Anunsteadyanalysisofnon-Newtonianbloodﬂowthrough ...bett0099/8390_1.pdf · fect. In an extensive study, Easthope and Brooks [18] found that Walburn and Schneck model [19]which reduces

International Journal of Non-Linear Mechanics 40 (2005) 151–164

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An unsteady analysis of non-Newtonian blood flow throughtapered arteries with a stenosis

Prashanta Kumar MandalDepartment of Mathematics, Krishnath College, Berhampore, Dt., Murshidabad 742101, WB, India

Received 13 October 2003; received in revised form 24 June 2004

Abstract

The problem of non-Newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non-Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of thetime-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the invivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made withnonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier–Stokesequations in cylindrical coordinates system governing flow assuming axial symmetry under laminar flow condition so thatthe problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rateof flow, the resistive impedance and the wall shear stress with minor significance of computational complexity by exploitingthe appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle,wall deformation, severity of the stenosis within its fixed length, steeper stenosis of the same severity, nonlinearity and non-Newtonian rheology of the flowing blood on the flow field. An extensive quantitative analysis is performed through numericalcomputations of the desired quantities having physiological relevance through their graphical representations so as to validatethe applicability of the present model.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Generalised power law; Tapered arteries; Steeper stenosis; Wall shear stress

1. Introduction

There are considerable evidences that vascular fluiddynamics play important role in the development andprogression of arterial stenosis, one of the most widespread diseases in human beings leading to the mal-function of the cardiovascular system. Although the

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exact mechanisms responsible for the initiation ofthis phenomena are not clearly known, it has beenestablished that once a mild stenosis is developed, theresulting flow disorder further influences the develop-ment of the disease and arterial deformity, and changethe regional blood rheology[1,2]. Understanding ofstenotic flow has proceeded from quite a good num-ber of theoretical and computational and experimentalefforts. Steady flow through an axisymmetric stenosis

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152 P.K. Mandal / International Journal of Non-Linear Mechanics 40 (2005) 151–164

has been investigated extensively by Smith[3] usingan analytical approach indicating that the flow patternsstrongly depend on the geometry of the stenosis andthe upstream Reynolds number. Deshpande et al.[4]considered the steady flow through an axisymmetricstenosis using a finite difference technique.

Realising the fact that the pulsatile nature of theblood flow cannot be neglected, many theoreticalanalyses and experimental measurements on the flowthrough stenosis have been performed[5–14]. In mostof the studies mentioned above, the flowing blood isassumed to be Newtonian. The assumption of Newto-nian behaviour of blood is acceptable for high shearrate flow, i.e. the case of flow through larger arteries.It is not, however, valid when the shear rate is lowas is the flow in smaller arteries and in the down-stream of the stenosis. It has been pointed out that insome diseased conditions, blood exhibits remarkablenon-Newtonian properties. Hemorheological studieshave documented three types of non-Newtonian bloodproperties: Thixotropy, Viscoelasticity and Shear thin-ning. Thixotropy, a transient property of blood, isexhibited at low shear rates and has a fairly long timescale. This suggests that thixotropy is of secondaryimportance in physiological blood flow. Thurston[15,16] has shown conclusively that blood, being asuspension of enumerable number of cells, possessessignificant viscoelastic properties in the frequencyrange of physiological importance. Studies pertainingto the viscoelasticity of blood are of great interestbecause of three main reasons. To medical scientists,an accurate knowledge of the mechanical propertiesof whole blood and the erythrocytes can suggest anew diagnostic tool. For specialists in fluid mechan-ics, detailed informations of the complex rheologicalbehaviour of the system is of utmost importance inany attempt towards establishing the equations thatgovern the flow of blood in various parts of the circu-latory system in different states. To rheologists, blood(whose biochemical and cellular compositions arewell-known in other respects) is an excellent modelfor correlating its rheological behaviour with the un-derlying molecular or cellular structures. However,the viscoelasticity of blood diminishes very rapidly asshear rate rises and at physiological hematocrit values(∼ 45%) [17]. This suggests that viscoelasticity hasa secondary impact on normal pulsatile blood flow atphysiological hematocrit values. The shear thinning

properties of blood, however, are not transient and areexhibited in normal blood at all shear rates upto about100 s−1. The purely viscous shear thinning nature ofblood is, therefore, the dominant non-Newtonian ef-fect. In an extensive study, Easthope and Brooks[18]found that Walburn and Schneck model[19] whichreduces to familiar Power-law relationship, is highlyeffective in modelling blood flow. The Power-lawfluid showed far more non-Newtonian influence asevident from the experimental findings of Perktoldet al. [20]. A number of researchers have studied theflow of non-Newtonian fluids[21–32] with variousperspectives.

In most of the investigations relevant to the do-main under discussion, the flow is mainly consideredin cylindrical pipes of uniform cross-section. But, itis well known that blood vessels bifurcate at frequentintervals and the diameter of the vessels varies withthe distance as propounded by Whitemore[33]. Hencethe concept of flow in a varying cross-section formsthe prime basis of a large class of problems in un-derstanding blood flow. Manton[34], Hall [35] andPorenta et al.[36] pointed out that most of the vesselscould be considered as long and narrow, slowly taper-ing cones. Thus the effects of vessel tapering togetherwith the non-Newtonian behaviour of the streamingblood seem to be equally important and hence cer-tainly deserve special attention.

With the above motivation, an attempt is made in thepresent theoretical investigation to develop a mathe-matical model in order to study the notable character-istics of the non-Newtonian blood flow through a flex-ible tapered arteries in the presence of stenosis subjectto the pulsatile pressure gradient. The non-Newtonianbehaviour of the streaming blood is characterised bythe generalised Power-law model. Malek et al.[37,38]extensively studied the existence and uniqueness aswell as the stability characteristics of such flow prob-lems. Although the present paper does not deal withthe existence of the flows characterised by generalisedPower-law fluids, the already cited references[37,38]bear the foundation to make an attempt safely to ex-plore the flow characteristics of the streaming blood.Although the general problem such as the present oneis of major physiological significance, due attention isalso paid to the effect of arterial wall motion on lo-cal fluid mechanics but not on the stresses and strainsin the vessel wall. The consideration of a time-variant

P.K. Mandal / International Journal of Non-Linear Mechanics 40 (2005) 151–164 153

geometry of the stenosis has not however been ruledout from the present analysis. An extensive quantita-tive analysis is carried out by performing large scalenumerical computations of the desired quantities hav-ing more physiological significance to explore the ef-fects of vessel tapering, the severity of the stenosis,the wall deformability, the steeper stenosis of sameseverity, the nonlinearity and the non-Newtonian be-haviour of the flowing blood on the physiological flowphenomena which are extensively quantified throughtheir graphical representations presented at the endof thepaper with appropriate scientific discussions. Afew comparisons are also made with the other exist-ing results so as to substantiate the applicability of thepresent model under study.

2. Formulation of the problem

The tapered blood vessel segment having a stenosisin its lumen is modelled as a thin elastic tube witha circular cross-section containing an incompressiblenon-Newtonian fluid characterised by generalisedPower-law model. Let (r, �, z) be the coordinates ofa material point in the cylindrical polar coordinatessystem where thez-axis is taken along the axis ofthe artery whiler, � are taken along the radial andthe circumferential directions, respectively. The ge-ometry of the time-variant stenosed arterial segment(seeFig. 1) is constructed mathematically as

R(z, t)

=

[(mz+ a)− �m sec�(z−d)

�2m sin �2− l20

4

{l0 − (z− d)}]a1(t), d� z� d + l0,

(mz+ a)a1(t), otherwise,

(1)

whereR(z, t) denotes the radius of the tapered arterialsegment in the stenotic region,a the constant radius ofthe non-tapered artery in the non-stenotic region,�,the angle of tapering,l0, the length of the stenosis,d,the location of the stenosis and�m sec� is taken to bethe critical height of the stenosis for the tapered arteryappearing atz = d + l0

2 + �m sin� andm(= tan�)represents the slope of the tapered vessel. The

Fig. 1. Geometry of the stenosed tapered artery for different taperangle.

time-variant parametera1(t) is given by

a1(t)= 1 − b(cos�t − 1)e−b�t , (2)

in which � represents the angular frequency where� = 2�fp, fp being the pulse frequency andb is aconstant. The arterial segment is taken to be of finitelengthL. One can explore the possibility of differ-ent shapes of the artery viz., the converging tapering(�<0), non-tapered artery (� = 0) and the divergingtapering (�>0) as shown inFig. 1.

Let us consider the stenotic blood flow in the taperedartery to be two-dimensional, unsteady, axisymmetricand fully developed, where the flowing blood is treatedto be non-Newtonian characterised by the generalisedPower-law model. The governing equations for thez and r components of momentum together with theequation of continuity, in the cylindrical coordinatesystem may be written as

�w�t

+ u�w�r

+ w�w�z

= − 1

��p�z

− 1

�

[1

r

��r(r�rz)− �

�z(�zz)

], (3)

�u�t

+ u�u�r

+ w�u�z

= − 1

��p�r

− 1

�

[1

r

��r(r�rr )+ �

�z(�rz)

](4)

and

�u�r

+ u

r+ �w

�z= 0, (5)

where the relationships between the shear stress andshear rate in case of two-dimensional motion are as


follows:

�zz=−2

m

∣∣∣∣∣[(

�u�r

)2

+(ur

)2 +(

�w�z

)2

+(

�u�z

+ �w�r

)2]1/2

∣∣∣∣∣∣n−1

(

�w�z

), (6)

�rz = −{. . . . . . .}(

�w�r

+ �u�z

)(7)

and

�rr = −2{. . . . . .}(

�u�r

). (8)

Herew(r, z, t) andu(r, z, t) are the axial and the radialvelocity components, respectively,p is the pressureand�, the density of blood.

Since the lumen radius,R, is sufficiently smallerthan the wavelength�, of the pressure wave i.e.R/�>1, the radial Navier–Stokes equation simplyreduces to�p/�r = 0 (cf. [39]) and hence Eq. (4)can be omitted. It is then reasonable and convenientto assume that the pressure is independent of radialcoordinate[5,40] and eventually the pressure gradient�p/�z appearing in (3), the form of which has beentaken following Burton[41] for human beings as

− �p�z

= A0 + A1 cos�t, t >0, (9)

whereA0 is the constant amplitude of the pressuregradient,A1 is the amplitude of the pulsatile compo-nent giving rise to systolic and diastolic pressure.

3. Boundary conditions

On the symmetry axis, the normal component ofthe velocity, the axial velocity gradient and the shearstress vanish. These may be stated mathematically as

u(r, z, t)= 0,�w(r, z, t)

�r= 0 and �rz = 0

on r = 0. (10)

The velocity boundary conditions on the arterial wallare taken as

u(r, z, t)= �R�t, w(r, z, t)= 0 on r =R(z, t). (11)

It is further assumed that initially no flow takes placewhen the system is at rest, that means

u(r, z,0)= 0 = w(r, z,0). (12)

4. Method of solution

Let us introduce a radial coordinate transformation[40], given by

x = r

R(z, t), (13)

which has the effect of immobilizing the vessel wallin the transformed coordinatex.

Using this transformation, Eqs. (3), (5)–(7) togetherwith the prescribed conditions (10)–(12) take the fol-lowing form:

�w�t

={x

R

�R�t

− u

R+ x

R

�R�z

w

}�w�x

− 1

��p�z

−w �w�z

− 1

�

{1

xR�xz + 1

R

��xz�x

− ��zz�z

+ x

R

�R�z

��zz�x

}, (14)

1

R

�u�x

+ u

xR+ �w

�z− x

R

�R�z

�w�x

= 0, (15)

�zz=−2

m

∣∣∣∣∣[(

1

R

�u�x

)2

+( u

xR

)2

+(

�w�z

− x

R

�R�z

�w�x

)2

+(

�u�z

− x

R

�R�z

�u�x

+ 1

R

�w�x

)2]1/2

∣∣∣∣∣∣n−1

×(

�w�z

− x

R

�R�z

�w�x

), (16)

�xz = −{. . . . . . .}(

�u�z

− x

R

�R�z

�u�x

+ 1

R

�w�x

)(17)

with

u(x, z, t)= 0,�w(x, z, t)

�x= 0,

�xz = 0 on x = 0 (18)


u(x, z, t)= �R�t, w(x, z, t)= 0 on x = 1, (19)

and

u(x, z,0)= 0 = w(x, z,0). (20)

Multiplying Eq. (15) byxR and integrating with re-spect tox from the limits 0 tox one finds,

u(x, z, t)=x �R�z

w − − R

x

∫ x

0x

�w�z

dx

− 2

x

�R�z

∫ x

0xw dx. (21)

This equation takes the following form by making useof the boundary conditions (19) as

−∫ 1

0x

�w�z

dx

=∫ 1

0x

[2

R

�R�z

w + 1

R

�R�t

f (x)

]dx. (22)

Since the choice off (x) is, of course, arbitrary, letf (x) be of the form

f (x)= −4(x2 − 1) satisfying∫ 1

0x f (x)dx = 1.

Taking the approximation of considering the equalitybetween the integrals to integrands, we have from (22)

�w�z

= − 2

R

�R�z

w + 4

R(x2 − 1)

�R�t. (23)

Introducing (23) into (21) one gets

u(x, z, t)= x

[�R�z

w + �R�t(2 − x2)

]. (24)

5. Finite difference approximations

The finite difference scheme for solving Eq. (14) isbased on the central difference approximations for allthe spatial derivatives in the following manner:

�w�x

= (w)ki,j+1 − (w)ki,j−1

2�x= wfx,

�w�z

= (w)ki+1,j − (w)ki−1,j

2�z= wfz, (25)

while the time derivative in (14) is approximated by

�w�t

= wk+1i,j − wk

i,j

�t. (26)

Similar expressions can also be obtained foru, �xz and�zz. Herew(x, z, t) is discretised tow(xj , zi, tk) andin turn, towk

i,j where we definexj = (j −1)�x, (j =1,2, . . . N + 1) such thatx(N+1) = 1.0, zi = (i −1)�z, (i = 1,2, . . .M + 1) andtk = (k− 1)�t, (k=1,2, . . .) for the entire arterial segment under studywith �x,�z are the increments in the radial and theaxial directions, respectively, and�t is the small timeincrement.

Using (25) and (26), Eq. (14) may be transformedto the following difference equation:

wk+1i,j =wk

i,j + �t

[− 1

�

(�p�z

)k+1

+{xj

Rki

(�R�t

)ki

− uki,j

Rki

+ xj

Rki

(�R�z

)ki

wki,j

}(wf x)

ki,j

−wki,j (wf z)

ki,j − 1

�

{1

xjRki

(�xz)ki,j

+ 1

Rki

[(�xz)f x]ki,j − [(�zz)f z]ki,j

+ xj

Rki

(�R�z

)ki

[(�zz)f x]ki,j}]

(27)

while Eqs. (16) and (17) have their discretised form as

(�zz)ki,j=−2

m

∣∣∣∣∣∣∣( 1

Rki

(uf x)ki,j

)2

+(uki,j

xjRki

)2

+((wf z)

ki,j − xj

Rki

(�R�z

)ki

(wf x)ki,j

)2

+((uf z)

ki,j − xj

Rki

(�R�z

)ki

(uf x)ki,j

+ 1

Rki

(wfx)ki,j

)2

1/2∣∣∣∣∣∣∣n−1

[(wf z)

ki,j

− xj

Rki

(�R�z

)ki

(wf x)ki,j

], (28)


(�xz)ki,j=−{. . . . . . .}[(uf z)

ki,j

− xj

Rki

(�R�z

)ki

(uf x)ki,j + 1

Rki

(wfx)ki,j

].

(29)

Also the prescribed conditions (18)–(20) have theirfinite difference representations, given by

uki,1 = 0, wki,1 = wk

i,2, (�xz)ki,1 = 0, (30)

wi,N+1 = 0, uki,N+1 =(

�R�t

)ki

, (31)

u1i,j = 0 = w1

i,j . (32)

The difference equation (27) is solved forw by mak-ing use of (28)–(29) together with the prescribedconditions (30)–(32) throughout the arterial segmentunder consideration. After having obtained the axialvelocity of the flowing blood, the radial velocity canbe calculated directly from (24).

Now with the help of the axial and the radial veloc-ity of the streaming blood, one can easily determinethe volumetric flow rate(Q), the resistance to flow(∧) and the wall shear stress(�w) from the followingrelations, given by

Qki = 2�(Rki )

2∫ 1

0xjw

ki,j dxj , (33)

∧ki =

∣∣L(�p/�z)k∣∣Qki

, (34)

(�w)ki=�

[1

Rki

(wfx)ki,j + (uf z)

ki,j

− xj

Rki

(uf x)ki,j

(�R�z

)ki

]x=1

×cos

[arctan

(�R�z

)ki

]. (35)

The present analysis bears the potential to explore sev-eral case studies by means of which the effects ofwall distensibility, the non-Newtonian rheology of theflowing blood and the nonlinearity on the flow pat-tern can be estimated both analytically and numeri-cally through the following cases of reduction of thepresent system to the particular consideration.

CaseI: When the wall motion is withdrawn fromthe present system, the analysis can be reduced to aform by consideringR=R(z) only and consequently�R/�t = 0 so that the effect of wall distensibility canbe estimated directly.

CaseII: The generalised Power-law model is quitedifferent from the other form as it does not simplyreduce to Newtonian model forn = 1 and hence toestimate the effects of non-Newtonian rheology of thestreaming blood, the following set of equations havebeen solved using the same methodology as in the caseof non-Newtonian model but for the shake of brevity,the detailed derivations are not presented here:

�w�t

+ u�w�r

+ w�w�z

= − 1

��p�z

+ ��

[�2w

�r2 + 1

r

�w�r

+ �2w

�z2

],

�u�t

+ u�u�r

+ w�u�z

= − 1

��p�r

+ ��

[�2u

�r2 + 1

r

�u�r

− u

r2 + �2u

�z2

],

and�u�r

+ u

r+ �w

�z= 0.

CaseIII: When the convective acceleration termsviz. the nonlinear terms are disregarded from thegoverning equations mentioned in Case II, one findsthe system reducing to a linearized (Newtonian) onewhere the termsu(�w/�r), w(�w/�z), u(�u/�r),w(�u/�z) are totally absent. The effect of nonlin-earity can therefore be measured through the directcomparison of the present system and its linearized(Newtonian) version.

CaseIV: Instead of treating the arterial wall as elas-tic (moving wall), if one clamps the nonlinear visco-elastic wall property so that the variation of the resultscan be quantified and the necessary comparison canalso be made with those of the existing result by mak-ing use of the following pressure–radius relationship[10]:

p(z, t)= �w(�R/�t)+ 0(e(�−1)−1)

(rm/h)�2 − 1

2

,

where the input data for�w, 0, are taken to besimilar to those of Ref.[10].


6. Numerical results and discussion

For the purpose of numerical computations of thedesired quantities of major physiological significance,the following parameter values have been made useof [21,41–43]: a= 0.8 mm,L= 50 mm,L0 = 16 mm,d = 20 mm, b = 0.1, m = 0.1735P , � = 0.035P ,n= 0.639 � = 1.06× 103 kg m−3, fp = 1.2 Hz,A0 =100 kg m−2 s−2, A1 = 0.2A0, �m = 0.4a, �x = 0.025,�z= 0.1.

The iterative method has been found to be quite ef-fective in solving the equations numerically for dif-ferent time periods. The results appeared to convergewith an accuracy of the order∼ 10−7 when the timestep was chosen to be 0.00001.

The computed results obtained following the above-mentioned method for various physical quantities ofmajor physiological significance in order to have theirquantitative measures are all exhibited through theFigs. 2–14 and discussed at length.

Fig. 2 illustrates the results for the axial velocityprofile of the flowing blood characterised by the gen-eralised Power-law model (non-Newtonian) at a spe-cific location ofz = 28 mm in the stenotic region ofthe tapered artery at an instant oft=0.45 s comprisingof six distinct curves for different perspectives withdistinguishable marks. The curves are all featured tobe analogous in the sense that they do decrease fromtheir individual maxima at the axis as one moves awayfrom it and finally drop to zero on the wall surface.Examining the behaviour of the results of the presentfigure, one observes that the axial velocity profile as-sumes a flat shape in the presence of a convergingtapering (� = −0.1◦) instead of a parabolic one fornon-tapered (� = 0◦) artery when both are treatedunder stenotic conditions. This observation can beinterpreted physically that if the tube is tapered, theninertial forces associated with the convective accelera-tions manifest themselves in an amount of the same or-der as viscous forces while the former compel the axialvelocity profile to attain a flat shape. This observationagrees qualitatively well with that of Belardinelli andCavalcanti[9] though their observations were basedon the Newtonian rheology of flowing blood past ataperedartery. Likewise, the effect of tapering withoutany arterial constriction can be visualised and quan-tified from the first and second curves from the topof the present figure where again the vessel tapering

15

10

5

00 0.2 0.4

x0.6 0.8 1

Fig. 2. Axial velocity profile atz=28 mm fort=0.45 s. (�m=0.4a,d = 20 mm, l0 = 16 mm).

diminishes the flow velocity significantly. Thus onemay conclude that the axial flow velocity is reducedto some extent with vessel tapering irrespective of thepresence of any arterial constriction. The correspond-ing results for a diverging tapering (�=0.1◦) with ar-terial stenosis represented by the third curve from thetop differ from all the remaining curves and becomehigher than those for a converging tapering as antici-pated. The present figure also includes the result for asteeper stenosis (of same severity) for the same axialposition for a non-tapered artery where the axial ve-locity profile gets reduced to a considerable extent incase of a steeper stenosis which can be quantified bycomparing the relevant curves of the present figure.Thus, analysing the behaviour of all the curves of thepresent figure, one may conclude that the presence ofstenosis, tapering and steeper stenosis affect the axialvelocity of the streaming blood past a stenosed taperedartery significantly.

Unlike the characteristics of the axial velocity pro-file, the results of the radial velocity component vary-ing radially at the same critical location ofz= 28 mmexhibited inFig. 3at the same instant oft=0.45 s, arefound to be negative. All the curves appear to declinefrom zero on the axis as one moves from away fromit and finally to increase towards the wall to attendsome finite value on the wall surface which clearly


x

Fig. 3. Radial velocity profile atz=28 mm fort=0.45 s. (�m=0.4a,d = 20 mm, l0 = 16 mm).

reflects the presence of wall motion encountered in thepresent model. Analysing all the curves of the presentfigure, one can easily observe that the effect of taper-ing on the radial velocity profile is more prominent incase of stenosed artery than for a non-stenosed one,however, for non-stenosed artery, the maximum devi-ation occurs near the wall. The corresponding resultfor a steeper stenosis is almost unperturbed which canbe visualised if one go through the relevant curves ofthe present figure. Even though the magnitudes of theradial velocity are smaller than those of the axial ve-locity, the noted feature is directly responsible for thelocal storage and is manifested in the convective ac-celeration of the nonlinear flow phenomena.

Fig. 4 displays the variation of the axial velocityprofile at the same critical location ofz= 28 mm in atapered artery for different times spread over a singlecardiac period. The curves do shift towards the originwhen the time is allowed to increase from 0.1 to 0.45 sand eventually att = 0.7 s, the curve is found to shiftaway from the origin. Such behaviour is believed to bedirectly responsible for the pulsatile pressure gradientproduced by the heart as it comes into play. It is inter-esting to note that the rate of decrease of the axial ve-locity in the systolic phase appears almost the same asthe rate of increase in the diastolic phase over a singlecardiac cycle, as anticipated. Moreover, the velocityis reduced to a considerable extent if the wall motion

7

6

5

4

3

2

1

00 0.2 0.4 0.6 0.8 1

x

Fig. 4. Axial velocity profile for different times (z = 28 mm,�m = 0.4a, d = 20 mm, l0 = 16 mm,� = −0.1◦).

is totally withdrawn from the system under consider-ation as observed from the bottom two curves of thepresent figure. Here too, the observations that the re-sults obtained by Newtonian model of the streamingblood are dramatically much higher than the non-Newtonian values. Thus the vascular wall deforma-bility and the non-Newtonian characteristics of theflowing blood affect the axial velocity profile whichcan be estimated by the relevant curves of the presentfigure.

The results indicating the unsteady behaviour of theflowing blood over a single cardiac cycle, presentedin theFig. 5 at the same critical location are found tobe quite interesting to note. The radial velocity profileassumes positive values with time advancement from0.1 to 0.3 s in the systolic phase while with the passageof time from 0.45 to 0.7 s, the radial velocity profileassumes to continue with negative values only causingback flow. Such typical nature of the curves reflectsvery closely the radial motion of the arterial wall fora single cardiac cycle whose graphical presentation isnot shown here for brevity. When the wall motion istotally disregarded, the radial velocity profile takes acomplete symmetrical shape as recorded in the thirdcurve from the top of the present figure. The presentfigure also displays the results for the flowing bloodhaving Newtonian rheology and it turns out that New-tonian characteristics of the flowing blood affects the


0.2

0.1

0

-0.1

-0.20 0.2 0.4 0.6 0.8 1

x

Fig. 5. Radial velocity profile for different times (z = 28 mm,�m = 0.4a, d = 20 mm, l0 = 16 mm,� = −0.1◦).

14

12

10

8

6

4

2

00 0.2 0.4 0.6 0.8 1

x

Fig. 6. Axial velocity profile for different axial positions att = 0.45 s. (d = 20 mm,�m = 0.4a, l0 = 16 mm,� = −0.1◦).

radial velocity pattern less significantly than the axialvelocity profile.

In order to analyse the flow-field intensively alongthe arterial segment under study,Figs. 6exhibits theaxial velocity profiles at five distinct axial locationsfor � = −0.1◦ at t = 0.45 s. The axial velocity pro-file is parabolic at the upstream(z = 15 mm) while

0.05

0

-0.05

-0.1

-0.15

-0.20 0.2 0.4 0.6 0.8 1

x

Fig. 7. Radial velocity profile for different axial positions att = 0.45 s. (d = 20 mm,�m = 0.4a, l0 = 16 mm,� = −0.1◦).

a flattening trend is followed at the converging sec-tion (z = 24 mm) and subsequently it becomes muchblunter at the specific location(z = 28 mm) than atthe entry. The velocity appears to be enhanced at thediverging section(z = 34 mm) and finally at the off-set of the stenosis, the axial flow velocity profile getsback again into the parabolic pattern. This result agreesqualitatively well with those of Tu et al.[7] thoughtheir studies were based on the stenotic blood flow inwhich the streaming blood was treated as Newtonianfluid.

The results of the radial velocity component att =0.45 s exhibited inFig. 7 are noted to be negative ex-cepting at the downstream. Most of the curves becomeconcave near the wall excepting that for the down-stream of the stenosis. It is interesting to record that atthe downstream, back flow occurs near the wall wherethe direction of the velocity changes from positive tonegative and that, in turn, causes separation in the flowfield. This observation is believed to be quite justifiedfrom the physiological point of view where the arte-rial tapering plays a key role in order to characterisethe flow phenomena under study. If one disregards thenon-Newtonian rheology of the flowing blood and alsoignores the convective acceleration terms in the mo-mentum (case II) i.e. in case of linear Newtonian flow,a meagre deviation is observed in the radial velocity ofthe flowing blood in the present mathematical model.


7

6

5

4

3

2

1

00 0.2 0.4 0.6 0.8 1

x

Fig. 8. Axial velocity profile for different taper angles att=0.45 s.(d = 20 mm,z= 28 mm,�m = 0.4a, l0 = 16 mm).

0

-0.03

-0.06

-0.09

-0.12

-0.150 0.2 0.4 0.6 0.8 1

x

Fig. 9. Radial velocity profile for different taper angles att=0.45 s.(d = 20 mm,z= 28 mm,�m = 0.4a, l0 = 16 mm).

Figs. 8and9 show how the constricted arterial ta-pering with varied taper angles influences the patternsof the flow-field at a particular instant oft = 0.45 s. Itis observed that the curves representing both the axialand the radial flow velocity do shift towards the originwhen the taper angle increases from−0.2◦ to −0.4◦(converging tapering) while they shift away from theorigin for a non-tapered (� = 0◦) and positively ta-

0.008

0.006

0.004

0.002

00 0.05 0.1 0.15 0.2

Fig. 10. Flow dependence on the pressure gradient att = 0.1 s(z= 28 mm,d = 20 mm, l0 = 16 mm,�m = 0.4a).

pered (� = 0.2◦) artery. Examining the behaviour ofthe results of the present figures, one can easily quan-tify the effect of arterial tapering on the flow-field.Moreover, by comparing the present results with ourprevious work[13], one may also take note that thereis no significant change in behaviour of the flow-fieldpatterns evaluated by Newtonian and non-Newtonianmodel of the streaming blood.

For the purpose of making a comparative study withthe existing results[9], several plots have been madein Fig. 10 just to characterize the behaviour of theflow rate with the pressure gradient for different taperangles at a particular instant oft=0.1 s where the inputpressure gradient data has been chosen to be similar tothose of Ref.[9], at a particular location ofz=28 mm.All the curves appear to be linear and they do shifttowards the origin with increasing taper angle from 0◦to −0.1◦ or in other words, the flow rate diminishes asthe artery gets narrowed gradually. The present figurealso includes the results for a diverging tapering (� =0.1◦) exhibiting an increasing trend as anticipated. Inthe absence of any constriction, the flow rate enhancesto a considerable extent which can be quantified fromthe present figure. Finally, the effects of vascular walldeformability, the steeper arterial stenosis and of thenon-Newtonian rheology of the flowing blood with thenoted range of the pressure gradient for a non-taperedartery (� = 0◦)are, however, not ruled out from thepresent investigation.


0.02

0

-0.02

-0.04

-0.060 0.2 0.4 0.6 0.8 1

x

Fig. 11. Axial velocity profile forz=37 mm att=0.1 s. (d=20 mm,l0 = 16 mm,�m = 0.05a).

The representations ofFig. 11 record how the be-haviour of the axial velocity profile is dramaticallychanged at the post-stenotic region(z = 37 mm) fora particular instant oft = 0.1 s when one takes intoaccount the arterial wall to be isotropic, incompress-ible, visco-elastic material with circular cross-section[10]. For a converging tapered artery (� = −0.03◦),one observes that the axial velocity profile varying ra-dially is always negative while for a diverging artery(� = 0.03◦), the axial flow velocity is always posi-tive when the arterial wall is treated as elastic (mov-ing wall). On the contrary, a vortex takes shape whenthe arterial wall deformation is treated to be similar tothose of Cavalcanti[10] which may cause irregulari-ties in the wall shear stress. This observation is in goodagreement with those of Cavlcanti[10] though thelater studied the flow phenomena in a mildly stenosedartery by considering the Newtonian rheology of theflowing blood. Perhaps the pressure–radius relationconsidered by Cavalcanti[10] gives more realistic re-sult, but it lacks tocompare the results with the rigidand straight circular arterial tube without any taperingand constriction.

Fig. 12includes more results showing the variationof flow rate at a specific location ofz= 28 mm forcertain distinct cases stretched over a period of nearlyfour cardiac cycles. The pulsatile nature of the flowrate has been found to be distributed for all the curves

5

4

3

2

1

00 0.5 1 1.5 2 2.5 3

Fig. 12. Variation of the rate of flow with time atz = 28 mm(�m = 0.4a, d = 20 mm, l0 = 16 mm).

throughout the time scale considered here. In the ab-sence of the constriction, the flow rate gets enhancedsignificantly for the entire time range. The deviationof the results thus obtained clearly estimates the ef-fect of stenosis quantitatively on the flow rate in thetapered artery. The flow rate for a non-tapered artery(� = 0◦) possesses relatively higher magnitudes thanthat of negatively tapered artery (� = −0.1◦). How-ever, the magnitude of the flow rate for a diverging ta-pered artery is alltime higher than those of non-taperedand negatively tapered artery. Further, the correspond-ing Newtonian model yields an analogous behaviourwith higher magnitudes. Hence the effect of taper an-gle and non-Newtonian rheology of the flowing bloodcan be quantified if one analyses the relevant curvesof the present figure. Moreover, when the wall motionis totally withdrawn from the present system by disre-garding the vessel wall distensibility, the flow rate de-clines keeping its pulsatile nature unaltered. One mayalso take note from the present figure that the flowrate diminishes to a considerable extent in the caseof a steeper stenosis where stenosis has been madesteeper just by reducing its length with thesame sever-ity. Studying all the results referred to the present fig-ure, one may conclude that the presence of stenosis,the taper angle, the steeper stenosis, the wall deforma-bility and the non-Newtonian rheology of the flowing


Fig. 13. Variation of the resistance to flow with time atz=28 mm(�m = 0.4a, d = 20 mm, l0 = 16 mm).

blood certainly bear the potential to influence the flowrate to a considerable extent in the realm of the phys-iological flow phenomena.

Fig. 13indicates how the resistances to flow are in-fluenced by the unsteady flow behaviour of blood aswell as by the vessel tapering, the vessel wall disten-sibility, the stenosis, the steeper stenosis and by thenon-Newtonian rheology of the streaming blood. Theresistances to flow follow a reverse trend from thoseof Fig. 12 in a way that the streaming fluid experi-ences higher resistance when the rate of flow in theconstricted tapered artery are correspondingly lowerand vice-versa. Unlike the characteristics of the flowrate, one may observe that the flowing blood experi-ences much higher resistances to flow in the presenceof arterial constriction, in the absence of vascular walldistensibility and in the presence of non-Newtoniancharacteristics of the flowing blood. However, the ef-fects of tapering and the steeper stenosis on the resis-tive impedances are, however, not ruled out from thepresent investigation.

Finally, the variation of the time-dependent wallshear stress at a specific location ofz = 28 mm cor-responding to a constricted zone of a tapered arteryhas been portrayed inFig. 14. The wall shear stressesrepresented by the curves of the concluding figure ap-pear to be compressive in nature. It appears that thewall shear stress declines from zero at the onset of thecardiac cycle and the rate of decline with negative val-

0

-0.02

-0.04

-0.06

-0.080 0.5 1 1.5 2 2.5 3

Fig. 14. Variation of the wall shear stress with time atz= 28 mm(�m = 0.4a, d = 20 mm, l0 = 16 mm).

ues gradually diminishes for rest of the pulse cycleswhen the streaming blood is Newtonian past a taperedartery (� = −0.1◦) which has a remarkable deviationwith the corresponding non-Newtonian result if onegoes through the relevant curves of the present figureand thereby the effect of non-Newtonian rheology ofthe flowing blood on the wall shear stress can be wellestablished. However, for the rest of the curves of thepresent figure, there is a remarkable variation of thestress characteristics almost immediately after the on-set of the first cardiac cycle where small fluctuationswith some fixed amplitudes keep the stress steady withthe advancement of time. The stress yields all timevalues higher for a diverging tapering than all otherexisting results corresponding to converging taperingand without tapering so far as their magnitudes areconcerned which is in good agreement with Sagaya-mary and Devanathan[44] who studied the steadyflow behaviour of couple stress fluid in a stenosed ta-pered tube and disregarded walldeformability. The de-viation in results for the rigid artery and for the con-stricted artery can be visualised and quantified theireffects on the stresses from the relevant curves of thepresent figure. According to Glagov et al.[45], hightensile stresses are associated with vessel wall thick-ening and alterations in composition. One may alsorecord from the present results that wall shear stressenhances to a considerable extent for steeper stenosis


of the same severity throughout the time under consid-eration which agrees well with that of Provenzano andRutl [46] who studied a model for wall shear stress instenosed arteries based on boundary layer theory. Thisobservation of the present results further highlights thevalidity of the present improved mathematical model.

Acknowledgements

The author would like to thank the referees for care-ful reading of the manuscript and for valuable sugges-tions.The author gratefully acknowledges ProfessorS. Chakravarty, Department of Mathematics, Visva-Bharati, INDIA for his valuable suggestions whilepreparing the manuscript.

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Anunsteadyanalysisofnon-Newtonianbloodﬂowthrough ...bett0099/8390_1.pdf · fect. In an extensive study, Easthope and Brooks [18] found that Walburn and Schneck model [19]which reduces