numerically for a given value of n,QS and 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 δP=0.1. Throughout the analysis, the steady flow rate QS 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 R).


### 4.1. Pressure Gradient

The variation of pressure gradient with axial distance for different fluid models in the core region is shown in Figure2. It has been observed that the pressure gradient for the Newtonian fluid (single fluid model) is 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 model (single phase fluid model) is 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.Figure 2
Variation of pressure gradient with axial direction for different fluids in the core region withδP=0.1.


### 4.2. Pressure Drop

The variation of pressure drop (Δp) (across the stenosis, i.e., from z=4 to z=6) in a time cycle for different values of A, θ, andδP with n=β=0.95 is depicted in Figure 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 δP 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 δP are held fixed. Figure 3 shows the simultaneous effects of the stenosis size and nonNewtonian nature of blood on pressure drop.Figure 3
Variation of pressure drop in a time cycle for different values ofA, θ and δP with n=β=0.95.


### 4.3. Plug Core Radius

The variation of plug core radius(RP) with axial distance for different values of the amplitude A and stenosis thickness δP (in the peripheral layer) with n=β=0.95, αH=0.5, θ=0.1, and t=60° is shown in Figure 4. It is noted that the plug core radius decreases as the axial variable z varies from 4 to 5 and it increases as z varies from 5 to 6. It is further observed that for a given value of δP, 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 of the amplitude A. Figure 4 depicts the effects of stenosis height on the plug core radius of the blood vessels.Figure 4
Variation of plug core radius with axial distance for different values ofA and δP with n=β=0.95, αH=0.5, θ=0.1 and t=60°.Figure5 sketches the variation of plug core radius in a time cycle for different values of the pulsatile Reynolds number αH of the Herschel-Bulkley fluid and yield stress θ with n=β=0.95, A=0.5, z=5, t=60°, andδP=0.1. It is noted that the plug core radius decreases as time 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 αH, the plug core radius increases as the yield stress θ increases. Also, it is noticed that for a given value of yield stress θ and with increasing values of the pulsatile Reynolds number αH, the plug core radius increases when t lies between 0° and 90° and also between 270° and 360° and decreases when t lies between 90° and 270°. Figure 5 depicts 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.Figure 5
Variation of plug core radius in a time cycle for different values ofαH and θ with n=β=0.95, δP=0.1, A=0.5, t=60° and z=5.


### 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 understanding of the effects of blood flow on the endothelial cells [36, 37]. The variation of wall shear stress in the axial direction for different values of yield stress θ and pulsatile Reynolds number αN of the Newtonian fluid with t=45°, n=β=0.95, A=0.5, andδP=0.1 is plotted in Figure 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 as z increases further from 5 to 6. For a given value of the pulsatile Reynolds number αN, the wall shear stress increases considerably with the increase in the values of the yield stress 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 αN, 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 6 shows 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 6
Variation of wall shear stress with axial distance for different valuesθ and αN with t=45°, n=β=0.95, A=0.5 and δP=0.1.Figure7 depicts the variation of wall shear stress in a time cycle for different values of the amplitude Aand peripheral stenosis height δP with n=β=0.95, θ=0.1, αN=0.5 and 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 minimum at 270°. Also, it may be noted that for a given value of the amplitude A the wall shear stress increases with increasing values of the stenosis thicknessδP. Further, it is noticed that for a given value of the stenosis size and increasing values of the amplitude A, the wall shear stress increases when t lies between 0° and 180° and decreases when t lies between 180° and 360°. This figure sketches the effects of the stenosis size and amplitude on the wall shear stress of the two-phase blood flow model.Figure 7
Variation of wall shear stress in a time cycle for different values ofA and δP with θ=0.1, n=β=0.95, z=5 and αN=0.5.


### 4.5. Velocity Distribution

The velocity profiles are of interest, since they provide a detailed description of the flow field. The velocity distributions in the radial direction for different values of the amplitudeA, pulsatile Reynolds number ratio α, pulsatile Reynolds number of Herschel-Bulkley fluid αH, the ratio of the central core radius to the tube radius β with n=0.95, z=5, θ=δP=0.1, and t=45° are shown in Figure 8. One can easily notice the plug flow around the tube axis in Figure 8. Also, it is found that the velocity increases as the amplitude A increases for a given set of values of α, αH and β. Further, it is observed that for a given set of values of A, α and αH, the velocity decreases considerably near the tube axis as the ratio β increases. The same behavior is observed for increasing values of the pulsatile Reynolds number ratio α and pulsatile Reynolds number αH for the given values of A and β, but there is only a slight decrease in the later case. Figure 8 depicts the effects of amplitude, pulsatility and stenosis size on velocity distribution of the two-phase model. The velocity distribution in the radial direction at different times is shown in Figure 9. It is observed that the velocity increases as time t (in degrees) increases from 0° to 90° and then it decreases as t increases from 90° to 270° and again it increases as t increases further from 270° to 360°. This figure shows the transient effects of blood flow on velocity of the two-phase model.Figure 8
Velocity distribution for different values ofA, α, αH and β with θ=δP=0.1, z=5, n=0.95 and t=45°.Figure 9
Velocity distribution at different times withn=β=0.95, θ=δP=0.1, α=αH=0.5, z=5 and A=0.5.


### 4.6. Resistance to Flow

The variation of resistance to flow with peripheral layer stenosis size for different values of the amplitudeA and yield stress θ with n=β=0.95, α=αH=0.25, and t=45° is plotted in Figure 10. Since δC=βδP, the stenosis size of the core region δC also increases when the peripheral layer stenosis height δP increases for a given value of β. It is seen that the resistance to flow increases gradually with increasing stenosis size while the rest of the parameters are kept fixed. It is to be noted that for a given value of yield stress θ, the resistance to flow decreases with increasing values of the amplitude A. It is also found that for a given value of the amplitude A, the resistance to flow increases with increase in the values of the yield stress θ when the other parameters held constant. Figure 10 illustrates the effects of the amplitude, stenosis size and the nonNewtonian nature of blood on resistance to flow of the two-phase model.Figure 10
Variation of resistance to flow with stenosis size for different values ofA and θ with n=β=0.95, α=αH=0.25 and t=45°.Figure11 sketches the variation of resistance to flow in a time cycle for different values of the power law index n and the pulsatile Reynolds number ratio α, pulsatile Reynolds number of the Herschel-Bulkley fluid αH with θ=δP=0.1,β=0.95 and A=0.2. It is clear that the resistance to flow decreases as time t (in degrees) increases from 0° to 90° and then it increases as t increases from 90° to 270° and then again it decreases as t increases further from 270° to 360°. The resistance to flow is minimum at 90° and maximum at 270°. It is found that for the fixed values of α and αH and the increasing values of the power law index n, the resistance to flow decreases when time t lies between 0° and 180° and increases when t lies between 180° and 360°. Further, it is noted that for a fixed value of the power law index n and with the increasing values of α and αH, the resistance to flow increases slightly when t lies between 0° and 90° and also between 270° and 360° and decreases slightly when t lies between 90° and 270°. Figure 11 shows the simultaneous effects of pulsatility of the flow and the nonNewtonian nature of blood on resistance to flow of the two-phase model.Figure 11
Variation of resistance to flow in a time cycle for different values ofα, αH and n with θ=δP=0.1, β=0.95, and A=0.2.


### 4.7. Quantification of Wall Shear Stress and Resistance to Flow

The wall shear stress(τw) and resistance to flow (Λ) are physiologically important quantities which play an important role in the formation of platelets [38]. High wall shear stress not only damages the vessel wall and causes intimal thickening, but also activates platelets, cause platelet aggregation, and finally results in the formation of thrombus [7].The wall shear stress increase factor is defined as the ratio of the wall shear stress of particular fluid model in the stenosed artery for a given set of values of the parameters to the wall shear stress of the same fluid model in the normal artery for the same set of values of the parameters. The estimates of the wall shear stress increase factor for two-phase Herschel-Bulkley fluid model and single-phase fluid model witht=45°, n=0.95, A=α=αH=0.5,β=0.985,andθ=0.1 are given in Table 1. It is observed that for the range of the stenosis size 0–0.15, the corresponding ranges of the wall shear stress increase of the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.074–1.594 and 1.156–2.848, respectively. It is found that the estimates of the wall shear stress increase factor are marginally lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model.Table 1
Estimates of the wall shear stress increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes withn=0.95,A=α=αH=0.5, β=0.985, θ=0.1, and t=45°.
Stenosis size(δP)Two-phase fluid modelSingle-phase fluid model0.0251.0741.1560.051.1571.3500.0751.2491.5950.11.3521.9070.1251.4672.3130.151.5972.848One can define the resistance to flow increase factor in a similar way as in the case of wall shear stress increase factor. The estimates of the increase in resistance to flow factor for two-phase Herschel-Bulkley fluid model and single-phase fluid model witht=45°, n=0.95,A=α=αH=0.5, β=0.985, and θ=0.1 are given in Table 2. It is noted that for the range of the stenosis size 0–0.15, the corresponding range of the increase in resistance to flow factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.050–1.393 and 1.104–2.189, respectively. It is found that the estimates of the wall shear stress increase factor are significantly lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model. Hence, it is clear that the existence of the peripheral layer is useful in the functioning of the diseased arterial system. It is strongly felt that the present model may provide a better insight to the study of blood flow behavior in the stenosed arteries than the earlier models.Table 2
Estimates of the resistance to flow increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes withn=0.95, A=α=αH=0.5, β=0.985, θ=0.1, and t=45°.
Stenosis size(δP)Two-phase fluid modelSingle-phase fluid model0.0251.0501.1040.051.1051.2320.0751.1661.3910.11.2331.5920.1251.3081.8500.151.3932.189Perturbation method is a very useful analytical tool for solving nonlinear differential equations. In the present study, it is used to solve the nonlinear coupled implicit system of partial differential equations to get an asymptotic solution. This method yields a closed form to the flow quantities which enables us to evaluate them at any particular instant of time and at any particular point in the flow domain. This facility is unavailable when we use the computational methods such as finite difference method, finite element method, finite volume method.


## 4.1. Pressure Gradient

The variation of pressure gradient with axial distance for different fluid models in the core region is shown in Figure2. It has been observed that the pressure gradient for the Newtonian fluid (single fluid model) is 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 model (single phase fluid model) is 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.Figure 2
Variation of pressure gradient with axial direction for different fluids in the core region withδP=0.1.


## 4.2. Pressure Drop

The variation of pressure drop (Δp) (across the stenosis, i.e., from z=4 to z=6) in a time cycle for different values of A, θ, andδP with n=β=0.95 is depicted in Figure 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 δP 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 δP are held fixed. Figure 3 shows the simultaneous effects of the stenosis size and nonNewtonian nature of blood on pressure drop.Figure 3
Variation of pressure drop in a time cycle for different values ofA, θ and δP with n=β=0.95.


## 4.3. Plug Core Radius

The variation of plug core radius(RP) with axial distance for different values of the amplitude A and stenosis thickness δP (in the peripheral layer) with n=β=0.95, αH=0.5, θ=0.1, and t=60° is shown in Figure 4. It is noted that the plug core radius decreases as the axial variable z varies from 4 to 5 and it increases as z varies from 5 to 6. It is further observed that for a given value of δP, 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 of the amplitude A. Figure 4 depicts the effects of stenosis height on the plug core radius of the blood vessels.Figure 4
Variation of plug core radius with axial distance for different values ofA and δP with n=β=0.95, αH=0.5, θ=0.1 and t=60°.Figure5 sketches the variation of plug core radius in a time cycle for different values of the pulsatile Reynolds number αH of the Herschel-Bulkley fluid and yield stress θ with n=β=0.95, A=0.5, z=5, t=60°, andδP=0.1. It is noted that the plug core radius decreases as time 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 αH, the plug core radius increases as the yield stress θ increases. Also, it is noticed that for a given value of yield stress θ and with increasing values of the pulsatile Reynolds number αH, the plug core radius increases when t lies between 0° and 90° and also between 270° and 360° and decreases when t lies between 90° and 270°. Figure 5 depicts 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.Figure 5
Variation of plug core radius in a time cycle for different values ofαH and θ with n=β=0.95, δP=0.1, A=0.5, t=60° and z=5.


## 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 understanding of the effects of blood flow on the endothelial cells [36, 37]. The variation of wall shear stress in the axial direction for different values of yield stress θ and pulsatile Reynolds number αN of the Newtonian fluid with t=45°, n=β=0.95, A=0.5, andδP=0.1 is plotted in Figure 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 as z increases further from 5 to 6. For a given value of the pulsatile Reynolds number αN, the wall shear stress increases considerably with the increase in the values of the yield stress 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 αN, 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 6 shows 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 6
Variation of wall shear stress with axial distance for different valuesθ and αN with t=45°, n=β=0.95, A=0.5 and δP=0.1.Figure7 depicts the variation of wall shear stress in a time cycle for different values of the amplitude Aand peripheral stenosis height δP with n=β=0.95, θ=0.1, αN=0.5 and 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 minimum at 270°. Also, it may be noted that for a given value of the amplitude A the wall shear stress increases with increasing values of the stenosis thicknessδP. Further, it is noticed that for a given value of the stenosis size and increasing values of the amplitude A, the wall shear stress increases when t lies between 0° and 180° and decreases when t lies between 180° and 360°. This figure sketches the effects of the stenosis size and amplitude on the wall shear stress of the two-phase blood flow model.Figure 7
Variation of wall shear stress in a time cycle for different values ofA and δP with θ=0.1, n=β=0.95, z=5 and αN=0.5.


## 4.5. Velocity Distribution

The velocity profiles are of interest, since they provide a detailed description of the flow field. The velocity distributions in the radial direction for different values of the amplitudeA, pulsatile Reynolds number ratio α, pulsatile Reynolds number of Herschel-Bulkley fluid αH, the ratio of the central core radius to the tube radius β with n=0.95, z=5, θ=δP=0.1, and t=45° are shown in Figure 8. One can easily notice the plug flow around the tube axis in Figure 8. Also, it is found that the velocity increases as the amplitude A increases for a given set of values of α, αH and β. Further, it is observed that for a given set of values of A, α and αH, the velocity decreases considerably near the tube axis as the ratio β increases. The same behavior is observed for increasing values of the pulsatile Reynolds number ratio α and pulsatile Reynolds number αH for the given values of A and β, but there is only a slight decrease in the later case. Figure 8 depicts the effects of amplitude, pulsatility and stenosis size on velocity distribution of the two-phase model. The velocity distribution in the radial direction at different times is shown in Figure 9. It is observed that the velocity increases as time t (in degrees) increases from 0° to 90° and then it decreases as t increases from 90° to 270° and again it increases as t increases further from 270° to 360°. This figure shows the transient effects of blood flow on velocity of the two-phase model.Figure 8
Velocity distribution for different values ofA, α, αH and β with θ=δP=0.1, z=5, n=0.95 and t=45°.Figure 9
Velocity distribution at different times withn=β=0.95, θ=δP=0.1, α=αH=0.5, z=5 and A=0.5.


## 4.6. Resistance to Flow

The variation of resistance to flow with peripheral layer stenosis size for different values of the amplitudeA and yield stress θ with n=β=0.95, α=αH=0.25, and t=45° is plotted in Figure 10. Since δC=βδP, the stenosis size of the core region δC also increases when the peripheral layer stenosis height δP increases for a given value of β. It is seen that the resistance to flow increases gradually with increasing stenosis size while the rest of the parameters are kept fixed. It is to be noted that for a given value of yield stress θ, the resistance to flow decreases with increasing values of the amplitude A. It is also found that for a given value of the amplitude A, the resistance to flow increases with increase in the values of the yield stress θ when the other parameters held constant. Figure 10 illustrates the effects of the amplitude, stenosis size and the nonNewtonian nature of blood on resistance to flow of the two-phase model.Figure 10
Variation of resistance to flow with stenosis size for different values ofA and θ with n=β=0.95, α=αH=0.25 and t=45°.Figure11 sketches the variation of resistance to flow in a time cycle for different values of the power law index n and the pulsatile Reynolds number ratio α, pulsatile Reynolds number of the Herschel-Bulkley fluid αH with θ=δP=0.1,β=0.95 and A=0.2. It is clear that the resistance to flow decreases as time t (in degrees) increases from 0° to 90° and then it increases as t increases from 90° to 270° and then again it decreases as t increases further from 270° to 360°. The resistance to flow is minimum at 90° and maximum at 270°. It is found that for the fixed values of α and αH and the increasing values of the power law index n, the resistance to flow decreases when time t lies between 0° and 180° and increases when t lies between 180° and 360°. Further, it is noted that for a fixed value of the power law index n and with the increasing values of α and αH, the resistance to flow increases slightly when t lies between 0° and 90° and also between 270° and 360° and decreases slightly when t lies between 90° and 270°. Figure 11 shows the simultaneous effects of pulsatility of the flow and the nonNewtonian nature of blood on resistance to flow of the two-phase model.Figure 11
Variation of resistance to flow in a time cycle for different values ofα, αH and n with θ=δP=0.1, β=0.95, and A=0.2.


## 4.7. Quantification of Wall Shear Stress and Resistance to Flow

The wall shear stress(τw) and resistance to flow (Λ) are physiologically important quantities which play an important role in the formation of platelets [38]. High wall shear stress not only damages the vessel wall and causes intimal thickening, but also activates platelets, cause platelet aggregation, and finally results in the formation of thrombus [7].The wall shear stress increase factor is defined as the ratio of the wall shear stress of particular fluid model in the stenosed artery for a given set of values of the parameters to the wall shear stress of the same fluid model in the normal artery for the same set of values of the parameters. The estimates of the wall shear stress increase factor for two-phase Herschel-Bulkley fluid model and single-phase fluid model witht=45°, n=0.95, A=α=αH=0.5,β=0.985,andθ=0.1 are given in Table 1. It is observed that for the range of the stenosis size 0–0.15, the corresponding ranges of the wall shear stress increase of the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.074–1.594 and 1.156–2.848, respectively. It is found that the estimates of the wall shear stress increase factor are marginally lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model.Table 1
Estimates of the wall shear stress increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes withn=0.95,A=α=αH=0.5, β=0.985, θ=0.1, and t=45°.
Stenosis size(δP)Two-phase fluid modelSingle-phase fluid model0.0251.0741.1560.051.1571.3500.0751.2491.5950.11.3521.9070.1251.4672.3130.151.5972.848One can define the resistance to flow increase factor in a similar way as in the case of wall shear stress increase factor. The estimates of the increase in resistance to flow factor for two-phase Herschel-Bulkley fluid model and single-phase fluid model witht=45°, n=0.95,A=α=αH=0.5, β=0.985, and θ=0.1 are given in Table 2. It is noted that for the range of the stenosis size 0–0.15, the corresponding range of the increase in resistance to flow factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model are 1.050–1.393 and 1.104–2.189, respectively. It is found that the estimates of the wall shear stress increase factor are significantly lower for the two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model. Hence, it is clear that the existence of the peripheral layer is useful in the functioning of the diseased arterial system. It is strongly felt that the present model may provide a better insight to the study of blood flow behavior in the stenosed arteries than the earlier models.Table 2
Estimates of the resistance to flow increase factor for the two-phase Herschel-Bulkley fluid model and single-phase Herschel-Bulkley fluid model for different stenosis sizes withn=0.95, A=α=αH=0.5, β=0.985, θ=0.1, and t=45°.
Stenosis size(δP)Two-phase fluid modelSingle-phase fluid model0.0251.0501.1040.051.1051.2320.0751.1661.3910.11.2331.5920.1251.3081.8500.151.3932.189Perturbation method is a very useful analytical tool for solving nonlinear differential equations. In the present study, it is used to solve the nonlinear coupled implicit system of partial differential equations to get an asymptotic solution. This method yields a closed form to the flow quantities which enables us to evaluate them at any particular instant of time and at any particular point in the flow domain. This facility is unavailable when we use the computational methods such as finite difference method, finite element method, finite volume method.


## 5. Conclusion

The present study analyzes the two-phase Herschel-Bulkley fluid model for blood flow through stenosed arteries and brings out many interesting fluid mechanical phenomena due to the presence of the peripheral layer. The results indicate that the 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 found that the velocity increases, plug core radius, and resistance to flow decrease as the amplitude increases. It is also observed that the difference between the estimates of increase in the wall shear stress factor of the two-phase fluid model and single-phase fluid model is substantial. A similar behavior is observed for the increase in resistance to flow factor. Thus, the results demonstrate that this model is capable of predicting the hemodynamic features most interesting to physiologists. Thus, the present study could be useful for analyzing the blood flow in the diseased state. From this study, it is concluded that the presence of the peripheral layer (outer phase) helps in the functioning of the diseased arterial system.



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*Source: 465835-2010-06-16.xml*