A Mathematical Study of Non-Newtonian Micropolar Fluid in...

Appl. Math. Inf. Sci.8, No. 4, 1567-1573 (2014) 1567

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/080410

A Mathematical Study of Non-Newtonian MicropolarFluid in Arterial Blood Flow Through CompositeStenosisR. Ellahi1,∗, S. U. Rahman1, M. Mudassar Gulzar2, S. Nadeem3 and K. Vafai4

1 Department of Mathematics and Statistics, IIUI, Islamabad, Pakistan2 NUST College of Electrical and Mechanical Engineering, Islamabad, Pakistan3 Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan4 Department of Mechanical Engineering, University of California, Riverside, USA

Received: 11 Jul. 2013, Revised: 13 Oct. 2013, Accepted: 14 Oct. 2013Published online: 1 Jul. 2014

Abstract: The unsteady and incompressible flow of non-Newtonian fluid through composite stenosis is investigated in the presentstudy. The micropolar fluid is treated as a blood flow model. Mild stenosis andslip velocity are also taken into account. The governingequations are given in cylindrical coordinates system. Analytic solutions of velocity and volumetric flow flux are developed interms ofmodified Bessel functions. The expressions for the impedance (flow resistance)λ , the wall shear stress distribution in the stenotic regionTw and the shearing stress at the stenosis throatTs are also given. Impact of involved pertinent parameters is sketched and examined bythe resistance of impedance and shear stress. The stream lines are alsomade for different sundry parameters.

Keywords: Micropolar fluid, Blood flow, Stenosed arteries, Exact solutions

1 Introduction

In the last three decades, the study of non-Newtonianfluids [1–7] has gained great importance and this ismainly due to their huge range of applications, such asblood flow, paints, melts of polymers, biological solutionsand glues but a careful review of the literature reveals thatnon-Newtonian micropolar fluid has yet received verylittle attention. The study of blood flow of non-Newtonianfluids in a stenosed artery is very important because of thefact that number of cardiovascular diseases in the bloodvessels such as hearts attacks and strokes are the leadingcause of death worldwide. Even though the considerableinventions for the diagnosis and treatment of thesedisorders have been made, but still this subject needsmore care and attention to avoid these diseases. Fewinvestigators have highlighted different aspects of bloodflow analysis in arteries.

Recently, Mekheimer and El Kot [8] have studied themathematical modeling of unsteady flow of such fluidsthrough an anisotropically tapered elastic arteries withtime variant overlapping stenosis. They [8] analytically

solved their mathematically model for mild stenosis case.Riahi et al [9] have examined the problem of blood flowin an artery in the presence of an overlapping stenosis. Amathematical study on three layered oscillatory bloodflow through stenosed arteries have been investigated byTripathi [10]. In a number has papers, Mekheimer and Elkot [11–14] have discussed the different aspects of bloodflow analysis in stenosed arteries. Very recently, Mishra etal [15] have studied the blood flow through a compositestenosis in an artery with permeable wall. Some relevantstudies on the topic can be seen from the list ofreferences [16, 17] and a number of references on thetopic can be found therein.

Moreover, getting an exact analytic solution of a givencoupled partial differential equation is often more difficultas compared to getting a numerical solution. However,results obtained by numerical methods give discontinuouspoints of a curve when plotted. Though numerical andanalytic methods for solving coupled partial differentialequations have limitations, at the same time they havetheir own advantages too. Therefore, we cannot neglecteither of the two approaches but usually it is pleasing to

∗ Corresponding author e-mail:[email protected], [email protected]

c© 2014 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/080410

1568 R. Ellahi et. al. : A Mathematical Study of Non-Newtonian Micropolar Fluid...

solve a coupled partial differential equationsanalytically [18–24].

Motivated by these facts, the present work has beenundertaken in order to get an exact solution of fullydeveloped flow of an incompressible blood flow ofmicropolar fluid through a composite stenosis in an arterywith permeable wall. Physical problem is first modelledand then simplified by using the nondimensionalvariables. The analytical solution of the simplifiedequations are found and the results for the resistanceimpedance, wall shear stress axial velocity, pressuregradient and stream functions are discussed throughgraphs for various physical parameters of the problem.After the introduction in Section 1, the outlines of thispaper are as follows. Section 2 contains mathematicalformulation. In Section 3 exact solutions of the problemsare presented. Discussion and results are given in Sections4. Finally Section 6 summaries the concluding remarks.

2 Mathematical formulation of problem

Consider an incompressible micropolar fluid is flowingthrough a composite stenosis in a circular artery withpermeable walls. We are considering cylindricalcoordinates(r,θ ,z) in such a way thatz−axis is takenalong the axis of the artery where asr, θ are the radialand circumferential directions respectively ( see Fig. 1).Let r = 0 is considered as the axis of the symmetry of thetube. The geometry of the stenosis is assumed to bemanifested in the arterial segment is described as

R(z)/R0 = 1−2δ

R0 L0(z−d) ; d ≤ z ≤ d +L0/2, (1)

= 1−2δR0

(

1+cos2πL0

(z−d −L0/2)

)

; d +L0/2≤ z ≤ d +L0, (2)

= 1; otherwise, (3)

whereR ≈ R(z) andR0 are the radius of the artery withand without stenosis, is respectively,L0 is the length ofthe stenosis and d indicates its location,δ is themaximum projection(maximum height) of the stenosis atz = d +L0/2. Consider the blood as micropolar fluid, theequations for the flow model are described as

∂v∂ r

+vr+

∂u∂ z

= 0 (4)

ρ(

∂u∂ t +u ∂u

∂ z + v ∂u∂ r

)

=− ∂ p∂ z +(µ + k)×

(

∂ 2u∂ r2 +

1r

∂u∂ r +

∂ 2u∂ z2

)

+ kr

∂ (r G)∂ r ,

(5)

ρ(

∂u∂ t + v ∂v

∂ z + v ∂v∂ r

)

=− ∂ p∂ r +(µ + k)×

(

∂ 2v∂ r2 +

1r

∂v∂ r −

vr2

)

− k ∂ G∂ z ,

(6)

ρ j(

v ∂G∂ r +u ∂G

∂ z

)

=−2kG− k(

∂u∂ r −

∂v∂ z

)

+ γ(

∂∂ r

(

1r

∂ (rG)∂ r

)

+ ∂ 2G∂ z2

)

.

(7)

Fig. 1: Flow geometry of a composite stenosis in an artery withpermeable wall.

Introducing the following nondimensional variables

x′ = xb , r = r

d0, u′= u

u0, v = bv

u0δ , h′ = hd0, p′ =

d20 p

u0bµ , j′ = jd2

0, G′ = d0G

u0.

Making use of above nondimensional quantities and forthe case of mild stenosis, we arrive at

∂ p∂ z

=1

1−N

(

∂ 2u∂ r2 +

1r

∂u∂ r

+Nr

∂∂ r

(rG)

)

, (8)

2G =−∂u∂ r

+2−N

m2

∂∂ r

(

1r

∂∂ r

(rG)

)

, (9)

where

N =k

µ + k, m2 =

d20k(2µ + k)

γ(µ + k). (10)

The boundary conditions are

∂u∂ r

= 0 atr = 0, (11)

u = uB and∂u∂ r

=α

√Dα

(uB −uporous) at r = R(z) , (12)

where uporous = −Daµ

∂ p∂ z , uporous is the velocity in the

permeable boundary,Da is the Darcy number andα(called the slip parameter) is a dimensionless quantitydepending on the material parameters which characterizethe structure of the permeable material with in theboundary region.

3 Solution of the problem

Equation(8) can be written as

∂∂ r

(

r∂u∂ r

+NrG− (1−N)r2

2∂ p∂ z

)

= 0. (13)

After integrating, we get

∂u∂ r

= (1−N)

(

r2

∂ p∂ z

+Ar

)

−NG, (14)


Appl. Math. Inf. Sci.8, No. 4, 1567-1573 (2014) /www.naturalspublishing.com/Journals.asp 1569

whereA is constant of integration. Invoking Eq.(14) intoEq.(9), we arrive at

∂ 2G∂ r2 +

1r

∂G∂ r

− (m2+1r2 )G =

(1−N)m2

(2−N)

(

r2

∂ p∂ z

+Ar

)

.

(15)The general solution of above equation is defined as

G = BI1(mr)+CK1(mr)−1−N2−N

(

r2

∂ p∂ r

+Ar

)

, (16)

where BI1(mr) and CK1(mr) are modified Besselfunctions of first order of first and second kindrespectively. Substituting Eq.(16) into Eq. (14) andintegrating with the help of boundary conditions, weobtain

u =(

1−N2−N

)

(

−∂ p∂ z

)(

R2− r2− NRm

(

I0(mR)−I0(mr)I1(mR)

))

+uB.

(17)The slip velocityuB is calculated as

uB =−√

Da(1−N)

2α

(

∂ p∂ z

)(

R−2α

√Da

1−N

)

. (18)

The volumetric flow flux,Q is thus calculated as

Q = 2π∫ R

0rudr,

or

Q = π(1−N)4(2−N)

∂ p∂ z R2

(

2√

Da(N−1)(N−2)R+4Da(N−1)+(N−1)R2α(N−1)α

+4Nm2 −

2NI0(mR)mI1(mR) R

)

.

(19)Note that whenN → 0, the results of Mishra et al [15] canbe recovered as a special case of our problem

Q =π(1−N)

4(2−N)

∂ p∂ z

F (z) , (20)

F (z) = R2

(

2√

Da(N−1)(N−2)R+4Da(N−1)+(N−1)R2α(N−1)α

+4Nm2 −

2NI0(mR)mI1(mR) R

)

,

(21)

∇p =

∫ L

0

(

−∂ p∂ z

)

, (22)

λ =4(2−N)

π (1−N)Ψ , (23)

where

Ψ =d∫

0

1F(z)R/R0=1

dz+d+L0/2∫

d

1F(z)R/R0 f rom(1)

dz

+d+L0∫

d+L0/2

1F(z)R/R0 f rom(2)

dz+L∫

d+L0

1F(z)R/R0=1

dz.(24)

The expressions for the impedance (flow resistance)λ ,the wall shear stress distribution in the Stenotic regionTwand the shearing stress at the stenosis throatTs in their non-dimensional form can be calculated as

λ = 1− L0L + η

L

∫ d+L0/2d

dz

a4(

1+ 2√

Da(N−2)

αR20a2

(

R0a+ 2α√

DaN−1

)

+ 4Na2m2R2

0

)

+ηL0

2πL

∫ π

0

1

θ 4(

1+ 2√

Da(N−2)αR2

0θ2

(

R0θ + 2α√

DaN−1

)

+ 4Nθ2m2R2

0

) ,

(25)

Tw =1

(

(R0/R)3+ 2√

Da(N−2)αR2

0

(

R0 (R/R0)2+ 2α

√Da(R/R0)N−1

)) ,

(26)

Ts =1

(

b3+ 2√

Da(N−2)αR2

0

(

R0b2+ 2α√

DabN−1

)) , (27)

where

a ≈ a(z) = 1−2δ (z−d)/R0L0, b = 1−δ/R0, c =−δ/2R0,

θ ≈ θ(β ) = b+cosβ , β = π − (2π/L)(z−d−L0/2), η = 1+ 2√

Da(N−2)αR2

0

(

R0+2α

√Da

N−1

)

.

4 Discussion and results

To observe the quantitative effects of the micropolarparameterm. The coupling numberN, computer codesare developed for the numerical evaluation of theanalytical results obtain for flow impedanceλ , the wallshear stressTw, for parameters valuesL = 1,2,3, L0 = 1,d = 0 and for various values of slip parameterα, darcynumber

√Da and stenosis heightδ/R0. It is noticed from

Fig. 2 that the flow impedanceλ increases by increasingthe stenosis heightδ/R0, while a decrease in impedanceis observed by increasing the slip parameterα and tubelengthL by keeping all other parameter fixed. The effectsof m, stenosis heightδ/R0 and Darcy’s number

√Da on

the impedanceλ are plotted in Fig. 3. It is depicted thatimpedance decreases with the increase inm. Moreover,impedance increases with the increases in stenosis heightδ/R0 and Darcy’s number

√Da. The effects of tube

lengthL and micro-rotation viscosityN on the impedanceare shown in Fig. 4. It is observed that impedancedecreases with the increase of bothL andN. The effectsof impedance againstm for various values ofN aresketched in Fig. 5. It is seen that with the increase inNimpedance decreases for all values ofm The wall shearstressTw against axial distancez for various values ofαand

√Da are given in Fig. 6. It is concluded that the wall

shear stress increases with the increase inα and decreaseswith the increase in

√Da. The wall shear stressTw for

different values ofm andN are shown in Fig. 7. It is seen


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Fig. 2: variation of resistance to flowλ with√

Da.

Fig. 3: variation of resistance to flowλ withα .

that shear stressTw increases by increasingN anddecreasingm. The streams lines for different parametersare shown in Figs. 8 to 11. It is seen that with theincreases inm, the size of trapping bolus decreases (seeFig. 8). Fig. 9 shows the streams lines for different valuesof N. It is observed that with the increases inN, the sizeof trapping bolus slightly decreases. The streams lines fordifferent values ofα are given in Fig. 10. Here thetrapping bolus increases with the increases inα. Thelarge Darcy’s number

√Da, causes to decrease the

trapping bolus (see Fig. 11).

5 Conclusion

A mathematical model for the blood flow of micropolarfluid through a composite stenosis in an artery withpermeable wall has been studied. The physical problem isfirst modelled and then simplified by using thenondimensional variables. The exact solution of thegoverning equations are solved analytically. The resultsfor the resistance impedance, wall shear stress, axial

Fig. 4: variation of resistance to flowλ withδ/R0.

Fig. 5: variation of resistance to flowλ with m.

Fig. 6: variation ofTw with z for various values ofα and√

Da.

velocity, pressure gradient and stream functions arediscussed through graphs for various physical parametersof the problem. To the best of our knowledge, no suchanalysis is available in the literature which can describethe effects of non-Newtonian micropolar fluid in arterialblood flow through composite stenosis simultaneously.



Fig. 7: variation ofTw with z for various values of m and N.

a b c

Fig. 8: Stream lines for(a) m = 0.5, (b) m = 1.0, (c) m = 1.4.

a b c

Fig. 9: Stream lines for(a) N = 0.1, (b) N = 0.2, (c) N = 0.3.

a b c

Fig. 10: Stream lines for(a) α = 0.03, (b) α = 0.04, (c) α =0.05.

a b c

Fig. 11: Stream lines for(a)√

Da = 0.34, (b)√

Da = 0.36, (c)√Da = 0.38.

The results presented in this paper are now available forfurther experimental verification to give confidence forthe well-posedness of this boundary value problem. Thefollowing main results are observed.

–The stenosis itself and the tapering effect change theflow pattern.

–The flow impedanceλ increases by increasing thestenosis heightδ/R0, while a decrease in impedanceis observed by increasing the slip parameterα andtube lengthL.

–Impedance decreases with the increase inm.–With the increase inN impedance decreases for allvalues ofm.

–The shear stressTw increases by increasingN anddecreasingm.

–By increasingm, the size of trapping bolus decreases.–Trapping bolus increases with the increases inα.–The large Darcy’s number

√Da, causes to decrease the

trapping bolus.–When N → 0, the results of Mishra et al [15] arerecovered as a special case of our problem.

References

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[2] A. Zeeshan and R. Ellahi, Series solutions of nonlinearpartial differential equations with slip boundary conditionsfor non-Newtonian MHD fluid in porous space, J. of Appl.Math. & Inf. Sci.,7, 253 (2013).

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[7] W. C. Tan and T. Masuoka, Stokes first problem for anOldroyd−B fluid in a porous half space. Phys. Fluids,17,023101 (2005).

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[9] D. N. Riahi, R. Roy and S. Cavazos, On arterial blood flowin the overlapping stenosis, Math. & Comp. Model.,54,2999 (2011).

[10] D. Tripathi, A mathematical study on three-layeredoscillatory blood flow through stenosed arteries,. J. BionicEng.,9, 119 (2012).

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[14] Kh. S. Mekheimer and M. A. Elkot, Influence of Magneticfield and Hall currents on blood flow through stenoticartery,.Appl. Math. Mech. Eng. Ed.,29, 1093 (2008).

[15] S. Mishra, S. U. Siddiqui and A. Medhavi, Blood flowthrough a composite stenosis in an artery with permeablewall, Application and Appl. Math.,6, 1798 (2011).

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R. Ellahi is a FulbrightFellow in the UniversityOf California Riverside, USAand also founder Chairpersonof the Department ofMathematics and Statisticsat IIUI, Pakistan. He hasseveral awards and honors onhis credit: Fulbright Fellow,Productive Scientist of

Pakistan, Best University Teacher Award by highereducation commission (HEC) Pakistan, Best BookAward, and Valued Reviewer Award by Elsevier. He isalso an author of six books and editor of five internationaljournals.

S. U. Rahman isworking as a PhD Scholarin the department ofMathematics at InternationalIslamic University, Islamabadunder the supervision ofDr. R. Ellahi. He obtained hisMS from IIUI. His researchwork has been publishedin internationally refereed

journals.

M. Mudassar Gulzardid his PhD degree from thedepartment of Mathematics,Quaid-i-Azam University,Islamabad, Pakistan.He is an Associate Professorin the Department of BasicSciences and Humanities,NUST College of Electricaland Mechanical Engineering

Islamabad. His field of research is Fluid Mechanics. Hisresearch articles have been published in the journals ofinternational repute.

S. Nadeem isan Associate Professorin the department ofMathematics, Quaid-i-AzamUniversity Islamabad. Heis recipient of Razi-ud-Dingold medal by PakistanAcademy of Sciencesand Tamgha-i-Imtiaz by thePresident of Pakistan. He is

also young fellow of TWAS, Italy.



Kambiz Vafai isProfessor of MechanicalEngineering at Universityof California, Riverside(UCR), where he startedas the Presidential Chair inthe department of MechanicalEngineering. He joinedUCR from The Ohio StateUniversity, where he receivedoutstanding research awards

as assistant, associate, and full professor. He is a Fellowof American Association for Advancement of Science,American Society of Mechanical Engineers, WorldInnovation Foundation, and an Associate Fellow of theAmerican Institute of Aeronautics and Astronautics. He isamong the ISI highly cited category. He has carried outvarious sponsored research projects through companies,governmental funding agencies and national labs. He hasalso consulted for various companies and national labsand has been granted eleven U.S. patents. He was therecipient of the ASME Classic Paper Award in 1999 andhad received the 2006 ASME Heat Transfer MemorialAward. He is also the recipient of the highest award of theInternational Society of Porous Media. Dr. Vafai receivedhis B.S. degree from the University of MinnesotaMinneapolis, and the M.S. and Ph.D. degrees from theUniversity of California, Berkeley.



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