Mathematical Analysis of non-Newtonian Fluid … Vol 13...ISSN 1749-3889 (print), 1749-3897 (online)...

ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.13(2012) No.1,pp.15-27

Mathematical Analysis of non-Newtonian Fluid Flow through Multiple StenoticArtery in the Presence of Catheter-A Pulsatile Flow

D. Srikanth ∗, Kebede TaddesseDepartment of Applied Mathematics, Defence Institute of Advanced Technology (DU), Pune - 411 025, India.

(Received 1 June 2011, accepted 6 September 2011)

Abstract: Mathematical analysis of blood flow through an artery with multiple stenosis in the presenceof catheter is investigated. Blood flow is assumed to be represented by an incompressible Non-Newtonian(Micropolar and Couple stres fluid characterized by pulsatile flow in the constricted artery. The non lineargoverning equations are solved and the closed form of solutions are obtained for velocity and microrotationcomponents in terms of Bessel functions of the first and second kind. The impedance (resistance to the flow)and wall shear stress are calculated and the effect of various parameters on them are discussed. It is observedthat the resistance to the flow increase as the height of any one of the multiple stenosis increases and sameobservation is found when catheter size increases. However the trend is reversed in case of micropolar pa-rameter m and couple stress fluid parameters φ and σ.

Keywords: Miropolar fluid; couple stress fluid; multiple stenosis; catheter; pulsatile flow

1 IntroductionDiseases in the blood vessels and in the heart, such as heart attack and stroke, are the major causes of mortality worldwide. The underlying cause for these events is the formation of lesions, known as atherosclerosis, in the large and medium-sized arteries in the human circulation. Atherosclerosis is a vascular pathology that has become a prominent disease inWestern society. The term comes from the Greek words athero (gruel or paste) and sclerosis (hardness), and the disorderis characterized by progressive narrowing and occlusion of blood vessels. When fatty substances, cholesterol, cellularwaste products, calcium, and fibrin build up in the inner lining of an artery, this causes a narrowing of the lumen of thevessel and also an increase in the wall stiffness or a decrease in compliance of the vessel. The buildup that results iscalled plaque. Some level of stiffening of the arteries and narrowing is a normal result of aging. Eventually, the plaquecan block an artery and restrict flow through that vessel, resulting in a heart attack if the vessel being blocked is one thatsupplies blood to the heart. Atherosclerosis can also produce blood clot formation, sometimes resulting in a stroke. Whena piece of plaque breaks away from the arterial wall and flows downstream, it can also become lodged in smaller vesselsand block flow, also resulting in the formation of thrombosis which cause stroke. Atherosclerosis usually affects medium-sized or large arteries. The common feature in the location for the development of the lesion is the presence of curvature,branching, and bifurcation present in these sites. The fluid dynamics at these sites can be anticipated to be vastly differentfrom other segments of the arteries that are relatively straight and devoid of any branching segments. As mentioned above,atherosclerosis occurs when the nature of blood flow changes from its usual state to a disturbed flow condition due to thepresence of a stenosis in an artery. The initiation and development of atherosclerotic plaques is depicted in Figure 1.Several researchers [[1],[2],[3]] have studied the flow of blood in stenosed artery by considering it as a Newtonian fluid.It is well known that blood, at low shear rates and during its flow through narrow blood vessels, behaves like a non-Newtonian fluid. Although there are many models to describe non-Newtonian behavior of the fluids, the micropolar fluidintroduced by Eringen [4] has a special importance, as it exhibits some microscopic effects arising from the local structureand micromotion of the fluid elements. Further, they can sustain couple stresses. The model of micropolar fluid representsfluids consisting of rigid randomly oriented (or spherical) particles suspended in a viscous medium where the deformationof the particles is ignored.The fluids containing certain additives, some polymeric fluids and animal blood are examples

∗corresponding author. E-mail address: sri [email protected], [email protected]

Copyright c⃝World Academic Press, World Academic UnionIJNS.2012.02.15/570

16 International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 15-27

of micro polar fluids. Kumar and Verma [5] discussed Effects of Stenosis on non-Newtonian Flow of Blood in Bloodvessels. The fluids containing certain additives, some polymeric fluids and animal blood are examples of micro polarfluids. Devanathan and Parvathamma [6] presented mathematical model for the steady flow of non-Newtonian micropolarfluid through stenotic region. Philip and Chandra [7] have studied the flow of blood, which has been modelled by a simplemicrofluid in the core region with a Newtonian fluid peripheral layer, in a tube in the presence of very mild stenosis.The mathematical theory of equations of micropolar fluids and applications of these fluids to the theory of lubricationand to the theory of porous media is presented by Lukaszewicz [8]. Srinivasacharya and Srikanth [9] investigated thepulsatile flow of micropolar fluid through catheterized mild stenotic artery. The couple stress fluid theory developed byStokes [10] represents the simplest generalization of the classical viscous fluid theory that sustains couple stresses andthe body couples. The important feature of these fluids is that the stress tensor is not symmetric and their accurate flowbehavior cannot be predicted by the classical Newtonian theory. The main effect of couple stresses will be to introduce asize dependent effect that is not present in the classical viscous theories. This model has been widely used because of itsrelative mathematical simplicity compared with other models developed for the couple stress fluid. The fluids consistingof rigid ,randomly oriented particles suspended in a viscous medium, such as blood fluids, lubricants containing smallamount of high polymer additive, electro-rheological fluids and synthetic fluids are examples of these fluids. An analysisof the effects of couple stresses on the blood flow through thin artery with mild stenosis has been carried out by Sinha andSingh [11]. Srivastava [12] considered the flow of couple stress fluid through stenotic blood vessels. Chaturani [13] hasanalyzed the problems of pulsatile flow of couple stress fluid with application to blood flow.

The flow through an annulus with mild constriction at the outer wall can be used as a model for the blood flowthrough the catheterized stenotic artery. The insertion of a catheter (a long flexible cylindrical tube) into a constrictedtube (i.e. stenosed artery) results in an annular region between the walls of the catheter and artery. This will alter theflow field, modify the pressure distribution and increase the resistance. Even though the catheter tool devices are used forthe measurement of arterial blood pressure or pressure gradient and flow velocity or flow rate, X-ray angiography and in-travascualar ultrasound diagnosis and coronary balloon angioplasty treatment of various arterial diseases, a little attentionhas been given in the literature to the flow in catheterized arteries. Roose and Lykoudis [14] studied the fluid mechanicsof the uretor with an inserted catheter by considering the peristaltic wave moving along the stationary cylinder.Sankarand Hemalatha [15] studied steady blood flow in catheterized artery by considering the blood as non- Newtonian fluidsin particular Hershley Berkley fluid. MacDonald [16] considered the pulsatile blood flow in a catheterized artery andobtained theoretical estimates for pressure gradient corrections for catheters, which are positioned eccentrically, as wellas coaxially with the artery. The effect of catheterization on various flow characteristics in an artery with or withoutstenosis was studied by Karahalios [17]. Dash et al. [18] considered the steady and pulsatile flow of the Casson fluidin a narrow artery when a catheter is inserted into it and estimated the increase in frictional resistance in the artery dueto catheterization. Srinivasacharya and Srikanth [19] analyzed pulsatile flow of couple stress fluid through catheterizedarteries.

In all the above analysis it was either multiple stenosis without catheter or simple stenosis in the presence of catheterare considered. Hence, in the present study, we have mathematically modelled pulsatile flow of blood through multiplestenotic artery in the presence of catheter by assuming the blood as non-Newtonian fluid. It will help us to understandthe behavior of physiological parameters under various parameters in a much better way. The velocity, microrotationcomponent, resistance to the flow (impedance) and shearing stress are calculated. The variation of impedance is analyzedfor various values of geometric, micropolar and couple stress parameters.

2 Formulation of the problemConsider the flow of an incompressible micropolar and couple stress fluid between an axi-symmetric rigid tube (artery)of radius ’a’ with a mild constrictions (stenosis) and a coaxial flexible tube (a catheter) of radius ’ka’ (k < 1). Assumingthat the flow is axi-symmetric and the stenosis over a length of the artery being assumed to have been developed in an axi-symmetric manner. Let the length of the tube be L, the magnitude of the distance along the artery over which the stenosisis spread out be Li, the locations of the stenosis be indicated by di and the maximum heights of the stenosis be hi(wherei=1,2,3). The schematic diagram is shown in Figure 2. The problem has been studied in cylindrical coordinate systems(r, θ, z), where the z-axis is taken along the axis of the artery while r and θ are along the radial and the circumferentialdirections, respectively. Since the flow is axi-symmetric all the variables are independent of θ. Hence for the Micropolar

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D. Srikanth, K. Taddesse: Mathematical Analysis of non-Newtonian Fluid Flow through Multiple Stenotic Artery · · · 17

Figure 1: Development of atherosclerotic plaques Figure 2: Development of atherosclerotic plaques

fluid the velocity vector is given by→q= (u (r, z, t) , 0, w (r, z, t)) and the microrotation vector is

→v= (0, u (r, z, t) , 0)

and also→q= (u (r, z, t) , 0, w (r, z, t)) for Couple stress fluids.

The radius of the stenosed artery, rs(Z) is given as

rs(z) =

a 0 ≤ Z ≤ d1

a− h1

2

(1 + cos 2π

L1

(z − d1 − L1

2

))d1 ≤ Z ≤ d1 + L1

a d1 + L1 ≤ Z ≤ d2

a− h2

2

(1 + cos 2π

L2

(z − d2 − L2

2

))d2 ≤ Z ≤ d2 + L2

a d2 + L2 ≤ Z ≤ d3

a− h3

2

(1 + cos 2π

L3

(z − d3 − L3

2

))d3 ≤ Z ≤ d3 + L3

a d3 + L3 ≤ Z ≤ L

(1)

2.1 Micropolar fluidThe equations governing the flow of an incompressible Micropolar fluid in the absence of body force and body couple are:

div→q= 0 (2)

ρ

[∂

→q

∂t+

(→q .∇

) →q

]= −grad (p) + κcurl

→v − (µ+ κ) curlcurl

→q (3)

ρj

[∂

→v

∂t+(→q .∇

) →v

]= −2κ

→v +κcurl

→q −γcurlcurl →

v +(α1 + β2 + γ) graddiv→v (4)

where→q is a velocity vector,

→v is the microrotation vector, p is the fluid pressure, ρ is fluid density, j is microgyration

parameter. Further the material constants µ,κ,α1,β1 and γ satisfy the following inequalities.

κ ≥ 0, 2µ+ κ ≥ 0, 3α1 + β1 + γ ≥ 0, γ ≥ |β1| (5)

It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in a tubewith mild stenosis. In this case the equations (2), (3), and (4) becomes:

−∂p∂r

= 0 (6)

ρ∂w

∂t= −∂p

∂z+κ

r

∂ (rv)

∂r+ (µ+ κ)

[1

r

∂

∂r

(r∂w

∂r

)](7)

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ρj∂v

∂t= −2κv − κ

∂w

∂r+ γ

[∂

∂r

1

r

∂

∂r(rv)

](8)

The non-dimensional variables are defined as

r = ar w = w0w p =w0µp

aα2 =

a2Ωρ

µz = Lz v =

w0v

aj = a2j rs (Z) = ars t =

t

Ω(9)

where, w0 is typical axial velocity, L is typical length, Ω is the circular(angular) frequency, µ is blood viscosity, and α2 isthe Womersley number.Introducing non-dimensional variables into equations (1),(6),(7) and (8) and dropping tildes, we get,

α2 ∂w

∂t= −∂p

∂z+

N

1−N

1

r

∂ (rv)

∂r+

(1

1−N

)D2w (10)

jα2 1−N

N

∂v

∂t= −2v − ∂w

∂r+

(2−N

m2

)∂

1r∂(rv)

r

∂r

(11)

rs(Z) =

1 0 ≤ Z ≤ d1

1− ϵ12

(1 + cos 2π

γ1

(z − d1 − γ1

2

))d1 ≤ Z ≤ d1 + γ1

1 d1 + γ1 ≤ Z ≤ d2

1− ϵ22

(1 + cos 2π

γ2

(z − d2 − γ2

2

))d2 ≤ Z ≤ d2 + γ2

1 d2 + γ2 ≤ Z ≤ d3

1− ϵ32

(1 + cos 2π

γ3

(z − d3 − γ3

2

))d3 ≤ Z ≤ d3 + γ3

1 d3 + γ3 ≤ Z ≤ 1

(12)

where ϵi = hi

a , γi = Li

L , N = kµ+k is coupling number (0 ≤ N ≤ 1), m2 = a2k(2µ+k)

γ(µ+k) is the Micropolar fluid parameter,D2 = 1

r∂∂r

(r ∂∂r

)(i = 1, 2, 3)

2.2 Couple stress fluidThe equations governing the flow of an incompressible Couple stress fluid in the absence of body force and body coupleare:

div→q= 0 (13)

ρ

[∂

→q

∂t+

(→q .∇

) →q

]= −gradp− µcurlcurl

→q −ηcurlcurlcurlcurl

→q (14)

where ρ is the density,→q is the velocity vector, η is the couple stress fluid parameter, p is the fluid pressure and µ is the

fluid viscosity.The force stress tensor τ and the couple stress tensor M that arises in the theory of couple stress fluids are given by

τ =(−

→q +λ1div

→q)I + µ

[grad

→q +

(grad

→q)T

]+I

2∗ [divM + ρC] (15)

andM = mI + 2η1grad

(curl

→q)+ 2η1

(grad

(curl

→q))T

(16)

where m is 1/3rd trace of M and ρ C is the body couple tensor. The quantity λ1 is the material constant and η is theconstant associated with couple stresses. The dimensions of the material constant λ1 is that of viscosity whereas thedimensions of η and η

′are those of momentum. These material constants are considered by the inequalities.

µ ≥ 0, 3λ+ 2µ ≥ 0, η′≤ η, η ≥ 0 (17)



It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds number flow in a tubewith mild stenosis. In this case Eqs. (13) and (14) become

−∂p∂r

= 0 (18)

ρ∂w

∂t= −∂p

∂z+ µD2w − ηD4w (19)

Introducing non-dimensional variables from Eq. (9) into Eq. (18) and (19) and dropping tildes, we get

−∂p∂r

= 0 (20)

α2 ∂w

∂t= −∂p

∂z+D2w − 1

φ2D4w (21)

where φ = a(

µη

) 12

is the Couple stress fliud parameter. The term − 1φ2D

4w in Eq. (21) gives the effect of couple stresses.Hence, as φ increases, the effect of couple stress decreases.

3 Solution of the problemSince the flow is pulsatile, we seek the solution of the form,

w = w∗ expit, p = p∗ expit, v = v∗ expit (22)

3.1 Micropolar fluidSubstituting Eq. (22) in to Eq. (10) and (11), we get

iα2w∗ = −∂p∗

∂z+

(N

1−N

)1

r

∂(rv∗)

∂r+

(1

1−N

)D2W ∗ (23)

(N − 1

N

)ijα2v∗ = 2v∗ +

∂w∗

∂r−

(2−N

m2

)[∂

∂r

1

r

∂(rv∗)

∂r

](24)

Differentiating Eq. (23) with respect to r and substituting in Eq. (24),we get

[2N + ijα2m2 (1−N)

]m2v∗ = −

[(Nm2 − (2−N) (1−N) iα2

) ∂w∗

∂r+ (2−N)

∂(D2w∗)∂r

](25)

Substituting Eq. (25) in Eq. (23) we get a differential equation of the form

(D2 − η21

) (D2 − η22

)w∗ = −η

21η

22

iα2

∂p∗

∂z(26)

where

η21 + η22 = m2 + (1−N) iα2 +m2ijα2 (1−N)

N (2−N)(27)

η21η22 =

(1−N) iα2m2(2N + (1−N)ijα2

)N(2−N)

(28)

The corresponding boundary conditions are

w∗ = 0&v∗ = 0 at r = rs(z) , w∗ = 0&v∗ = 0 at γ = k (29)

The solution of the Eq. (26) is given by

w∗(r) = C1(z)I0(η1r) + C2(z)K0(η1r) + C3(z)I0(η2r) + C4(z)K0(η2r) +i

α2

∂p∗

∂z(30)



where I0(ηir) and K0(ηi, r), (i=1,2) are Modified Bessel’s functions of first and second kind of zeroth order and C1(z),C2(z), C3(z) and C4(z) are arbitrary functions of z and determined by using boundary conditions. Substituting Eq. (30)into Eq. (25), we get the microrotation component as

v∗ = Cη1 [C1(z)I1(η1r)− C2(z)K1(η1r)] + Cη2 [C3(z)I1(η2r)− C4(z)K1(η2r)] (31)

where I1(ηir) and K1(ηir)(i=1,2) are Modified Bessel’s functions of the first order, first and second kind and

Cη1 =

[(2−N)

((1−N)iα2 − η21

)−Nm2

]η1

[2N + ijα2(1−N)]m2, Cη2 =

[(2−N)

((1−N)iα2 − η22

)−Nm2

]η2

[2N + ijα2(1−N)]m2(32)

Using the boundary conditions (29), we can obtain the values of C1(z), C2(z), C3(z), and C4(z).Hence the complete solution of the axial velocity is w(r, z, t) = realpartof

[expit w∗(r, z)

]and microrotation compo-

nents is v(r, z, t) = realpartof[expit v∗(r, z)

].

The dimensionless flux, defined as Q =∫ rs(Z)

k2rw∗dr can be be obtained from equation (30) in the following form

Q =dp∗

dzF [rs(z), k] (33)

where

F [rs(z), k] =2

η1[d1(z) (rs(z)I1(η1rs(z))− kI1(η1k))− d2(z) (rs(z)k1(η1rs(z))− kK1(η1k))]

+2

η2[d3(z) (rs(z)I1(η2rs(z))− kI1(η2k))− d4(z) (rs(z)K1(η2rs(z))− kK1(η2k))]

+i

α2

((rs(z))

2 − k2)

(34)

with dp∗

dz di(z) = Ci(z), i=1,2,3,4The pressure drop ∆p (is p1 at z = 0 and p0 at z = L) across the tube is obtained from equation (33) as

∆p = Q

∫ L

0

dz

F [rs(z), k](35)

The dimensionless resistance to the flow (resistive impedance), λ is given by

λ =∆p

Q=

∫ 1

0

dz

F [rs(z), k](36)

which can be written as

λ =

∫ d1

0

dz

F [rs(z), k]+

∫ d1+γ1

d1

dz

F [rs(z), k]+

∫ d2

d1+γ1

dz

F [rs(z), k]+

∫ d2+γ2

d2

dz

F [rs(z), k]+

∫ d3

d2+γ2

dz

F [rs(z), k]

+

∫ d3+γ3

d3

dz

F [rs(z), k]+

∫ 1

d3+γ3

dz

F [rs(z), k](37)

Since rs(z) = 1 in the regions 0 ≤ z ≤ d1, d1 + γ1 ≤ z ≤ d2, d2 + γ2 ≤ z ≤ d3 and d3 + γ3 ≤ z ≤ 1, the resistance tothe flow (γ) simplifies to

λ =1− γ1 − γ2 − γ3F [rs(z), k]rs(z)=1

+ γ1

∫ 1

0

dψ1

F [rs(ψ1), k]+ γ2

∫ 1

0

dψ2

F [rs(ψ2), k]+ γ3

∫ 1

0

dψ3

F [rs(ψ3), k](38)

where ψi =(z−di)

γi, i=1,2,3.

The shearing stress at the wall is given by

τrz = − Q

F [rs(z), k]

1

1−N[η1 d1(z)I1(η1rs(z))− d2(z)K1(η1rs(z))+ η2 d3(z)I1(η2rs(z))− d4(z)K1(η2rs(z))]

(39)which is same as in case of simple stenosis. Further it is to be noted that whether it is simple stenosis or multiple stenosisthe results in case of shear stress are not effected.



3.2 Couple stress fluidSubstituting equation (22) in to equation (21) for p and w and noting that p is a function of z only from Eq. (20), Eq. (21)simplified to the form (

D2 − α21

) (D2 − α2

2

)w = −φ2 ∂p

∗

∂z(40)

whereα21 + α2

2 = φ2andα21α

22 = iα2φ2 (41)

The corresponding non-dimensional boundary conditions are

w = 0 at r = rs(z)&r = k (42)

∂2w∗

∂r2− σ

r

∂w∗

∂r= 0 at r = rs(z)&r = k (43)

where σ = η′

η is a couple stress fluid parameter. Boundary condition (43) shows that couple stresses (16) vanish at the

tube wall and catheter wall. All the effects of couple stresses will be absent in a material for which η′= η i.e. σ = 1.

This is equivalent to requiring that the couple stress tensor be symmetric. If couple stress tensor is symmetric, then all itseffects will be absent.Using the separation of variables, the solution of Eq. (40) is

w∗(r) = C1(z)I0(α1r) + C2(z)K0(α1r) + C3(z)I0(α2r) + C4(z)K0(α2r) +i

α2

∂p∗

∂z(44)

where I0(αi, r) and K0(αi, r), (i = 1, 2) are called modified Bessel’s functions of first and second kind of zeroth orderrespectively while C1(z), C2(z), C3(z) and C4(z) are arbitrary functions of z.The dimensionless flux, defined as Q =

∫ rs(z)

k2rw∗dr can be be obtained from Eq. (44) in the following form

Q =dp∗

dzF [rs(z), k] (45)

where

F [rs(z), k] =2

α1[d1(z)rs(z)I1(α1rs(z))− kI1(α1k) − d2(z)rs(z)K1(α1rs(z))− kK1(α1k)]

+2

α2[d3(z)rs(z)I1(α2rs(z))− kI1(α2k) − d4(z)rs(z)K1(α2rs(z))− kK1(α2k)]

−(α21 + α2

2

)α21α

22

((rs(z))

2 − k2). (46)

with dp∗

dz di(z) = Ci(z), i = 1, 2, 3, 4The pressure drop ∆ p across the tube is obtained from Eq. (45) as

∆p = Q

∫ 1

0

dz

F [rs(z), k](47)

The dimensionless resistance to the flow (resistive impedance), λ is given by

λ =∆p

Q=

∫ 1

0

dz

F [rs(z), k](48)

Considering stenosed and non-stensoed region of the tube, Eq. (48) is written as

λ =1− γ1 − γ2 − γ3F [rs(z), k]rs(z)=1

+ γ1

∫ 1

0

dς1F [rs(ς1), k]

+ γ2

∫ 1

0

dς2F [rs(ς2), k]

+ γ3

∫ 1

0

dς3F [rs(ς3), k] 0

dz (49)



where ςi =(z−di)

γi, i = 1, 2, 3.

The shearing stress τrz at the wall is given by

τrz = α1

(1− α2

1

4 (α21 + α2

2)

)[d1(z)I1 (α1rs(z))− d2(z)K1 (α1rs(z))]

+α2

(1− α2

2

4 (α21 + α2

2)

)[d3(z)I1 (α2rs(z))− d4(z)K1 (α2rs(z))] . (50)

which is same as in case of simple stenosis.

4 Results and discussionsThe system of equations in terms of C1(z), C2(z), C3(z) and C4(z) are obtained by using the boundary conditions (29)in case of Micropolar and (42), (43) in case of Couple stress fluid. The dimensionless impedance and shear stress (38),(39), (49) and(50) for micropolar and couple stress fluids are evaluated numerically using Mathematica for various valuesof nondimensional geometric parameters and fluid parameters. The results are graphically presented in Figures 3 to 13 incase of Microrpolar fluid and 14 to 24 in case of Couple stress fluid. It is interesting to notice from these figures that theimpedance curves are intersecting at a particular value of t. Further, it is observed that the impedance distributions aredecreasing as t increases and then starts increasing after certain value of t. The distributions are negative for some periodof time and become positive. The time for which this change takes place corresponds to separation of flow [4].

The variation of impedance with t for different values of k, the size of the catheter and for the fixed values of othergeometric and micropolar parameters is shown in figure 3. It can be observed from this figure that the presence of catheterin a stenosed artery increases the impedance. Further, as the size of the catheter k increases the impedance increasessignificantly. Figure 4 shows the effect m, micropolar parameter on impedance for fixed value of other parameters. Hereit is observed that as m increases the impedance is slightly decreasing i.e. the effect of m on the impedance is not muchsignificant. The effect of N on the impedance is depicted in the Figure 5. It can be seen from this figure that as thevalue of N increases impedance increases significantly. Since in the limit N −→0, equations (3) and (4) reduce to thecorresponding relations for a viscous fluid, it is for one’s observation, that the impedance in the case of micropolar fluid ishigher than that of viscous fluid. Figure (6), (7) and (8) showed the effect of ϵ1, ϵ2, ϵ3. From these figures, it is observedthat impedance increases even if the height of one of the stenosis is slightly increased. This implies that each stenosis hastheir own effect for the increment of the impedance. Simultaneous effect of each stenosis on impedance for fixed valueof other parameters is presented in figure (9). It is observed that impedance is increased as the height of the stenosis isincreased from 0.1 to 0.3 simultaneously. In this case impedance is more than that of individual increment of the stenosissize which is shown in figures (6) to (8) respectively. Figures (10),(11)and(12) shows the effect of γ1, γ2, γ3 . In thiscase it is observed that as the length of one of the stenosis increases, the impedance is also increases. This indicates thateach stenosis has a very high effect for the increment of the impedance. Figure (13) shows that simultaneous effect ofall three stenosis on impedance for fixed values of other parameters. It is observed that as γ1, γ2 and γ3 are increasingsimultaneously, impedance is also increasing. It is also also worth noting that in case of multiple stenosis impedenceincreases by almost four to five times when compared to simple stenosis.

The variation of impedance with t for different values of the catheter size k, for the fixed values of couple stressparameters and geometric parameters is depicted in figure(14). k −→0 corresponds to the case when there is no catheter.It can be observed from this figure that the presence of catheter in a multiple stenosed artery increases the impedance.Further, as the size of the catheter, k increases the impedance increases significantly. Further it is also observed that itis increasing very significantly when compared to simple stenosis. Figure (15) shows the effect of φ, the couple stressparameter on the impedance. As φ increases impedance decreases rapidly. φ −→ ∞ corresponds to Newtonian fluidcase. It can be noted that the impedance in case of couple stress fluid is more than that of a Newtonian fluid case i.e., thepresence of couple stresses in the fluid increases the impedance. Figure (16) shows the effect of σ on impedance. Hereit is worth noting that as σ increases impedance decreases. The couple stresses will be absent for the case σ= 1. Fromthis figure also it can be observed that the impedance is more for couple stress fluids when multiple stenosis is considered. Figure (17), (18) and (19) shows that the effect of ϵ1,ϵ2 and ϵ3 on impedance for fixed value of other parameters. It isobserved that even as the height of one of the stenosis increases impedance increases and from figure (20), it is observedthat as the height of the stenosis is increased simultaneously, impedance is also increasing significantly. The effect ofγ1,γ2 and γ3 are shown in figure(21),(22) and (23). From these figures, it is observed that as the length of one of thestenosis increases impedance is also increases. Similarly, from figure (24) it is observed that simultaneous increment ofthe length of stenosis increases the impedance. The analysis for shear stress is not included in the present work as it is



same as simple stenosis.Graphical results in case of Micropolar fluid : Fig 3-13 and in case of Couple Stress fluid: Fig 14- 24

0 5 10 15 20 25 30 35−80

−60

−40

−20

0

20

40

60

80

t

Impe

danc

e

k=0.05k=0.1k=0.15k=0.2

Figure 3: Effect of k on Impedance with N =0.75,m = 10, j = 0.1, α = 1.5, ϵ1 = 0.1, ϵ2 =0.15, ϵ3 = 0.2, γ1 = 0.1, γ2 = 0.15, γ3 = 0.2.

0 5 10 15 20 25 30 35−60

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−20

0

20

40

60

t

Impe

danc

e

m=10m=25m=50m=100

Figure 4: Effect of m on Impedance with N =0.75, k = 0.1, j = 0.1, α = 1.0, ϵ1 = 0.1, ϵ2 =0.15, ϵ3 = 0.2, γ1 = 0.1, γ2 = 0.15, γ3 = 0.2.

0 5 10 15 20 25 30 35−80

−60

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−20

0

20

40

60

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t

Impe

danc

e

N=0.2N=0.4N=0.6N=0.8

Figure 5: Effect of N on Impedance withm=10,k=0.1, j=0.1, α=1.0 , ϵ1=0.1,ϵ2 =0.15,ϵ3=0.2, γ1 =0.1 ,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−60

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0

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60

t

Impe

danc

e

εi=0.1,0.1,0.1

εi=0.15,0.1,0.1

εi=0.2,0.1,0.1

εi=0.3,0.1,0.1

Figure 6: Effect of ϵ1 on Impedance withm=10,k=0.1, j=0.1,N=0.75, α=1.5 , ϵ2 =0.15,ϵ3=0.2, γ1 =0.1 ,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−60

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−20

0

20

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60

t

Impe

danc

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εi=0.1,0.1,0.1

εi=0.1,0.15,0.1

εi=0.1,0.2,0.1

εi=0.1,0.3,0.1


0 5 10 15 20 25 30 35−60

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0

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t

Impe

danc

e

εi=0.1,0.1,0.1

εi=0.1,0.1,0.15

εi=0.1,0.1,0.2

εi=0.1,0.1,0.3




0 5 10 15 20 25 30 35−80

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−20

0

20

40

60

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t

Imp

ed

an

ce

εi=0.1,0.1,0.1

εi=0.15,0.15,0.15

εi=0.2,0.2,0.2

εi=0.3,0.3,0.3

Figure 9: Effect of ϵ1,ϵ2,ϵ3 on Impedance withm=10,k=0.1, j=0.1,N=0.75, α=1.5 , γ1 =0.1 ,γ2=0.15,γ3=0.2.

0 5 10 15 20 25 30 35−50

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−20

−10

0

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30

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t

Imp

ed

an

ce

γi=0.1,0.1,0.1

γi=0.2,0.1,0.1

γi=0.3,0.1,0.1

γi=0.4,0.1,0.1

Figure 10: Effect of γ1 on Impedancewith m=10,k=0.1, j=0.1,N=0.75, α=1.0,ϵ1=0.1,ϵ2=0.15,ϵ3=0.2, γ2=0.15, γ3=0.2.

0 5 10 15 20 25 30 35−60

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t

Imp

ed

an

ce

γi=0.1,0.1,0.1

γi=0.1,0.2,0.1

γi=0.1,0.3,0.1

γi=0.1,0.4,0.1

Figure 11: Effect of γ2 on Impedance with m =10, k = 0.1, j = 0.1, N = 0.75, α = 1.0, ϵ1 =0.1, ϵ2 = 0.15, ϵ3 = 0.2, γ1 = 0.1γ3 = 0.2.

0 5 10 15 20 25 30 35−60

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t

Imp

ed

an

ce

γi=0.1,0.1,0.1

γi=0.1,0.1,0.2

γi=0.1,0.1,0.3

γi=0.1,0.1,0.4

Figure 12: Effect of γ3 on Impedance with m =10, k = 0.1, j = 0.1, N = 0.75, α = 1.0, ϵ1 =0.1, ϵ2 = 0.15, ϵ3 = 0.2, γ1 = 0.1, γ2 = 0.15.

0 5 10 15 20 25 30 35−60

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t

Imp

ed

an

ce

γi=0.1,0.1,0.1

γi=0.2,0.2,0.2

γi=0.3,0.3,0.3

Figure 13: Effect of γ1,γ2,γ3 on Impedance withm = 10, k = 0.1, j = 0.1, N = 0.75, α =1.0, ϵ1 = 0.1, ϵ2 = 0.15, ϵ3 = 0.2.

0 5 10 15 20 25 30−600

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−200

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600

t

Imp

ed

an

ce

k =0.05k =0.1k =0.15k=0.2

Figure 14: Effect of k on Impedance with φ =1.0, σ = 0.5, α = 1.5, ϵ1 = 0.1, ϵ2 = 0.15, ϵ3 =0.2, γ1 = 0.1, γ2 = 0.15, γ3 = 0.2.



0 5 10 15 20 25 30 35−1500

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−500

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500

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t

Imp

ed

an

ce

φ=0.5φ =1.0φ =1.5φ =2.0

Figure 15: Effect of φ on Impedance with k =0.1, σ = 0.5, α = 1.5, ϵ1 = 0.1, ϵ2 = 0.15, ϵ3 =0.2, γ1 = 0.1, γ2 = 0.15, γ3 = 0.2.

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t

Imp

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σ =0.2σ =0.4σ =0.6σ =0.8

Figure 16: Effect of σ on Impedance with k =0.1, σ = 0.5, α = 1.5, ϵ1 = 0.1, ϵ2 = 0.15, ϵ3 =0.2, γ1 = 0.1, γ2 = 0.15, γ3 = 0.2.

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t

Imp

ed

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ce

εi =0.1,0.1,0.1

εi =0.15,0.1,0.1

εi =0.2,0.1,0.1

εi =0.3,0.1,0.1

Figure 17: Effect of ϵ1 on Impedance with k=0.1,φ=1.5,α=1.5,σ=0.5, ϵ2=0.15,ϵ3=0.2,γ1=0.1,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−250

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t

Imp

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an

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εi =0.1,0.1,0.1

εi =0.1,0.15,0.1

εi =0.1,0.2,0.1

εi =0.1,0.3,0.1

Figure 18: Effect of ϵ2 on Impedance with k=0.1,φ=1.5, α=1.5,σ=0.5, ϵ1=0.1,ϵ3=0.2,γ1=0.1,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−250

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t

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an

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εi =0.1,0.1,0.1

εi =0.1,0.1,0.15

εi =0.1,0.1,0.2

εi =0.1,0.1,0.3

Figure 19: Effect of ϵ3 on Impedance with k=0.1,α=1.5,φ=1.5,σ=0.5, ϵ1=0.1,ϵ2=0.15,γ1=0.1,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−300

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t

Imp

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εi =0.1,0.1,0.1

εi =0.15,0.15,0.15

εi =0.2,0.2,0.2

εi =0.3,0.3,0.3

Figure 20: Effect of ϵ1,ϵ2,ϵ3 on Impedance withk = 0.1, φ = 1.5, α = 1.5, σ = 0.5, γ1 =0.1, γ2 = 0.15, γ3 = 0.2.



0 5 10 15 20 25 30 35−200

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t

Imp

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ce

γi =0.1,0.1,0.1

γi =0.2,0.1,0.1

γi =0.3,0.1,0.1

γi =0.4,0.1,0.1

Figure 21: Effect of γ1 on Impedance with k=0.1,α=1.5,σ=0.5,φ=1.5,ϵ1=0.1,ϵ2=0.15,ϵ3=0.2,γ2 =0.15,γ3=0.2.

0 5 10 15 20 25 30 35−200

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t

Imp

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an

ce

γi =0.1,0.1,0.1

γi =0.1,0.2,0.1

γi =0.1,0.3,0.1

γi =0.1,0.4,0.1

Figure 22: Effect of γ2 on Impedance with k=0.1,α=1.5,σ=0.5,φ=1.5, ϵ1=0.1,ϵ2=0.15,ϵ3=0.2,γ1 =0.1,γ3=0.2.

0 5 10 15 20 25 30 35−200

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Imp

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ce

γi =0.1,0.1,0.1

γi =0.1,0.1,0.2

γi =0.1,0.1,0.3

γi =0.1,0.1,0.4

Figure 23: Effect of γ3 on Impedance with k=0.1,α=1.5,σ=0.5,φ=1.5, ϵ1=0.1,ϵ2=0.15,ϵ3=0.2,γ1 =0.1,γ2=0.15.

0 5 10 15 20 25 30 35−250

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t

Imp

ed

an

ce

γi =0.1,0.1,0.1

γi =0.2,0.2,0.2

γi =0.3,0.3,0.3

Figure 24: Effect of γ1,γ2,γ3 on Impedance withk=0.1, α=1.5,σ=0.5,φ=1.5,ϵ1=0.1,ϵ2=0.15,ϵ3=0.2.

i = 1, 2, 3 which represent 1st, 2nd and 3rd stenosis respectively.

5 ConclusionsThis work presents an analytical solution for the micropolar and couple stress fluid through an annulus with mild constric-tions at the outer wall. The resistive impedance and wall shear stress are calculated and the effects of geometric, micropolarand couple stress fluid parameters on the impedance are studied. The presence of catheter in a multiple stenosed arteryand the presence of microstructure and couple stresses in the fluid increases the impedance and shear stress. From thepresent work, it is observed that the impedance is more in case of couple stress fluid than the micropolar fluids.It is alsoobserved that as the number of stenosis increases, impedence increases or decreases very significantly. The shear stress isanother important physiological factor which is calculated at maximum height of the stenosis. All the results have beencompared with earlier analysis by other authors and it has found that this work is giving good results. For fixed radius andsymmetric shape of stenosis, shear stress is independent of number of stenosis and the results same as in case of simplestenosis and are in [9] and [19] and hence not included in the present analysis.

AcknowledgmentsThe authors would like to thank Mr. J V Ramana Reddy and Mr. Prasad Arulkar, Research Scholars of the department fortheir effort in preparing the LaTex file.



References[1] Young DF. Effect of time dependent stenosis on flow through a tube. 90 (1968) pp. 248-254. J. Eng. For Ind., Trans

of ASME., 90 (1968): 248-254.[2] Lee JS and Fung YC. Flow in locally constricted tubes at low Reynolds numbers. J. Appl. Mech.Trans of ASME.37

(1970): 9-16.[3] Padmanabhan. Mathematical model of arterial stenosis. Med. Biol.Eng. Comput., 8 (1980):281-286.[4] Eringen AC. Theory of micropolar fluids. J. Math. Mech., 16 (1966): 1-16.[5] B.Kumar, M.Verma. Effects of Stenosis on non-Newtonian Flow of Blood in Blood vessels.. Australian Journal of

Basic and Applied Sciences.,4(4) (2010):588-601.[6] Devanathan and Parvathamma. Flow of Micropolar Fluid through a tube with stenosis. Med.&Bio.Eng. & Comp.21

(1983), pp.438-445.[7] Philip D and Chandra. Flow of Eringen fluid (simple microfluid) Through an artery with mild stenosis. Int. J. Eng.

Sci. 34 (1996): 87-99.[8] Lukaszewicz G. Micropolar fluids-theory and applications. Birkhauser Boston(1999).[9] D. Srinivasacharya and D. Srikanth. Pulsatile Flow of a Micropolar Fluid Through Constricted Annulus. Int. Journal

of Applied Math. And Mechanics 3(3) ( 2007):36-48.[10] V.K.Stokes. Couple stresses in fluids. Phys. Fluids 9 (1966) 1710.[11] P.Sinha, C.Singh. Effects of couple stresses on the blood flow through an artery with mild Stenosis. Biorheology 21

(3) (1984) 303.[12] L.M.Srivastava. Flow of couple stress fluid through stenotic blood vessels. J. Biomech. (1985).[13] P.Chaturani, Upadhya. Pulsatile flow of a couple stress fluid through circular tubes with applications to blood flow.

Biorheology 15 (3-4) (1978) 193.[14] Roose Rudolf and Lykodis S Paul. The fluid mechanics of the ureter with an inserted catheter. J.Fluid Mech. 46

(1971): 625-630.[15] Sankar Sankar,D.S. and Hemalatha, K. Pulsatile flow of Herschel-Bulkley fluid through catheterized arteries - A

mathematical model. Applied mathematical modeling 31 (2007):1497-1517.[16] MacDonald DA. Pulsatile flow in a catheterized artery. J. Biomech. 19 (1986):239-249.[17] G.T.Karahalios. Some possible effects of a catheter on the arterial wall. Med. Phys. 17 (1990) 922.[18] Dash.RK, Jayaraman.G and Mehta.KN. Estimation of increased flow resistance in a narrow catheterized artery - a

theoretical model. J. Biomech. 29 (1996): 917-930.[19] D.Srinivasacharya and D. Srikanth. Effect of Couple stresseson the pulsatile flow through constricted annulus.

Comptes Rendus Mecanique 336 (2008):820-827.[20] Abramowitz M, Stegun IA. Handbook of Mathematical Functions. Applied Mathematics Series No.55, National

Bureau of Standards,Washington, DC, 1964.


Mathematical Analysis of non-Newtonian Fluid … Vol 13...ISSN 1749-3889 (print), 1749-3897 (online)...

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Transcript of Mathematical Analysis of non-Newtonian Fluid … Vol 13...ISSN 1749-3889 (print), 1749-3897 (online)...