Physiological flow of shear-thinning viscoelastic fluid past an … · 2013-08-27 · Physiological...

© 2013 The Korean Society of Rheology and Springer 163

Korea-Australia Rheology Journal, Vol.25, No.3, pp.163-174 (August 2013)DOI: 10.1007/s13367-013-0017-6

www.springer.com/13367

Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction

Sarifuddin1, Santabrata Chakravarty2 and Prashanta Kumar Mandal2,*1Department of Mathematics, Raiganj Surendranath College, Raiganj, W.B., India.

2Department of Mathematics, Visva-Bharati, Santiniketan – 731235, W.B., India.

(Received July 29, 2012; final revision received April 5, 2013; accepted April 6, 2013)

The present investigation deals with the effect of the shape of a stenosis on the flow characteristics of blood,having shear-thinning viscoelastic rheological properties by using a suitable mathematical model. Keepingthe relevance of the physiological situation, the mathematical model is developed by treating blood as anon-Newtonian shear-thinning viscoelastic fluid characterised by unsteady Oldroyd-3-constant modelthrough an axisymmetric irregular arterial stenosis obtained from casting of a mildly stenosed artery (cf.Back et al., 1984). Comparison with the well-known cosine-shaped stenosis, in order to estimate the effectof surface roughness on the flow characteristics of blood, has however not been ruled out from the presentstudy. Numerical illustrations are presented for a physiological flow, as well as for an equivalent simple pul-satile flow with equal stroke volume to that of the physiological flow, and the differences in their flowbehaviour are recorded and discussed. The Marker and Cell method is developed in cylindrical co-ordinatesystem in order to tackle the highly nonlinear governing equations of motion. The effects of the quantitiesof significance such as Reynolds number, Deborah number, blood viscoelasticity and flow pulsatility, aswell on the velocity components, pressure drop, wall shear stress and patterns of streamlines are quan-titatively investigated graphically. Comparison of the results reveals that although the behaviour of two dif-ferent pulses are similar at the same instant of time, there exist some important deviations in the flowpattern, pressure drop and wall shear stress as well. The present results also predict that the excess pressuredrop across the cosine stenosis compared with the irregular one is consistent with several existing resultsin the literature which substantiate sufficiently to validate the applicability of the model under consideration.

Keywords: shear-thinning, viscoelastic, pulsatile, irregular stenosis, unsteady, MAC method.

1. Introduction

Atherosclerosis is an arterial disease causing major con-cerns to health in the form of heart attacks and strokes. Itusually affects large and medium sized artery (cf. Ross,1993; Waters et al., 2011). The role of blood flow dynam-ics on the formation of atherosclerosis, its subsequent pro-gression to plaque rupture and thrombosis have beenestimated experimentally by Brunette et al. (2008). Math-ematical models of blood flow constitute an alternativeand useful tool for supporting experiments and detectingminor constriction phenomena including local featureswhich are not always obvious by measurement. The lineartheory of Navier-Stokes equations is frequently used tomodel the flow of blood in larger arteries, but the role ofmaterial nonlinearity and the influence of red cells on theblood viscosity becomes more important at low shear ratesand in smaller vessels (cf. Chien et al., 1984). Keepingthis view in mind many successful studies have been car-ried out by treating blood as a non-Newtonian fluid (cf.Mann and Tarbell, 1990; Usha and Prema, 1999; Pontrelli,2001; Khanafer et al., 2006; Mandal et al., 2007; Luka-

cova-Medvidova and Zauskova, 2008; Sarifuddin et al.,2008, 2009; Ikbal et al., 2010; Sankar and Lee, 2010).

Thurston (1972) was among the earliest to recognise theviscoelastic nature of blood which is less prominent withincreasing shear rate. As flow proceeds, the sliding ofcells requires a continuous input of energy. This is dis-sipated through viscous friction (cf. Thurston, 1979).These effects induce the blood to behave like a viscoelas-tic fluid. In addition, several pathologies are accompaniedby significant changes in the mechanical properties ofblood which result in alteration in blood viscosity and vis-coelastic properties, as reported in the recent review arti-cles by Robertson et al. (2008, 2009). As red blood cells(RBCs) form rouleaux, they tumble while flowing throughvessels. Such tumbling disturbs the normal flow patternand requires the consumption of energy resulting in anincrease in blood viscosity at low shear. Other major fac-tors in the viscoelasticity of streaming blood are primarilydue to the elastic energy that is stored in the deformed redblood cells as the heart pumps the blood through the body.Red blood cells by themselves have been shown to exhibitviscoelastic properties. These properties are the largestcontributing factors to the viscoelastic behaviour of blood(cf. Thurston, 1989; Thurston and Henderson, 2006).*Corresponding author: [email protected]

Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal

164 Korea-Australia Rheology J., Vol. 25, No. 3 (2013)

Other factors contributing to the viscoelastic properties ofblood are the plasma viscosity, plasma composition, tem-perature and the shear rate. Together, these factors makethe streaming blood viscoelastic (cf. Thurston, 1979). Inthis case, the viscosity µ cannot be considered merely asa constant, but as a decreasing function of the shear rate

(shear-thinning fluid) (cf. Chmiel et al., 1990; Mannand Tarbell, 1990; Politsis et al., 1991). The analyticalfunction which offers the best fit for experimental datawith a large range of flows has been found in the findingsof Yeleswarapu (1996) and we have made use of it in ourpresent simulation.

It is rather surprising to find that blood viscoelasticityhas received much less attention in the literature than itsNewtonian counterpart and/or other non-Newtonianmodels even though interest has been growing in recenttimes due to its implications in clinical medicine andphysiology. Etter and Schowalter (1965) and Waters andKing (1971) studied the unsteady flow of Oldroyd-Bfluid in a circular tube. Philips and Deutsch (1975)applied a general state equation for human blood byusing a four constant Oldroyd model. Experimentalinvestigation highlighting the influence of viscoelasticityof human blood has been carried out by Gijsen et al.(1999). Pontrelli (2000) studied the unsteady flow of aOldroyd-B fluid in a straight, long and rigid pipe drivenby a suddenly imposed pressure gradient. The influenceof the shear-thinning (Cross model) viscoelasticity on theshape of the flow domain has been successfully studiedby Leuprecht and Perktold (2000). A viscoelastic modelof blood within a generalised thermodynamic frameworkis developed successfully by Anand and Rajagopal(2004). Anand et al. (2006) proposed the mechanics of aparticular type of clot, formed from human plasma,within a thermodynamic framework by describing bloodas a viscoelastic fluid. Arada and Sequeira (2005) exten-sively studied the existence and uniqueness of such flowproblems. Nadau and Sequeira (2007) made an attemptto study the shear-dependent viscoelastic flow problemby using a combined finite element-finite volumemethod. The coupling of a generalised Newtonian fluidby taking into account the shear-thinning behaviour ofblood, with an elastic structure describing the vessel wallhas been successfully made to capture the pulse wavedue to the interaction between blood and vessel wall, andto estimate the energy for the coupling (cf. Janela et al.,2010). Very recently, Badnar et al. (2010) concluded thatthe shear-thinning effects related to fluid velocity, pres-sure and wall shear stress are more pronounced than theviscoelastic ones.

Most experimental and numerical studies of pulsatileflow through a stenosed artery are based on the assump-tion of simple periodic variation of the inflow with time inthe form of single harmonic pulse (cf. O’Brien and Ehr-

lich, 1985; Stergiopulos et al., 1996), but available data oncanine and human arteries reveal that the arterial fluxwaves are different from a single harmonic pulse. Forphysiological flow, the waveform given by Daly (1976)has been used in the present investigation. For the purposeof making comparison, a simple pulsatile flow having thesame stroke volume as the physiological flow is consid-ered. Zendehboodi and Moayeri (1999) concluded that forbetter understanding of pulsatile flow behaviour instenosed arteries, the actual physiological flow should besimulated.

Most of the studies relating to stenotic flow have beenperformed with the basic assumption representing thestenosis by the cosine curve. However, arterial constric-tion contains many small valleys and ridges, analogous toa mountain range. For the purpose of deeper investigationinto this problem one step closer to the real situation, thepublished clinical data (cf. Back et al., 1984) were used todefine the outline of the stenosis. Some attempts havealready been made to investigate the flow characteristicsthrough irregularly occluded vessels (cf. Johnston and Kil-patrick, 1991; Andersson et al., 2000; Yakhot et al., 2005;Chakravarty et al., 2005; Mustapha et al., 2010) by treat-ing blood as a Newtonian fluid with constant viscositywhich perhaps ignores some important rheological aspectsof physiological flow under stenotic conditions in smallerarteries.

Keeping all these in mind, an attempt is made in thepresent investigation to explore the influence of pulsatileas well as physiological flow through a rigid irregularlyconstricted artery in which the streaming blood is treatedto be a generalised viscoelastic fluid which accommodatesviscoelasticity (cf. Phan-Thien and Huilgel, 1985) andshear-thinning (cf. Yeleswarapu, 1996) characteristics.The assumption of wall rigidity may not seriously affectthe flow since the development of atherosclerosis in arter-ies causes a significant reduction in vessel wall disten-sibility (cf. Nerem, 1992). The governing equations ofmotion of the unsteady flow phenomena are successfullysolved numerically by MAC method primarily introducedby Harlow and Welch (1965) and achieve the desireddegree of accuracy. The primary objective of the presentstudy is to explore the effects of some essential issues likeflow unsteadiness, different stenosis shapes, different inletwave forms, Reynolds number and different viscoelasticparameters on the velocity profile of the blood stream andon the wall shear stresses quantitatively by using a rel-atively simple finite-difference scheme in rather complexgeometries. The novelty of the present study is the treat-ment of shear-thinning viscoelastic fluid characterisingblood rheology in general, and the choice of differentlyshaped stenosis models and different pulsatile flow waveforms which more closely resemble the physiological sit-uation.

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Korea-Australia Rheology J., Vol. 25, No. 3 (2013) 165

3. Governing Equations

The streaming fluid representing blood is generally con-sidered as a viscoelastic fluid having shear-thinning char-acteristic (cf. Yeleswarapu, 1996). The axisymmetric flowof blood through an axisymmetric irregular arterial steno-sis can be regarded as two-dimensional by making use ofcylindrical co-ordinate system under laminar flow con-ditions. The momentum and the continuity equations foran unsteady, incompressible, Oldroyd-3-constant modelproposed by Phan-Thien and Huilgel (1985) can be writ-ten as

(1)

and (2)

where u = (ur, 0, uz) is the velocity vector, , the density,and P, are the pressure and the extra stress respectively inwhich the extra stress consists of two parts: n from theNewtonian ( solvent stress) and s from the non-Newtonianrheology of the flowing blood (particle stress) as

Following Leuprecht and Perktold (2001), the shear-thinning behaviour of the viscoelastic fluid is realised byapplying a shear rate dependent viscosity on the solventfluid. Thus for the shear- thinning generalised Oldroyd-3-constant viscoelastic model, the fluid viscosity (µ) is nolonger a constant rather it possesses a functional depen-dence of shear rate ( ). Hence we have

, (3)

and

. (4)

where , ,

with > 0, a material constant with the dimension of timerepresenting degree of shear- thinning. 0 and (0

) are two asymptotic values of viscosity at and respectively. D is the strain rate tensor ,

I, the unit tensor, , the fluid relaxation time, 1, a dimen-sionless parameter p, the viscosity contribution from theparticle part and stands for the upper convective deriv-ative defined by

(5)

We now introduce non-dimensional variables as

, , , , ,

and

where r0 and U0 are the unconstricted radius and cross-sec-tional average velocity over the inlet section respectively.

The non-dimensionalised equations of motion for two-dimensional unsteady flow are given by

(6)

and (7)

in which , and

, (8)

with ,

, (9)

where the Reynolds number , the Deborah

number and . The complex viscosity of

blood is approximated here with a three-parameter model,where the apparent viscosity decreases dramatically asshear rate increases (cf. Fig. 1).

4. Stenosis Model

Two differently shaped models of stenosis each shownin Fig. 2 have been examined. The first profile of thestenosis considered here is the straight axisymmetricmodel of Back et al. (1984) mimicking real surface irreg-

ut------ u u+

⎩ ⎭⎨ ⎬⎧ ⎫

P– +=

u 0=

n s+=

·

n · u u T+ =

s s+ p D 1 D D –

12---1 D : D I+

⎩ ⎭⎨ ⎬⎧ ⎫

=

· 0 – 1 1 ·+ elog+

1 ·+------------------------------------+= · 1

2--- D :D =

· 0· u u T+

s

s · s

u s s u– u T s–+=

rrr0

---- zzr0

---- ur

ur

U0

------ uz

uz

U0

------ ttU0

r0

-------- PP

U02

---------

U0

r0

---------- ij

ij

U02

---------

ut------ u u+ P– n s+ +=

u 0=

n · Re

---------- u u T+ =

s Des+

1Re------ D 1De D D –

12---1De D : D I+

⎩ ⎭⎨ ⎬⎧ ⎫

=

· 1 – 1 1 ·+ elog+

1 ·+------------------------------------+=

· 12--- D : D =

ReU0r0

p

--------------=

DeUo

ro

---------=

o

------=

Fig. 1. Shear rate dependent viscosity function () with * =0.068 for four values of .



ularities since the actual variation of the cross-sectionalarea of a left circumflex coronary artery casting from ahuman cadaver is retained. The second geometrical modelof the stenosis encountered here is the most commonlyused cosine curve

,

and , elsewhere (10)

where z0 is the half-length, , the maximum width and z1

(= d + z0), the centre of the stenosis with = 0.276r0, r0

being the unconstricted radius of the stenosed artery.

5. Boundary Conditions

As the arterial wall is treated to be rigid, the velocityboundary conditions of the blood stream on the wall arethe usual no-slip conditions given by

on , (11)

while zero transverse velocity gradient and zero cross flowon the axis of symmetry are taken as

on . (12)

A pulsatile parabolic velocity profile at the inlet of thestenosed arterial lumen may be assumed as

at ,

where (13)

with (0) = 0, is the angular frequency having the samestroke volume as the physiological flow. The physiologicalflow waveform [cf. Daly (1976)] for the mean velocity (t) in the canine femoral artery is shown in Fig. 3. Using

the condition (0) = 0, the value of

can be obtained from the expression of (t). The velocitygradients at the outlet of the arterial segment of finite lengthL may be taken to have the traction-fee conditions as

at for , (14)

and finally

, , , for ,

and , , , at .

Also , , ,

for and (15)

6. Solution Procedure

In order to avoid interpolation error while discretising the gov-erning equations, a suitable radial co-ordinate transformation

transformation defined by has been made use of. The governing equations along with the set of initial andboundary conditions, duly transformed, are solved numer-ically by finite difference method. Control volume-basedfinite-difference discretisation of those equations is carriedout in non-uniform staggered grids, usually known as MAC(Marker and Cell) method proposed initially by Harlow andWelch (1965). In this type of grid alignment, the velocities,pressure, viscosity and stresses are calculated at differentlocations of the control volume as indicated in Fig. 4. Thedifference equations have been derived in three distinctcells corresponding to the continuity equation, the axial

R z 1

2ro

------- 1 z z1–

z0

-------------------⎩ ⎭⎨ ⎬⎧ ⎫

cos+–⎩ ⎭⎨ ⎬⎧ ⎫

= d z d 2z0+

R z 1=

uz r z t 0 ur r z t = = r R z =

uz r z t r

----------------------- 0 ur r z t = = r 0=

uz r z t 1r2

R2-----–⎝ ⎠

⎛ ⎞uz t = z 0=

uz t 1 uz1 t 0+ cos+ –=

uz

uz

uz 0

1 uz– uz

------------------⎩ ⎭⎨ ⎬⎧ ⎫1–cos=

uz

uz r z t z

----------------------- 0ur r z t

z-----------------------= = z L= 0 r R z

rr

r-------- 0=

rz

r-------- 0=

zz

r-------- 0=

r

--------- 0= r R z =

rr 0= rz 0= zz 0= 0= r 0=

rr

z-------- 0=

rz

z-------- 0=

zz

z-------- 0=

z

--------- 0=

z 0= z L=

xr

R z -----------=

Fig. 2. Comparison of two different stenosis models. Fig. 3. Time variation of the mean velocity for the physiologicalflow (Daly,1976) and for an equivalent simple pulsatile flow.



momentum and the radial momentum equations. The dis-cretisation of the time derivative terms are based on the firstorder accurate two-level forward time differencing formulawhile those for the convective terms in the momentumequations are accorded with a hybrid formula consisting ofcentral differencing and second order upwinding. The dif-fusive terms however, are discretised by second order accu-rate three-point central difference formula.

The Poisson equation for pressure is derived from thediscretised momentum and continuity equations. Theadvantage in using MAC cell is that the pressure boundarycondition is not needed at boundaries where the velocityvector is specified, because the domain boundaries arechosen to fall on velocity nodes. The Poisson equation forpressure is solved iteratively by Successive-Over- Relax-ation (S.O.R.) method with the chosen value of over relax-ation parameter as 1.2, in order to get the intermediatepressure field using the velocity field. Subsequently themaximum cell divergence of the velocity field is calcu-lated and checked for its tolerance of 510-8. If the tol-erance limit is not satisfied, then the pressure at each cellof the flow domain is corrected and the velocities at eachcell are adjusted accordingly by repeating the process. Nostandard package for the finite difference scheme has beenused in the present computations. However, the compu-tational code based on the following algorithm has beensuccessfully programmed using FORTRAN language.

The MAC method consists of the following two stages:Stage 1:

(i) Velocities and are initialized at each

cell (i, j). This is done either from result of the previouscycle or from the prescribed initial conditions.

(ii) Time step (t) is calculated from stability criteria.(iii) The Poisson equation for pressure is solved to get

the intermediate pressure field using velocities

and of the nth time step.

(iv) The momentum equations are solved to get inter-

mediate velocities and in an explicit

manner using the previously known velocities and pres-sure.

Step 2:(v) The maximum cell divergence of the velocity field is

calculated and checked for its limit. If satisfied, steady-state convergence is checked for whether to stop calcu-lation. If the maximum divergence is not achieved withdesirable value, it goes to step (vi).

(vi) The pressure at each of the flow domain is correctedand subsequently the velocities at each cell are adjusted to get

, and . Then step (v) is again performed.

This completes the necessary calculations for advancingthe flow field through one cycle in time.

The process is to be repeated until steady-state con-vergence is achieved.

The working procedure of the MAC methodology fordetermining the numerical solution of the problem underconsideration is primarily based upon the flow chart [cf.Scheme 1] presented herein.

6.1. Numerical stability: Time-stepping procedureAmsden and Harlow (1970) suggested that the number

of calculation cycles and hence the running time could bereduced by the use of an adaptive time stepping routine

uz i

12--- j+

n ur i j

12---+

n

pi j

uz i

12--- j+

n ur i j

12---+

n

uz i

12--- j+

ur i j 1

2---+

uz i

12--- j+

n 1+ ur i j

12---+

n 1+ pi jn

Fig. 4. A typical MAC cell.

Scheme 1. Flow chart for MAC algorithm.



which, at a given cycle, would automatically choose thetime step most appropriate to the velocity field at thatcycle. Welch et al. (1966) discussed the stability and accu-racy requirements for the MAC method. They suggestedthat two stability restrictions are required. The first is akinto the Courant condition which will only be appropriatefor selected class of problems. The second stability restric-tion involves the Reynolds number:

. (16)

This stability condition is related to viscous effect (cf.Hirt, 1968) which can be applied directly to select anappropriate time step.

A more appropriate treatment used by Markham andProctor (1983), among others, is to require that no particleshould cross more than one cell boundary in a given timeinterval i.e.,

. (17)

We shall now discuss the implementation of this adap-tive time stepping procedure. The time step to be used ata given point in the calculation will be

, (18)

where 0 < a 1 ; the reason for this extra added factor a in (18)leads to a considerable computational savings (cf. Markhamand Proctor, 1983) and our experience concurs with them.

7. Result and Discussions

For the purpose of numerical computation of the desiredquantities of major physiological significance, the fol-lowing parameters have been ranged around some typicalvalues in order to obtain results of physiological interest(cf. Pontrelli, 2000; Anand and Rajagopal, 2004):

x = 0.025, L = 60.0, 1 = 0.05, = 0.068,

= 48.0, = 0.5108.

The computational domain has been confined with a finitenon-dimensional arterial length of 60.0 in which the upstreamand downstream lengths have been selected to be 8 and 31.4times the non-dimensional arterial radius respectively. For thiscomputational domain, solutions are computed through thegeneration of staggered grid with a size of 81640 while theinsertion of additional points whatsoever needed between anytwo consecutive original irregular stenosis data of Back et al.(1984) by means of interpolation has been made for the pur-pose of generating finer mesh adequately.

7.1. Model verification and validationThe simulation concerning the grid independence study

was performed for the purpose of examining the errorassociated with the grid sizes used and is presented inFigs. 5(a) and 5(b). One may notice from these figuresthat the profiles concerning three distinct grid sizes almostoverlap one another for Re=100 and De=0.1. Thus thegrid independence study in the present context of numer-ical simulation has its own importance to establish the cor-rectness of the results obtained.

Fig. 6a displays the results of an axial velocity profilefor an irregular stenosis at the throat as well as at the inletand outlet of the stenosed arterial segment. All the curvesof the present figure have a common feature that the max-imum velocity at the axis of the artery diminishes at a ratedepending upon the axial positions to become zero on thewall surface. It is noted that the usual parabolic profile ismaintained at both the inlet and outlet of the arterial tubethroughout the computation in conformity to the fullydeveloped flow downstream of the stenosis. Moreover, atthe throat of the stenosis, the streamwise velocity accel-erates with increasing Deborah number and decelerates

t1 MinRe2

------zi

2x2

zi2 x2+

------------------------

i j

t2 Minzi

uz

------ xur

------i j

t a Min t1 t2 =

Fig. 5a. Cross-sectional velocity profiles of the axial velocity fordifferent grid size at the stenosis throat (48% stenosis, cosinegeometry, Re = 100, De = 0.1).

Fig. 5b. Centerline axial velocity profiles for different grid size(48% stenosis, cosine geometry, Re = 100, De = 0.1).



with increasing Reynolds number in the vicinity of thecenterline whereas a reverse trend is observed near thewall.

In order to validate the applicability of the model underconsideration, we compared our numerical results of thenormalized pressure drop obtained for an irregular stenosiswith the experimental results of Young and Tsai (1973) inFig. 6b. The comparative study of this figure shows that thedimensionless pressure drop diminishes considerably owingto Newtonian rheology of the flowing blood together withthe absence of surface roughness of the stenosis for 56%area occlusion. The present figure also depicts that as theDeborah number increases, the dimensionless pressure dropincreases for all Reynolds numbers under considerationirrespective of the shape of the stenosis.

7.2. Pressure dropThe dimensionless pressure drop, measured across a

stenosis of 48% area reduction, for simple pulsatile andphysiological flow are depicted in Figs. 7a-7e. The pres-

sure drop decreases with increasing Reynolds numberwhereas it increases with increasing Deborah number forboth simple pulsatile and physiological flow (cf. Figs. 7aand 7c) which may be justified in the sense that as Rey-nolds number increases, the fluid gets accelerated which,

Fig. 6a. Cross-sectional profiles of the axial velocity for irregularstenosis at the stenosis throat and at the inlet and outlet.

Fig. 6b. Comparison of pressure drop [ ] for different Deborah

number, different stenosis models and stenosis size.

p

Uo2

---------

Fig. 7a. Pressure drop across the stenosis for irregular model indifferent situations for pulsatile inlet.

Fig. 7b. Pressure drop across the stenosis for different geometryfor pulsatile inlet at De = 0.20.

Fig. 7c. Pressure drop across the stenosis for irregular model indifferent situations for physiological inlet.



in turn, reduces the resistance to flow and eventually thepressure drop as they are linearly proportional (cf.McDonald, 1974). Moreover, as Deborah number increases,the fluid relaxation time increases and hence the pressuredrop increases. Furthermore, the cosine-shaped stenosismodel experiences excess pressure drop due to higher areacovering than the irregular stenosis (cf. Figs. 7b, 7d).Although the general behaviour of the pressure drop astime progresses with the variation of parameters in termsof both pulsatile and physiological flow conditions is anal-ogous, it is worthwhile to note that the velocity profilescorresponding to both pulsatile and physiological flow areappreciably reflected in the pressure drop causing signif-icant deviations of the results and hence their influences onthe pressure drop can be estimated. Moreover, it is worth-while to record the influence of a viscoelastic fluid rep-resentation of blood on the pressure drop as evident fromFig. 7e, the viscoelastic fluid causes a larger pressure dropthan that of the Newtonian counterpart. This may result inhigher resistive impedances experienced by the viscoelasticfluid flow through the arterial tube under consideration.

7.3. Wall shear stressThe variations of the dimensionless wall shear stress

(WSS) distributions over the entire arterial segment forthe two different shapes of the present constricted arteryunder the influence of both the physiological and the sim-ple pulsatile flow are recorded in Figs. 8a and 8b in thepresence of both the peak forward and the peak backwardstream. One may observe from the results of Fig. 8a thatthe irregular stenosis model experiences higher stress thanthat of the cosine model towards the throat of the stenosiswhile the deviation of the stress distributions becomesmeager in the rest of the artery irrespective of the con-sideration of both the peak forward and the peak backwardflows. Of the two inlet conditions, the pulsatile inlet influ-ences higher wall shear stress distribution than that of itscounterpart in the event of peak backward flow only asappeared in Fig. 8b. No significant deviations of the stressdistribution between these two are observed for the peakforward flow situation. It may be further noted that allother critical locations corresponding to the peak back-

Fig. 7d. Pressure drop across the stenosis for different geometryfor physiological inlet at De = 0.20.

Fig. 7e. Pressure drop across the stenosis for irregular model indifferent situations for physiological inlet.

Fig. 8a. Variation of the wall shear stress through each of thestenosis geometries for pulsatile inlet at Re 100, De = 0.10.

Fig. 8b. Variation of the wall shear stress of irregular stenosisgeometry for different inlet at Re = 100, De = 0.10.



ward stream, where both flows occur in the reverse direc-tion, experience a much larger magnitude of shear stressfor simple pulsatile flow than for physiological flow. Thisis due to the fact that at the bottom most location, themagnitude of the flow rate corresponding to the simplepulsatile flow appears to have a three- fold enhancementover that of the physiological flow under consideration.All of these findings agree with the findings of Zende-boodi and Moayeri (1999) who studied Newtonian flowthrough cosine-shaped arterial constriction. The rheolog-ical characteristics of the streaming blood on the distri-bution of the wall shear stress have however not beenruled out from the present investigation. Fig. 8c exhibitsthe results of the stress distribution over the entire arterialsegment having an irregular shaped constriction in itslumen corresponding to three different rheological prop-erties of blood at a specific Re=100 in the case of peak for-ward flow only. It appears that, in the case of Newtonianfluid, there is an appreciable increase in the peak stressvalue at the throat of the stenosis to the tune of more thanten times that of the viscoelastic one. The Newtonian fluidand the viscoelastic fluid having constant viscosity as wellmaintain the increasing trend of the wall shear stress dis-tribution. Studying the behaviour of all these results of thepresent figure, one can conclude that the consideration ofthe shear-thinning viscoelastic fluid representing blood inthe generated wall shear stress distribution arising from theconstricted flow phenomena is quite significant and hencethe inclusion of blood rheology in the model is justified.

7.4. Patterns of streamlinesFigs. 9a–9f represent the results showing various pat-

terns of the streamlines in the stenosed arterial segmentcorresponding to shear-thinning viscoelastic model ide-alisation of blood. It is interesting to observe that severalflow lines are attracted towards the stenotic wall upstreamwith the formation of several recirculation zones. The

notable features are that for both pulsatile and physio-logical flow past an irregular or a cosine-shaped stenosis,the onset of flow-detachment from a point upstream of the

Fig. 8c. Variation of the wall shear stress of irregular stenosisgeometry for physiological inlet at peak forward velocity.

Fig. 9a. Pattern of streamlines for pulsatile inlet and irregularmodel at which the velocity positive to instantaneous zero (Re =100, De = 0.10).

Fig. 9b. Pattern of streamlines for physiological inlet and irreg-ular model at which the velocity positive to instantaneous zero(Re = 100, De = 0.10).

Fig. 9c. Pattern of streamlines for pulsatile inlet and cosine modelat which the velocity positive to instantaneous zero (Re = 100,De = 0.10).

Fig. 9d. Pattern of streamlines for physiological inlet and cosinemodel at which the velocity positive to instantaneous zero (Re =100, De = 0.10).

Fig. 9e. Pattern of streamlines for physiological inlet and irreg-ular model at which the velocity positive to instantaneous zero inthe case of Newtonian flow (Re = 100).



stenosis continues for the entire downstream of the irreg-ular stenosis (cf. Figs. 9a and 9b) and a long recirculationzone is observed downstream of a cosine-shaped stenosis(cf. Figs. 9c and 9d) in contrast to multiple recirculationzones downstream of the irregular stenosis (cf. Figs. 9aand 9b). Moreover, for physiological flow correspondingto a Newtonian model, eddies, both upstream and down-stream of the constriction, are visible but no downstreamflow detachment takes place for an irregular stenosismodel (cf. Fig. 9e). The formation of such eddies iscaused primarily by the adverse pressure. The pattern offlow lines gets perturbed further with the formation of asmall separation zone downstream of the stenosis whenthe fluid is considered to be viscoelastic having constantviscosity as shown in Fig. 9f.

The nature of the streamlines at the end of pulse rate forshear-thinning viscoelastic fluid for both physiological aswell as pulsatile flow past both irregular and cosine-shaped arterial stenosis is captured in the concluding Figs.10a-10d. The general observations of the present flowlines are almost analogous to those of Figs. 9 so far as theflow behaviour is concerned. Thus, the irregular stenosismodel bears the potential to measure the flow disturbancesone step closer to the real situation compared to the usualcosine-shaped stenosis model. Moreover, the influence ofthe present rheological behaviour of the streaming bloodon the constricted flow phenomena may be measuredquantitatively through a direct comparison of the resultsexhibited by the corresponding Newtonian model.

8. Concluding Remarks

At present, inclusion of the shear–thinning viscoelasticcharacterisation of the streaming blood and both pulsatileand physiological flow waveforms in the arterial flow, understenotic conditions, with both irregular and cosine geometryof the constriction, certainly contributes much to the domainof constricted flow phenomena in the arterial system. Unlikethe previous investigations, the present updated modelappears to have its own importance in the context of thequantitative estimation of the influences of blood rheology,the shape of the constriction and different flow waveformsas well on the unsteady constricted flow characteristics of

blood in terms of the pressure drop, the distribution of wallshear stress and the patterns of the streamlines.

The present study makes an observation that, althoughthe behaviour of both the pulsatile and the physiologicalflow are analogous at certain instances of time, theyshould always be treated as two completely differentflows. This observation agrees well with those of Zen-dehboodi and Moayeri (1999) and hence the real phys-iological flow needs to be modeled for the purpose ofdeeper understanding of the pulsatile flow through con-stricted arteries.

The irregular outline of the stenosis is often found to bemore realistic than that of the cosine geometrical shapewhere the flow experiences a greater number of separationzones both upstream and downstream of the constriction,

Fig. 9f. Pattern of streamlines for physiological inlet and irregularmodel at which the velocity positive to instantaneous zero forconstant viscosity (Lamda = 1.0, Re = 100, De = 0.10).

Fig. 10a. Pattern of streamlines for physiological inlet and irreg-ular model at the end of the pulse rate.

Fig. 10b. Pattern of streamlines for pulsatile inlet and cosinemodel at the end of the pulse rate.

Fig. 10c. Pattern of streamlines for physiological inlet and cosinemodel at the end of the pulse rate.

Fig. 10d. Pattern of streamlines for physiological inlet and irreg-ular model at the end of the pulse rate in the case of Newtonianflow.



in the case of irregular stenosis, than that of the cosinestenosis irrespective of the use of both pulsatile and phys-iological flow wave forms.

Moreover, apart from the considerable influence ofblood viscoelasticity on the distribution of wall shearstresses, it has also been observed from the present quan-titative analysis that the rheology of blood affects the peakpressure drop significantly at all times, irrespective of pul-satile and physiological flow wave forms at the inlet, overthe entire physiological range of Reynolds numbers underconsideration. The behaviour of the pressure drop withdifferent Reynolds numbers appears to be closer to that ofthe experimental results of Young and Tsai (1973) whenthe rheological parameter (De) value is continuouslyreduced. Such characteristics of the pressure drop basedon the present rheology of the streaming blood influencethe pattern of the velocity profile, the wall shear stress andthe flow lines significantly and hence its incorporationappears to be quite useful in order to update the appro-priate simulation and its validation in the realm of arterialbiomechanics.

Acknowledgements

The final form of the paper owes much to the helpfulsuggestions of the referees, whose careful scrutiny we arepleased to acknowledge. The present work is part of theSpecial Assistance Programme (SAP-II) sponsored by theUniversity Grants Commission (UGC), New Delhi, India.The authors are thankful to Dr. Martin Rowlands, Uni-versity Division of Anaesthesia & Intensive Care, QueensMedical Centre Campus (Nottingham University Hospi-tals NHS Trust), University of Nottingham, Derby Road,Nottingham, NG7 2UH, United Kingdom, email:[email protected], for evaluating and correcting theEnglish language of this paper.

List of Symbols

L dimensionless length of the arterial segmentp dimensionless fluid pressurer dimensionless radial coordinater0 dimensional radius of the artery [cm]Re Reynolds number [ ]De Deborah number [ ]ur dimensionless radial velocityU0 cross-sectional average velocity [cm s-1]uz dimensionless axial velocity0 asymptotic viscosity at [g cm-1 s-1]

asymptotic viscosity at [g cm-1 s-1]p viscosity for the particle part [g cm-1 s-1]

material constant [s]z dimensionless axial coordinatezz half-length of the stenosis [cm]

z1 length of the stenosis centre [cm] angular frequency [s-1] density [g cm-3] maximum width of the stenosis [cm]

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