Two-Fluid Nonlinear Mathematical Model for Pulsatile Blood Flow Through Stenosed Arteries

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Two-Fluid Nonlinear Mathematical Model for Pulsatile Blood Flow Through Stenosed Arteries

D. S. SANKAR

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Pulau Pinang, Malaysia sankar [email protected]

Abstract. Pulsatile flow of blood through mild stenosed narrow arteries is analyzed by treating the blood in the core region as Casson fluid and the plasma in the peripheral layer as Newtonian fluid. Perturbation method is used to solve the coupled implicit system of non-linear differential equations. The expressions for velocity, wall shear stress, plug core radius, flow rate and resistance to flow are obtained. The effects of pulsatility, stenosis, pe- ripheral layer and non-Newtonian behavior of blood on these flow quantities are discussed.

It is found that the pressure drop, plug core radius, wall shear stress and resistance to flow increase with the increase of the yield stress or stenosis size while all other parameters held constant.

2010 Mathematics Subject Classification: 35Q92, 76Z05

Keywords and phrases: Two-fluid model, pulsatile blood flow, stenosed arteries, pertur- bation method.

1. Introduction

Hemodynamics plays a vital role in the development, progression and treatment of arterial stenosis [4, 13, 16]. Arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting arterial stenosis. When an obstruction 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. Thus, the presence of a stenosis leads to the serious circulatory disorder. Hence, the mathematical modeling of blood flow through stenosed arteries is very important.

Several researchers studied the blood flow characteristics in the presence of stenosis [1, 5, 7, 10]. The assumption of Newtonian behavior of blood is acceptable for high shear rate flow through larger arteries [13]. But, blood, being a suspension of cells in plasma, exhibits non-Newtonian behavior at low shear rate ˙γ<10

sec

in small diameter arteries (0.02 mm–0.1 mm) [6, 9]. In diseased state, the actual flow is distinctly pulsatile [8, 9].

Several researchers have studied the non-Newtonian behavior and pulsatile flow of blood through stenosed arteries [3, 7, 10, 13].

Srivastava and Saxena [15] and Misra and Pandey [6] propounded that for blood flowing through small vessels, there is an erythrocyte-free plasma (Newtonian) layer adjacent to the

Received: June 22, 2009; Revised: April 30, 2010.

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vessel wall and a core layer of a suspension of all erythrocytes (non-Newtonian). Accepting this idea, several studies [2, 6, 15] revealed that the existence of the peripheral layer has some significance in the flow characteristics of the arterial system. Chaturani and Ponnala- gar Samy [3] and Scott Blair [12] pointed out that Casson fluid model is well suited and simple to apply for blood flow problems. Many researchers [2, 3, 6, 15] have used Casson fluid model for mathematical modeling of blood flow through narrow arteries at low shear rates for different flow situations. Hence, in this paper, we have studied the pulsatile flow of a two-fluid model for blood through stenosed narrow arteries (of diameters 0.02mm – 0.2mm) at low shear rates(γ˙<10/sec), assuming the suspension of all the erythrocytes in the core region of the blood vessel as a Casson fluid and the plasma in the peripheral layer as a Newtonian fluid.

2. Mathematical formulation

Consider an axially symmetric, laminar, pulsatile and fully developed flow of blood in the axial direction(¯z)through a circular artery with an axially symmetric mild stenosis. It is assumed that the blood is represented by a two-fluid model with a central layer (core region) of suspension of all the erythrocytes as a Casson fluid and a peripheral layer of plasma as a Newtonian fluid. In order to idealize the present two-fluid model, we have assumed the wall of the artery to be rigid. The geometry of the arterial stenosis is shown in Figure 1. The cylindrical polar coordinates ¯r,φ¯,¯z

are used to study the flow, where ¯r and ¯φare the radial coordinate and the azimuthal angle, respectively. Since, the stenosis present in the artery is considered to be mild, the radial transport of the blood is negligible [3] and thus, we have neglected the radial velocity of the blood in this study. Hence, in the present study, the flow of blood is considered to be unidirectional and is in the axial direction.

The principle of conservation of mass for one dimensional fluid flow in a deformable tube gives the following equations of continuity in the core region and peripheral layer region [8]:

(2.1) ∂^R¯1

∂^{¯t +} R¯₁

2

∂^u¯C

∂^{¯z +u¯C}

∂^R¯1

∂^{¯z =0, in 0≤¯r≤R¯1(¯z),}

(2.2) ∂^R¯

∂¯t+R¯ 2

∂^u¯N

∂^{¯z +u¯N}∂^R¯

∂^{¯z =0, in ¯R1(¯z)≤¯r≤R¯(¯z),}

where ¯R₁(=R¯₁(¯z))is the radius of the core region of the stenosed artery; ¯R(=R(¯z))¯ is the radius of the artery with peripheral layer; ¯u_C is the velocity of the fluid (Casson fluid) in the core region and ¯u_N is the velocity of the fluid (Newtonian fluid) in the peripheral layer region of the fluid flow. Since, the blood flow in narrow arteries at low shear rates is a slow flow, the viscous forces dominate over the inertial forces and thus, the magnitude of the convective terms are negligibly small. Hence, in the present study, we have neglected the convective terms in the momentum equations:

(2.3) ρ¯C∂^u¯C

∂^{¯t =−}

∂^p¯

∂^{¯z −} 1

¯r

∂

∂^{¯r(¯r ¯}τC) in 0≤¯r≤R¯₁(¯z),

(2.4) ρ¯N∂^u¯N

∂^{¯t =−}

∂^p¯

∂^{¯z −} 1

¯r

∂

∂^{¯r(¯r ¯}τN) in ¯R₁(¯z)≤¯r≤R¯(¯z),

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

where the shear stress ¯τ =|τ¯¯r ¯z|=−τ¯¯r¯z(since ¯τ=τ¯Cor ¯τ=τ¯N); ¯p is the pressure; ¯u_C and

¯

u_N are the axial velocity of the fluid in the core region and peripheral region, respectively;

τ¯Cand ¯τNare the shear stress of the Casson fluid and Newtonian fluid, respectively; ¯ρCand ρ¯N are the densities of the Casson fluid and Newtonian fluid respectively; ¯t is the time. The relations between the shear stress and strain rate of the fluids in motion in the core region (Casson fluid) and peripheral region (Newtonian fluid) are given by

√τ¯C= r

−µ¯C

∂^u¯C

∂^{¯r +}

pτ¯y if τ¯C≥τ¯y and R¯p≤¯r≤R¯₁(¯z), (2.5)

∂^u¯C

∂^{¯r =0 if} τ^¯C≤τ¯y and 0≤¯r≤R¯_p, (2.6)

τ¯N=−µ¯N∂^u¯N

∂^{¯r if R¯1(¯z)≤¯r≤R(¯z),¯} (2.7)

where ¯µCand ¯µNare the viscosities of the Casson fluid and Newtonian fluid, respectively; ¯τy

is the yield stress; ¯RPis the plug core radius. The geometry of the stenosis in the peripheral region and core region are given by

(2.8) R(¯z) =¯

(R¯₀, in the normal artery region,

R¯₀−^δ¯2^P

n

1+cosh

2π

¯L₀

¯z−d¯−^¯L2⁰

io, in ¯d≤¯z≤d¯+¯L0,

(2.9) R¯₁(¯z) =

(β^R¯0, in the normal artery region,

β^R¯0−^δ¯₂^Cn

1+cosh_2π

¯L₀

¯z−d¯−^¯L₂⁰io

, in ¯d≤¯z≤d¯+¯L₀,

where ¯R(¯z)and ¯R₁are the radii of the stenosed artery with the peripheral region and core region respectively; ¯R₀andβR^¯₀ are the radii of the normal artery and core region of the normal artery respectively; β is the ratio of the central core radius to the normal artery radius; ¯L₀ is the length of the stenosis; ¯d indicates the location of the stenosis; ¯δP and δ¯C are the maximum projections of the stenosis in the peripheral region and core region

490

respectively such thatδ¯P/R¯₀

≪1 andδ¯C/R¯₀

≪1. The boundary conditions are τ¯Cis finite and∂^u¯C

∂^{¯r =0 at ¯r=0;}

¯

u_N=0 at r=R;¯

τ¯C=τ¯Nand ¯u_C=u¯_Nat ¯r=R¯₁. (2.10)

Since the pressure gradient is a function of ¯z and ¯t, we assume

(2.11) −∂^p¯

∂^{¯z =q(¯z)¯ f(¯t),}

where ¯q(¯z) =−(∂p/^¯ ∂¯z) (¯z,0). Since, any periodic function can be expanded in a Fourier sine series, it is reasonable to choose 1+A sin ¯ω¯t as a good approximation for f(¯t), where A and ¯ωare the amplitude and angular frequency of the flow respectively. We introduce the following non-dimensional variables

z= ¯z

R¯₀, R(z) =R(¯z)¯

R¯₀ , R₁(z) =R¯₁(¯z)

R¯₀ , r= ¯r

R¯₀, d= d¯ R¯₀, L₀= ¯L₀

R¯₀, q(z) =q(¯z)¯

¯

q₀ , εC=α_C²=R¯²₀ω¯ρ^¯C

µ¯C

, εN=α_N²=R¯²₀ω¯ρ^¯N

µ¯N

, (2.12)

R_P=R¯_P

R¯₀, δP=δ¯P

R¯₀, δC=δ¯C

R¯₀, u_C= u¯_C

¯

q₀R¯²₀/4 ¯µC, u_N= u¯_N

¯

q₀R¯²₀/4 ¯µN, τC= τ¯C

(q¯₀R¯₀/2), τN= τ¯N

(q¯₀R¯₀/2), θ= τ¯y

(q¯₀R¯₀/2), t=ω¯¯t,

where ¯q₀is the negative of the pressure gradient in the normal artery,αC andαN are the pulsatile Reynolds numbers of the Casson fluid and Newtonian fluid, respectively. Using the non-dimensional variables, Equations (2.2)–(2.7) are simplified to

(2.13) εC∂^uC

∂^{t =4q(z)f(t)−} 2 r

∂

∂^r(rτC) if 0≤r≤R₁(z) (2.14) √τC=

r

−1 2

∂^uC

∂r +√

θ ^if τC≥θ ^{and R}p≤r≤R₁(z),

(2.15) ∂^uC

∂^{r =0 if} τC≤θ ^{and 0}≤r≤R_p, (2.16) εN∂uN

∂t =4q(z)f(t)−²_r∂r^∂ (rτN) τ=−¹2∂uN

∂r

)

if R₁(z)≤r≤R(z), where

(2.17) f(t) =1+A sint.

The boundary conditions (in the dimensionless form) are τCis finite and∂^uC

∂^{r =0 at r=0,} τC=τNand u_C=u_Nat r=R₁, u_N=0 at r=R.

(2.18)

The geometry of the stenosis in the peripheral region and core region (in the dimensionless form) are given by

(2.19) R(z) =

(1, in the normal artery region,

1−^δ₂^Pn

1+cosh

2π L0

z−d−^L₂⁰io

, in d≤z≤d+L₀,

(2.20) R¯₁(¯z) =

(β, in the normal artery region,

β−^δ₂^Cn

1+cosh_2π

L₀

z−d−^L₂⁰io

, in d≤z≤d+L0. The non-dimensional volume flow rate Q is given by

(2.21) Q=4

Z R(z)

0 u(r,z,t)r dr, where Q=Q/¯ π^R¯4₀^q¯0/8 ¯µ0

and ¯Q is the volume flow rate.

3. Perturbation method of solution

Since it is not possible to find an exact solution to the system of nonlinear equations (2.13) – (2.16), the perturbation method is used to obtain the approximate solution to the unknowns u_C, u_N,τCandτN. When we non-dimensionalize the momentum Equations (2.3) and (2.4), εC andεN occur naturally and hence, it is more appropriate to expand Equations (2.13)–

(2.16) aboutεCandεN. Let us expand the plug core velocity u_p, and the velocity in the core region u_Cin the perturbation series ofεCas below (whereεC≪1):

up(z,t) =u_0P(z,t) +εCu_1P(z,t) +···, (3.1)

u_C(r,z,t) =u_0C(r,z,t) +εCu_1C(r,z,t) +···. (3.2)

Similarly, one can expand uN,τP,τC,τN and the plug core radius RPin powers ofεC

andεN, whereεN ≪1. Using the perturbation series in Equations (2.13) and (2.14) and then equating the constant terms andεC terms, the differential equations of the core region becomes

(3.3)

∂

∂^r(rτ0C) =2q(z)f(t)r,

−∂^u0C

∂^{r =2}

τ0C−2p

θτ0C+θ,

∂^u0C

∂^{t =−} 2 r

∂

∂^r(rτ1C),

−∂^u1C

∂^{r =2}τ1C

1−p

θ/τ0C

.

Similarly, using the perturbation series expansions in Equation (2.16) and then equating the constant terms andεN terms, the differential equations of the peripheral region reduced to

(3.4)

∂

∂^r(rτ0N) =2q(z)f(t)r,

−∂^u0N

∂r =2τ0N,

∂^u0N

∂^{t =−} 2 r

∂

∂^r(rτ1N),

−∂^u1N

∂r =2τ1N.

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Substituting the perturbation series expansions in Equation (2.18) and then equating the constant terms andεCandεNterms, we get

τ0pandτ1pare finite and∂^u0P

∂^{r =0,}∂^u1P

∂^{r =0 at r=0;}

τ0C=τ0N,τ1C=τ1N,u0C=u_0N,u_1C=u_1Nat r=R₁; u_0N=u_1N=0 at r=R.

(3.5)

Solving the system of Equations (3.3) and (3.4) by using Equation (3.5) for the unknowns u_0C, u_1C,τ0C,τ1C, u_0N, u_1N,τ0Nandτ1N, one can obtain

τ0P=∇R_0P, τ0C=∇r, τ0N=∇r, (3.6)

u_0N=∇R² 1−ξ², (3.7)

u_0C=∇R²

1−Ω²+Ω²

1−ξ1²−8

3σ₁^1/21−ξ₁^3/2+2σ1(1−ξ1)

, (3.8)

u_0P=∇R²

1−Ω²+Ω²

1−χ²−8

3σ₁^1/21−χ^3/2+2σ1(1−χ)

, (3.9)

τ1P=−∇BR³1

4σ(1−Ω²) +Ω³σ1

1 4−1

3σ₁^1/2+ 1 12σ₁², (3.10)

τ1C=−∇BR³1

4ξ(1−Ω²)−1 8Ω³

2ξ1−ξ₁³−σ1⁴ξ₁⁻¹

− 8

21σ₁^1/27ξ1−4ξ₁^5/2−3σ₁^7/2ξ₁⁻¹, (3.11)

τ1N=−∇BR²R₁ 1

4ξ1−1

8Ω²ξ₁⁻¹−1 8Ω²ξ1³

+ξ₁⁻¹Ω² 1 8−1

7σ₁^1/2+ 1 56σ1⁴

, (3.12)

u_1N=−∇BR³R₁ 1

4Ω⁻¹ 1−ξ²₋¹

4Ω³logξ⁻¹₋ ¹

16Ω⁻¹ 1−ξ⁴

−Ω³logξ¹ 4−2

7σ₁^1/2+ 1

28σ₁⁴ , (3.13)

u_1C=−∇BR³R₁ (

3

16Ω⁻¹−1 4Ω+ 1

16Ω³+1

4Ω³logΩ

−Ω³logΩ1 4−2

7

√σ1+ 1 28σ1⁴

+Ω 1−Ω² 1

4 1−ξ1²

−1 3

√σ1

1−

qξ₁³ +Ω³

"

1

4 1−ξ₁³−1 3

√σ1

1−q

ξ₁³− 1

16 1−ξ₁⁴ + 53

294

√σ1

1−q

ξ₁⁷−1

3 1−ξ₁²+4 9σ1

1−q

ξ₁³

− 8

63σ1 1−ξ1³

− 1

28σ1⁴logξ1+ 1 14

qσ₁^{9 1}− 1 pξ1

! #) , (3.14)

u_1P=−∇BR³R₁ 3

16Ω⁻¹−1 4Ω+ 1

16Ω³+1

4Ω³logΩ

−Ω³logΩ1 4−2

7

√σ1+ 1 28σ1⁴

+Ω 1−Ω² 1

4 1−σ1²

−1 3

√σ1

1−

qσ₁³ +Ω³

1

4 1−σ1²

−1 3

√σ1

1−

qσ₁³− 1

16 1−σ1⁴

− 53 294

√σ1

1−

qσ₁⁷−1

3 1−σ1²

+4 9σ1

1−

qσ₁³ +16

63σ1 1−σ₁³₋ ¹

28σ₁⁴logσ1+ 1 14

qσ₁⁹1− 1

√σ1

, (3.15)

where∇=q(z)f(t), k²=r|r_0P=θ=R_0P=θ/[q(z)f(t)], B= [1/f(t)](d f(t)/dt), ξ= r/R, ξ1=r/R1, Ω=R₁/R, σ=k²/R, σ1=k²/R1andχ=R_0P/R1. The wall shear stressτwcan be obtained as below:

τw= (τ0N+εNτ1N)_r=R=τ0w+εNτ1w

=∇

R−BR³εN

8 1−Ω⁴

−BR₁³εNΩ 8

1−8

7σ₁^1/2+1 7σ1⁴

. (3.16)

Using Equations (3.7)–(3.9) and (3.13)–(3.15) in Equation (2.21), the volume flow rate is obtained as

Q=∇R⁴

1−Ω²

1+3Ω² +Ω⁴

1−16

7 σ₁^1/2+4 3σ1− 1

21σ1⁴

(3.17)

−εC∇BR³R₁³ 3

8Ω⁻¹−1 2Ω+1

8Ω³+1

2Ω³logΩ

−Ω³logΩ1 2−4

7

√σ1+ 1

14σ₁⁴+Ω 1−Ω² 1

4−2 7

√σ1+ 1 28σ₁⁴ +Ω³1

6−30 77

√σ1+ 8 35σ1−1

3σ₁^5/2+ 1

14σ₁⁴+ 5 21σ₁^9/2

−41

770σ1⁶− 1

14σ1⁶logσ1+ 1

14σ1⁴ 1−σ1²

log k

−εN∇BR⁵R₁ 1

6Ω⁻¹−3 8Ω+ 5

24Ω⁵−1

2Ω³ 1−Ω² log R₁

+Ω⁴ 1−Ω²

(1+2 logR₁) 1

4−2 7

√σ1+ 1 28σ1⁴

. The shear stressτC=τ0C+εCτ1C at r=R_Pis given by

(3.18) |τ0C+εCτ1C|r=RP=θ.

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Using the Taylor series ofτ0Candτ1Cabout R_0Pand usingτ0C

_r=R

0p=θ^{, we get}

(3.19) R_1P=τ1C

_r=R

∇ oP.

Using Equations (3.6), (3.11) and (3.19) in the two term approximated perturbation series of RP, the expression for RPcan be obtained as

(3.20) R_p=k²−BεCR³ 4

σ^{2 1}−Ω² +Ω³

σ1−4

3σ₁^3/2+1 3σ1³

.

The resistance to flow is given by

(3.21) Λ=∆p f(t)

Q ,

where∆p is the pressure drop. When R₁=R, the present model reduces to the single fluid Casson model and in such case, the expressions obtained in the present model for velocity u_C, shear stressτC, wall shear stressτw, flow rate Q and plug core radius R_pare identical with those of Chaturani and Ponnalagar Samy [3].

4. Results and discussion

The objective of the present model is to analyze the effects of the pulsatility, non-Newtonian nature, peripheral layer and stenosis on various flow quantities in a blood flow through a stenosed artery when blood is modeled by a two-fluid model with a core region of suspen- sion of red cells represented by the Casson fluid and a peripheral layer of plasma treated as the Newtonian fluid. In this study, we have used the range 0–0.15 for the yield stress θ. Though the amplitude A varies from 0 to 1, the range 0.2–0.5 is used to pronounce its effect. The ratioα ^{( =}αC/αN) between the pulsatile Reynolds numbers of the Newtonian fluid and Casson fluid is called the pulsatile Reynolds number ratio. Though, the ratioα ranges from 0 to 1, the range 0.2 to 0.5 is used [11]. The same range is used for the pulsatile Reynolds numberαC [15]. Given the values ofα^andαC, the value ofαN is obtained from α=αN/αC. The value ofβis generally taken as 0.95 and 0.985 [15]. To deduce the present model to a single-fluid Casson model,β is assigned the value 1. The relations R₁=β^{R and} δC=βδPare used to estimate R₁andδC [15]. The range 0.1 to 0.15 is used forδP [14].

But, to compare the present model with the earlier results, the value 0.2 is used forδC. It is observed that in Equation (3.17), f(t), R and θ are known and, Q and q(z)are the unknowns to be determined. A careful observation of Equation (3.17) reveals the fact that q(z)is the pressure gradient of the steady flow. Thus, if steady flow is assumed, then Equation (3.17) can be solved for q(z)[10, 3]. For steady flow, Equation (3.17) reduces to

0= R⁴−4R²R₁²+3R₁⁴

y⁴−Q_Sy³ +

(R1y)⁴−16 7

√θ^pR₁y7

+4

3θ(R1y)³− 1 21θ⁴, (4.1)

where y=q(z)and Q_Sis the steady state flow rate. Equation (4.1) is solved numerically for y by using Newton-Raphson method with variation in the axial direction and yield stress withβ =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.

4.1. Pressure drop

The variation of pressure drop∆p (across the stenosis) in a time cycle for different values of A, θ ^andδP withβ =0.95 is depicted in Figure 2. 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 observed that for a given value of A, the pressure drop increases with the increase ofδP

or yield stressθwhen the other parameters held constant. It is found 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^◦whileθ ^andδPare held fixed. The pressure drop increases with the increase of the width of the peripheral layer thickness.

Figure 2. Variation of pressure drop in a time cycle for different values of A,θ,δPandβ 4.2. Plug core radius

The variation of plug core radius with axial distance for different values of A andδPwith β=0.95,αC=0.5,θ=0.1 and t=60^◦is shown in Figure 3. It is noted that the plug core radius decreases as z increases from 4 to 5 and it increases as z increases further from 5 to 6.

It is observed that for a given value ofδP, the plug core radius decreases with the increase of A and the same behavior is noted asδPincreases for a given value of A. It is also noticed that the plug core radius increases with the increase of the yield stress of the fluid in the core region. It is of interest to note that the plot of the single-fluid Casson model is in good agreement with Figure 4 of Chaturani and Ponnalagar Samy [3].

Figure 3. Variation of plug core radius with axial distance for different values of A,δP

andθwithβ=0.95,αC=0.95 and t=60^◦.

496

4.3. Velocity distribution

The velocity distributions for different values of A,α^,αC andβ ^{with z}=5,θ=δP=0.1 and t=45^◦are shown in Figure 4. One can notice the plug flow around r=0. It is found that for the given values ofα^,αH andβ, the velocity increases as A increases. Further, it is observed that for given values of A,α ^andαC, the velocity decreases considerably near r=0 asβ increases. The velocity decreases with the increase of the yield stress when all the other parameters are kept as constant. For given values of A andβ, the same behavior is noted for increasing values ofα^andαC, but the decrease is only a slight.

Figure 4. Velocity distribution for different values of A,α,αC,βandθwith t=45^◦,δP=0.1 and z=5.

4.4. Wall shear stress

The variation of wall shear stress with axial distance for different values ofθ ^andαN with t=45^◦, A=0.5,β=0.95 andδP=0.1 is shown in Figure 5. One can notice that the wall shear stress increases as z increases from 4 to 5 and then it decreases symmetrically as z increases further from 5 to 6. It is found that the wall shear stress increases with the increase of the amplitude of the flow whileθ ^andαH are kept as constants. Also, it is noticed that for a given value ofθ and increasing values ofαN, the wall shear stress decreases slightly while the other parameters were kept as constants.

Figure 5. Variation of wall shear stress with axial distance for different values of A,θ andαNwith t=45^◦,β=0.95 andδP=0.2.

4.5. Resistance to flow

Figure 6 depicts the variation of resistance to flow in a time cycle for different values of A,θ^,β^,α ^andαH withδp=0.1. It is clear that the resistance to flow decreases as time t increases from 0^◦ to 90^◦and then increases as t increases from 90^◦to 270^◦and again decreases as t increases further from 270^◦to 360^◦. The resistance to flow is minimum at 90^◦and maximum at 270^◦. It is observed that for fixed values ofβ^,α^andαHand increasing values of A, the resistance to flow decreases when t lies between 0^◦and 180^◦and increases when t lies between 180^◦and 360^◦. The resistance to flow increases with increasing values ofθ^andδpwhile all other parameters are held fixed. It is noticed that the resistance to flow decreases asβ increases when the other parameters held fixed.

Figure 6. Variation of resistance to flow in a time cycle for different values of A,θ,β,α,αCandδP.

5. Conclusions

The present mathematical analysis brings out many interesting fluid mechanical phenomena due to the presence of the peripheral layer. It is observed that the pressure drop, plug core radius, wall shear stress and resistance to flow increase as the yield stressθ or stenosis size δPincreases while all other parameters are held constant. It is also found that the velocity increases and the plug core radius decreases as A increases. Thus, the results demonstrate that the present model is capable of predicting the hemodynamic features most interesting to physiologists. Thus, the presence of the peripheral layer helps in the functioning of the diseased arterial system. The extension of the present study to the blood flow through arter- ies with elastic walls may have more applications in the medical field which would be done in the near future.

Acknowledgement. The author thanks the referees for their constructive comments and useful suggestions which helped to improve the presentation of the paper. This research work was supported by the research university grant of Universiti Sains Malaysia, Malaysia (RU Grant Ref. No: 1001/PMATHS/811177).

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