Research Article Numerical Simulation of Nonlinear Pulsatile Newtonian Blood Flow ... · 2019. 7....

Research ArticleNumerical Simulation of Nonlinear Pulsatile NewtonianBlood Flow through a Multiple Stenosed Artery

Satyasaran Changdar1 and Soumen De2

1 Institute of Engineering amp Management Saltlake Kolkata 700101 India2Department of Applied Mathematics University of Calcutta 92 Acharya Prafulla Chandra Road Kolkata 700009 India

Correspondence should be addressed to Satyasaran Changdar satyasaranchangdariemcalcom

Received 16 July 2015 Accepted 13 October 2015

Academic Editor Abdelouahed Tounsi

Copyright copy 2015 S Changdar and S De This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An appropriate nonlinear blood flow model under the influence of periodic body acceleration through a multiple stenosed arteryis investigated with the help of finite difference methodThe arterial segment is simulated by a cylindrical tube filled with a viscousincompressible Newtonian fluid described by the Navier-Stokes equation The nonlinear equation is solved numerically with theproper boundary conditions and pressure gradient that arise from the normal functioning of the heart Results are discussed incomparison with the existing models

1 Introduction

At present the investigation of blood flow analysis in astenosed artery is very important in the medical domainbecause of the fact that many of the diseases such as heartattacks and strokes are related to blood flow and the physicalcharacteristic of vessel wall Nowadays the leading causesof the death in the world are due to heart diseases such asatherosclerosis Atherosclerosis involves an accumulation oflow-density lipoprotein in the wall of large arteries typicallywhere the wall shear rate is low and oscillatory [1]

Investigation of blood flow modeling through arterialmultistenosis is very challenging Accuracy of the simulationdepends mainly on suitable numerical approach realisticmodel geometry and boundary conditions Many investi-gators have focused their attention on blood flow throughstenosed arteries with single stenosis by Mekheimer [2 3]Chakravarty andMandal [4] Lee andXu [5] who pointed outthat the mathematical model becomes more accurate in thepresence of an overlapping stenosis instead of amild one Angand Mazumdar [6] studied asymmetric arterial blood flowwith numerical solution in three dimensions and Ikbal etal [7] have worked on unsteady response of non-Newtonian

blood flow in magnetic field without considering periodicbody acceleration Khler et al [8] studied the wall shearstress with the help of magnetic resonance imaging (MRI)measurements of the velocity field and compared them withsimulation outputs Stroud et al [9] have studied a 2D plaquemodel usingmodeling and simulationwhile Fischer et al [10]worked on numerical method for the computational study ofarterial blood flow with turbulence The asymmetric flows ina symmetric sudden expansion channel have been studiedusing experimental and numerical techniques by Fearn etal [11] and Durst et al [12] Mahapatra et al [13] investi-gated unsteady laminar separated flow through constrictedchannel using finite difference technique in staggered griddistribution and suggested that the critical value of Reynoldsnumber depends on the area reduction and the length ofthe constriction Chakravarty and Sannigrahi [14] solvedblood flow model with body acceleration but they do notconsider the nonlinear terms in the model Blood showsa non-Newtonian behaviour at low shear rates in tubes ofsmaller diameters andTaylor [15] suggested that at high shearrates commonly found in larger arteries blood behaves like aNewtonian fluid

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2015 Article ID 628605 10 pageshttpdxdoiorg1011552015628605

2 International Scholarly Research Notices

x

h(z)

120579

y

d

r0

r-axis

z-axis

2l

Figure 1 Cylindrical coordinate systemwithmultiple stenosis alongthe axial direction where 2119897 is the length of the stenosis and 119889 isdistance of stenosis from the radial axis

With the above motivation in our mind we have workedon numerical simulations of nonlinear pulsatile unsteadyNewtonian blood flow in a rigid cylindrical tube throughcosine-shape stenosis under the influences of periodic bodyacceleration It appears that a few studies address the issueof that nonlinear terms present in Navier-Stokes equationgovern blood flowwith periodic body acceleration associatedwith an atherosclerotic plaque The numerical solutions areobtained of the nonlinear model using appropriate finitedifference method A comparison of the axial and radialvelocities wall shear stresses flux rates and the streamlineswith other existing model [7] has been studied also

2 Physical Assumptions andMathematical Model

The segment of the artery is modeled as an axisymmetriccylindrical tube with radius 119903

0 The blood is modeled as a

homogeneous incompressible viscous unsteady Newtonianfluid of density 120588 and kinematic viscosity ] Therefore theblood flow is governed by the incompressible Navier-Stokesequation

A cylindrical coordinate system (119903 120579 119911) is chosen where(119903 120579) is the coordinate in the radial and circumferentialdirection while 119911 is taken along the axis of artery as shownin Figure 1 The velocity components in the axial and radialdirections are 119906 and V respectively The flow is driven bya prescribed dimensional oscillatory axial pressure gradientgiven by [16]

minus120597119901

120597119911= 1199010+ 1199011cos (120596119905) 119905 gt 0 (1)

where1199010and1199011are the steady component of the pressure gra-

dient and pulsatile component respectively The frequency120596 = 2120587119891 and 119891 is the heart pulse frequency The pressuregradient in the radial direction is negligibly small as thelumen radius of artery is small compared to pressure wavelength so that 120597119901120597119903 ≃ 0 Also because of the human bodyacceleration the axial flow is subject to an external force 119865ext

0 2 4 6 8 10minus1

minus05

0

05

1

Axial direction z-axis

Radi

al d

irect

ion

r-ax

is

Geometry of the multistenosis

Figure 2 The schematic representation of the stenosis along theaxial direction as given by ℎ(119911) The geometrical parameters are asfollows 119897 = 200 119889 = 300 and 120582 = 0500

For the present model we consider the periodic accelerationforce given by

119865ext = 1198860cos (120596119905 + 120601) (2)

where 1198860is the amplitude of the pulse

According to the above assumptions the blood flowdynamics is governed by the equation of continuity

120597119906

120597119911+V119903+120597V120597119903

= 0 (3)

themomentum equation in the radial direction (flow velocityV)

120597V120597119905

= minus(119906120597V120597119911

+ V120597V120597119903) minus

120597119901

120597119903

+1

Re(1205972V1205971199032+1

119903

120597V120597119903

+1205972V1205971199112minus

V1199032)

(4)

and the axial direction (flow velocity 119906)

120597119906

120597119905= minus(V

120597119906

120597119903+ 119906

120597119906

120597119911) minus

120597119901

120597119911

+1

Re(1205972119906

1205971199032+1

119903

120597119906

120597119903+1205972119906

1205971199112) + 119865ext

(5)

In (4) and (5) Re = 1199030119906infin120588] is the Reynolds number and 119906

infin

is the average velocity of the bloodFinally to model multiple axisymmetric stenosis we

define the following mathematical function ℎ(119911)

ℎ (119911) =

1 minus 120582 cos(120587119911119897) 119889 le 119911 lt 119889 + 119897

1 + 120582 cos(120587119911119897) 119889 + 119897 le 119911 le 119889 + 2119897

1 otherwise

(6)

where 120582 is a dimensionless constant and the geometry of thisaxisymmetric stenosis in the cross section of the artery isshown in Figure 2

International Scholarly Research Notices 3

We numerically simulate (3)ndash(5) subject to the followinginitial condition

119906 (119903 119911 119905) = 0

V (119903 119911 119905) = 0

at 119905 = 0

(7)

and the no-slip boundary conditions

120597119906 (119903 119911 119905)

120597119903= 0

V (119903 119911 119905) = 0

at 119903 = 0

119906 (119903 119911 119905) = 0 = V (119903 119911 119905) at 119903 = ℎ (119911)

(8)

3 Numerical SimulationComputational Method

We use the finite difference scheme to study the dynamics ofblood flow through the cylindrical shape artery To employthis method first we transform our cylindrical domaininto the rectangular domain by using the following radialtransformation

119909 =119903

ℎ (119911) (9)

Under this transformation the equation of continuity (3) andthe equations of motion in the radial direction (4) and axialdirection (5) respectively are rewritten as

120597119906

120597119911+

V119909ℎ (119911)

+1

ℎ (119911)

120597V120597119909

minus119909

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911= 0 (10)

120597V120597119905

= minus(V

ℎ (119911)

120597V120597119909

+ 119906120597V120597119911

minus119909119906

ℎ (119911)

120597V120597119909

119889ℎ

119889119911)

+1

Re

1

ℎ2(119911)

(1205972V1205971199092+1

119909

120597V120597119909

minusV1199092) +

1205972V1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972V

120597119909120597119911+

119909

ℎ (119911)

120597V120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597V120597119909

+ 1199092 1205972V1205971199092)

(11)

120597119906

120597119905= minus(

Vℎ (119911)

120597119906

120597119909+ 119906

120597119906

120597119911minus

119909119906

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911) minus

120597119901

120597119911

+1

Re

1

ℎ2(119911)

(1205972119906

1205971199092+1

119909

120597119906

120597119909) +

1205972119906

1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972119906

120597119909120597119911+

119909

ℎ (119911)

120597119906

120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597119906

120597119909+ 1199092 1205972119906

1205971199092) + 119865ext

(12)

Initial condition (7) and no-slip boundary condition (8) dueto radial transformation (9) then become

119906 (119909 119911 119905) = 0

V (119909 119911 119905) = 0

at 119905 = 0

120597119906 (119909 119911 119905)

120597119909= 0

V (119909 119911 119905) = 0

at 119909 = 0

119906 (119909 119911 119905) = 0 = V (119909 119911 119905) at 119909 = 1

(13)

Let us first apply the finite difference discretization scheme tosolve nonlinearmodel equations (10)ndash(12)We use the centraldifference approximation to discretize the spatial derivativesand the explicit forward finite difference approximation todiscretize the time derivative in the following manner

120597119906

120597119911=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119911

1205972119906

1205971199112=

(119906)119899

119894+1119895minus 2 (119906)

119899

119894119895+ (119906)119899

119894minus1119895

Δ1199112

120597119906

120597119909=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119909

120597119906

120597119905=

(119906)119899+1

119894119895minus (119906)119899

119894119895

Δ119905

(14)

where

(119906)119899

119894119895= 119906 (119909

119895 119911119894 119905119899)

119911119894= (119894 minus 1) Δ119911 119894 = 1 2 119872 + 1

119909119895= (119895 minus 1) Δ119909 119895 = 1 2 119873 + 1

119905119899= (119899 minus 1) Δ119905 119899 = 1 2

(15)

Similarly we approximate all the partial derivatives of VThe axial velocity (119906)

119899

119894119895is obtained from (10) and (12)

by applying the above finite difference scheme at any point(119911119894 119909119895) in the domain of interest at any time 119905

119899with the help

of the following discretize initial and boundary conditions(discretization of (13))

(119906)1

119894119895= 0

(V)1119894119895= 0

(119906)119899

1198941= (119906)119899

1198942

(V)1198991198941= 0

(119906)119899

119894119873+1= (V)119899119894119873+1

= 0

(16)


0 02 04 06 08 10

002

004

006

008

01

012

x

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(a)

0

01

02

03

04

05

06

07

08

x

1

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

0 02 04 06 08

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(b)

Figure 3 Distribution of axial velocities at different 119911 for Re = 400 Axial velocity (a) without body acceleration and (b) with bodyacceleration

subject to the input pressure gradient and external force 119865extfrom relations (1) and (2)The radial velocity (V)119899

119894119895is obtained

from (10) and (11)Finally we determine the volumetric flow rate

119876 = 2120587int

ℎ

0

119903119906 119889119903 = 2120587ℎ (119911)2int

1

0

119909119906 119889119909 (17)

and the wall shearing stress

120591 = minus120583119889119906

119889119903

10038161003816100381610038161003816100381610038161003816119903=ℎ(119911)

= minus120583

ℎ (119911)

119889119906

119889119909(18)

in the rectangular domainwith the help of transformation (9)where 120583 is the viscosity The discretize version of 119876 and 120591 isgiven by the following equations

(119876)119899

119894= 2120587 (ℎ

119899

119894)2int

1

0

119909119895(119906)119899

119894119895119889119909119895

120591119899

119894= minus

120583

ℎ119899

119894

(119906119894119873+1

minus 119906119894119873

Δ119909)

(19)

4 Simulation Results and Discussions

In this section we shall discuss the numerical simulation ofthe nonlinear equations to study the influence of stenosesand body acceleration on the blood flow for different valuesof the physical parameters The simulation parameters areas follows [7 14] 119897 = 200 119889 = 300 120582 = 05 Re = 400600 and 800 119901

0= 01 and 119901

1= 02 times 119901

0 The results

obtained for axial velocity by solving explicit finite differencescheme with various grid sizes are taken in order to achieve

the convergence and stability We perform the experimentsfor grid size 60 times 60 and 100 times 100 with 119889119905 = 001 and 0001The results are found to be very similar in both cases

Figures 3(a) and 3(b) represent the behavior of the axialvelocity profile of the blood at time 119905 = 10 without andwith body acceleration respectively Both figures are drawnfor Re = 400 at different 119911 The comparative study betweenthe figures (Figures 3(a) and 3(b)) reveals that the bodyacceleration enhances the axial velocity The curves in thesetwo figures reveal that the velocity profile is constant for0 le 119909 le 119909

119898and then velocity decreases and finally goes

to zero on the constricted arterial wall The values of 119909119898

depend on both the body acceleration and the stenosed zoneAlso in the stenosed zone of the artery (3 le 119911 le 7)the velocity is low and in presence of body acceleration thevelocity sharply decreases in this stenosed zone Figure 4(a)shows the results for the distribution of axial velocity overthe stenosed artery for three different Reynolds numbers Wecan say from this figure that as 119911 enters into the stenosedzone the axial velocity starts decreasing from its maximumvalue in nonstenosed zone until the first constriction attaintits maximum value near 119911 = 4 and then it gradually increasesup to 119911 = 52 and again it starts decreasing till the maximumheight of the second stenosis and gradually starts increasingthereafter and finally again flows with maximum velocityin nonstenosis region The three curves here indicate thatthe axial velocity increases in the constricted part of theartery as Reynolds number increases under the influence ofbody acceleration Figure 4(b) represents the results for thedistribution of radial velocity in the multiple stenosed arteryfor three different Reynolds numbers From the figure wecan say that the direction of radial velocity is negative in the


0 1 2 3 4 5 6 7 8 9

1

z

0

01

02

03

04

05

06

07

09

08

u a

xial

vel

ocity

forx

=085

timet

=1000

Re = 800

Re = 600

Re = 400

(a)

0

01

02

03

04

05

minus01

minus02

minus03

minus04

minus051 2 3 4 5 6 7 8 9

z

Re = 800

Re = 600

Re = 400

r

adia

l velo

city

at ti

met=10000000

(b)

Figure 4 Distribution of (a) axial velocities for various Reynolds numbers and (b) radial velocities for different Reynolds number

minus01

minus02

minus03

minus04

minus045

minus035

minus025

minus015

minus005

0

x

Re = 800

Re = 600

Re = 400

r

adia

l vel

ocity

atz=3898305

10 02 04 06 08

Figure 5 The distribution of radial velocity at 119911 = 39

stenosis zone due to presence of multiple stenosis Thus themultiple stenosis and the Newtonian characteristics of theflowing blood affect the axial velocity profile which can beestimated by the relevant curves of the present figure

The curves in Figure 5 describe the nature of radialvelocity for three different Reynolds numbers The velocityinitially starts with zero and continues till 119909 = 08 and then itdecreases gradually in negative direction

To test the effects of body acceleration on axial andradial velocities profile several simulations have been carried

out using the contour plot as shown in Figures 6(a)ndash6(d)and Figures 7(a)ndash7(d) respectively The velocity profiles 119906and V are shown in different region with different colorsrepresenting the value of velocity with the help of color barOne can see from these plot that velocity profile is dividedinto different layers due to the constriction of the artery andchanges in the plot also can be observed in case of no bodyacceleration We have shown the distribution of axial andradial velocities in Figures 8(a) and 8(d) in entire upper halfsegment of the artery using 3D plot Figure 8(a) shows the


x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)

Figure 6 Contour distribution of (a) axial velocities for Reynolds number Re = 400 with body acceleration (b) axial velocities for Re = 400

without body acceleration (c) axial velocities for Reynolds number Re = 800 with body acceleration and (d) axial velocities for Re = 800

without body acceleration

axial velocity profile of the flow without body accelerationat time 119905 = 10 for Re = 400 One can see that the velocityprofile is constant in the nonstenosed zone and varies overthe constricted area In case of body acceleration the profileis shown in Figure 8(c)

Wall shear stress plays an important role in the creationand propagation of arteriosclerosis If the wall shear stress ishigh then it may damage the arterial wall and is the maincause of the intimal thickeningOn the other hand the plaqueformation in an artery is created in the regions of low arterialwall shear stress Atherosclerotic lesions are associated withlow and high wall shear stress So it is important to studythe wall shear stress distribution in the multistenosed artery

Figures 9(a) and 9(b) show the distribution ofwall shear stresson the arterial segment for three different Reynolds numbersThe wall shear stress increases rapidly near to the peak ofthe constriction Here the effects of Reynolds number canbe observed from the figure The wall shear stress increasesas Reynolds number increases Figures 10(a) and 10(b) showthe distribution of flux over the stenosed artery for differentReynolds number One can conclude that flux decreases nearthe picks of the stenosis

The streamlines of the blood flow in the artery with mul-tistenosis are found in the transformed rectangular domainwith grid 60 times 60 in the upper half zone and same grid alsotaken for lower half portion We have plotted the different


v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)

Figure 7 Contour distribution of (a) radial velocities for Reynolds number Re = 400with body acceleration (b) radial velocities for Re = 400

without body acceleration (c) radial velocities for Reynolds number Re = 800 with body acceleration and (d) radial velocities for Re = 800


types of streamlines in Figures 11(a) and 11(b) at Re = 400All the streamlines follow the straight line path near the axiswhich gradually get perturbed more towards the wall of thestenosed artery It is interesting to observe that several flowlines are attracted towards the stenotic wall upstreamwith theformation of circulation zones while others pass through theconstricted region directly following the main stream

Figure 12 shows the streamline patterns at Re = 400whenthe artery is free from stenosis It is observed that the lines areparallel to the axial direction

For the purpose of model validation the axial velocityprofile is compared with [7] and as shown in Figure 13 theresults are found to be in good agreement though their studies

were based on the stenotic blood flow in which the streamingblood was treated as non-Newtonian fluid in magnetic fieldAlso the result agrees qualitatively well for Newtonian fluidwith Tu et al [17]

5 Conclusions

A nonlinear mathematical model for blood flow in a multiplestenosed arterial segment has been developed under theinfluence of body acceleration The numerical simulation ofblood flow is investigated in this study As the Reynolds num-ber increases the wall shear stress increases The multiplestenosis has significant effect on the wall shear stress in such


020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)

Figure 8 3D plot of (a) axial velocities without body acceleration at Re = 400 (b) radial velocities without body acceleration at Re = 400(c) axial velocities with body acceleration at Re = 400 and (d) radial velocities with body acceleration at Re = 400

0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)

Figure 9 Distribution of (a) wall shear stress with body acceleration and (b) wall shear stress without body acceleration


0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300

(b)Figure 10 Distribution of (a) flux with body acceleration and (b) flux without body acceleration

0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60

(b)Figure 11 Distribution of (a) streamlines with body acceleration and (b) streamlines without body acceleration at time 119905 = 10 in the upperhalf segment of the artery

0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40

60Streamlines Re = 400 at time t = 545

Figure 12 The streamline patterns of the flow at Re = 400 when there is no constriction


0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300

Ikbal et al (Re = 300)

Figure 13 Validation of axial velocity at Re = 300

a way that it develops more at the constricted locations thanall other sites of the artery The streamline pattern showsthe distinct boundary layer characteristics in the arterialsegment This is validated by the flow visualisation observedby the simulation studies of [7] The results obtained herewould help researchers greatly in gaining better insight intoblood flowmodels through themultistenosis artery under theinfluence of body acceleration

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D M Wooton and D N Ku ldquoFluid mechanics of vascular sys-tems diseases and thrombosisrdquo Annual Review of BiomedicalEngineering vol 1 pp 299ndash329 1999

[2] K SMekheimer andMA El Kot ldquoSuspensionmodel for bloodflow through catheterized curved artery with time-variantoverlapping stenosisrdquo Engineering Science and Technology vol18 no 3 pp 452ndash462 2015

[3] K S Mekheimer F Salama and M A El Kot ldquoThe unsteadyflow of a Carreau fluid through inclined catheterized arterieshave a balloon (angioplasty) with time-variant overlappingstenosisrdquoWalailak Journal of Science and Technology vol 12 pp863ndash883 2014

[4] S Chakravarty and P K Mandal ldquoMathematical modelling ofblood flow through an overlapping arterial stenosisrdquoMathemat-ical and Computer Modelling vol 19 no 1 pp 59ndash70 1994

[5] KW Lee andX Y Xu ldquoModelling of flow andwall behaviour inamildly stenosed tuberdquoMedical Engineering andPhysics vol 24no 9 pp 575ndash586 2002

[6] K C Ang and J N Mazumdar ldquoMathematical modelling ofthree-dimensional flow through an asymmetric arterial steno-sisrdquo Mathematical and Computer Modelling vol 25 no 1 pp19ndash29 1997

[7] M A Ikbal S Chakravarty K K L Wong J Mazumdar and PK Mandal ldquoUnsteady response of non-Newtonian blood flowthrough a stenosed artery in magnetic fieldrdquo Journal of Compu-tational and Applied Mathematics vol 230 no 1 pp 243ndash2592009

[8] U Khler I Marshall M B Robertson Q Long X Y Xu andP R Hoskins ldquoMRI measurement of wall shear stress vectorsin bifurcation models and comparison with CFD predictionsrdquoJournal of Magnetic Resonance Imaging vol 14 no 5 pp 563ndash573 2001

[9] J S Stroud S A Berger and D Saloner ldquoNumerical analysisof flow through a severely stenotic carotid artery bifurcationrdquoJournal of Biomechanical Engineering vol 124 no 1 pp 9ndash202002

[10] P F Fischer F Loth S E Lee S-W Lee D S Smith andH S Bassiouny ldquoSimulation of high Reynolds number vascularflowsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 196 no 31-32 pp 3049ndash3060 2007

[11] R M Fearn T Mullin and K A Cliffe ldquoNonlinear flow phe-nomena in a symmetric sudden expansionrdquo Journal of FluidMechanics vol 211 pp 595ndash608 1990

[12] F Durst J C F Pereira and C Tropea ldquoThe plane symmetricsudden-expansion flow at low Reynolds numbersrdquo Journal ofFluid Mechanics vol 248 pp 567ndash581 1993

[13] T R Mahapatra G C Layek and M K Maiti ldquoUnsteady lam-inar separated flow through constricted channelrdquo InternationalJournal of Non-LinearMechanics vol 37 no 2 pp 171ndash186 2002

[14] S Chakravarty and A K Sannigrahi ldquoEffect of body accelera-tion on blood flow in an irregular stenosed arteryrdquoMathemati-cal and Computer Modelling vol 19 no 5 pp 93ndash103 1994

[15] MG Taylor ldquoThe influence of the anomalous viscosity of bloodupon its oscillatory flowrdquo Physics inMedicine and Biology vol 3no 3 pp 273ndash290 1959

[16] A C Burton Physiology and Biophysics of the Circulation Intro-ductory Text Year Book Medical Publishers Chicago Ill USA1966

[17] C Tu M Deville L Dheur and L Vanderschuren ldquoFiniteelement simulation of pulsatile flow through arterial stenosisrdquoJournal of Biomechanics vol 25 no 10 pp 1141ndash1152 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


x

h(z)

120579

y

d

r0

r-axis

z-axis

2l

Figure 1 Cylindrical coordinate systemwithmultiple stenosis alongthe axial direction where 2119897 is the length of the stenosis and 119889 isdistance of stenosis from the radial axis

With the above motivation in our mind we have workedon numerical simulations of nonlinear pulsatile unsteadyNewtonian blood flow in a rigid cylindrical tube throughcosine-shape stenosis under the influences of periodic bodyacceleration It appears that a few studies address the issueof that nonlinear terms present in Navier-Stokes equationgovern blood flowwith periodic body acceleration associatedwith an atherosclerotic plaque The numerical solutions areobtained of the nonlinear model using appropriate finitedifference method A comparison of the axial and radialvelocities wall shear stresses flux rates and the streamlineswith other existing model [7] has been studied also

2 Physical Assumptions andMathematical Model

The segment of the artery is modeled as an axisymmetriccylindrical tube with radius 119903

0 The blood is modeled as a

homogeneous incompressible viscous unsteady Newtonianfluid of density 120588 and kinematic viscosity ] Therefore theblood flow is governed by the incompressible Navier-Stokesequation

A cylindrical coordinate system (119903 120579 119911) is chosen where(119903 120579) is the coordinate in the radial and circumferentialdirection while 119911 is taken along the axis of artery as shownin Figure 1 The velocity components in the axial and radialdirections are 119906 and V respectively The flow is driven bya prescribed dimensional oscillatory axial pressure gradientgiven by [16]

minus120597119901

120597119911= 1199010+ 1199011cos (120596119905) 119905 gt 0 (1)

where1199010and1199011are the steady component of the pressure gra-

dient and pulsatile component respectively The frequency120596 = 2120587119891 and 119891 is the heart pulse frequency The pressuregradient in the radial direction is negligibly small as thelumen radius of artery is small compared to pressure wavelength so that 120597119901120597119903 ≃ 0 Also because of the human bodyacceleration the axial flow is subject to an external force 119865ext

0 2 4 6 8 10minus1

minus05

0

05

1

Axial direction z-axis

Radi

al d

irect

ion

r-ax

is

Geometry of the multistenosis

Figure 2 The schematic representation of the stenosis along theaxial direction as given by ℎ(119911) The geometrical parameters are asfollows 119897 = 200 119889 = 300 and 120582 = 0500

For the present model we consider the periodic accelerationforce given by

119865ext = 1198860cos (120596119905 + 120601) (2)

where 1198860is the amplitude of the pulse

According to the above assumptions the blood flowdynamics is governed by the equation of continuity

120597119906

120597119911+V119903+120597V120597119903

= 0 (3)

themomentum equation in the radial direction (flow velocityV)

120597V120597119905

= minus(119906120597V120597119911

+ V120597V120597119903) minus

120597119901

120597119903

+1

Re(1205972V1205971199032+1

119903

120597V120597119903

+1205972V1205971199112minus

V1199032)

(4)

and the axial direction (flow velocity 119906)

120597119906

120597119905= minus(V

120597119906

120597119903+ 119906

120597119906

120597119911) minus

120597119901

120597119911

+1

Re(1205972119906

1205971199032+1

119903

120597119906

120597119903+1205972119906

1205971199112) + 119865ext

(5)

In (4) and (5) Re = 1199030119906infin120588] is the Reynolds number and 119906

infin

is the average velocity of the bloodFinally to model multiple axisymmetric stenosis we

define the following mathematical function ℎ(119911)

ℎ (119911) =

1 minus 120582 cos(120587119911119897) 119889 le 119911 lt 119889 + 119897

1 + 120582 cos(120587119911119897) 119889 + 119897 le 119911 le 119889 + 2119897

1 otherwise

(6)

where 120582 is a dimensionless constant and the geometry of thisaxisymmetric stenosis in the cross section of the artery isshown in Figure 2



119906 (119903 119911 119905) = 0

V (119903 119911 119905) = 0

at 119905 = 0

(7)


120597119906 (119903 119911 119905)

120597119903= 0

V (119903 119911 119905) = 0

at 119903 = 0

119906 (119903 119911 119905) = 0 = V (119903 119911 119905) at 119903 = ℎ (119911)

(8)



119909 =119903

ℎ (119911) (9)


120597119906

120597119911+

V119909ℎ (119911)

+1

ℎ (119911)

120597V120597119909

minus119909

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911= 0 (10)

120597V120597119905

= minus(V

ℎ (119911)

120597V120597119909

+ 119906120597V120597119911

minus119909119906

ℎ (119911)

120597V120597119909

119889ℎ

119889119911)

+1

Re

1

ℎ2(119911)

(1205972V1205971199092+1

119909

120597V120597119909

minusV1199092) +

1205972V1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972V

120597119909120597119911+

119909

ℎ (119911)

120597V120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597V120597119909

+ 1199092 1205972V1205971199092)

(11)

120597119906

120597119905= minus(

Vℎ (119911)

120597119906

120597119909+ 119906

120597119906

120597119911minus

119909119906

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911) minus

120597119901

120597119911

+1

Re

1

ℎ2(119911)

(1205972119906

1205971199092+1

119909

120597119906

120597119909) +

1205972119906

1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972119906

120597119909120597119911+

119909

ℎ (119911)

120597119906

120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597119906

120597119909+ 1199092 1205972119906

1205971199092) + 119865ext

(12)


119906 (119909 119911 119905) = 0

V (119909 119911 119905) = 0

at 119905 = 0

120597119906 (119909 119911 119905)

120597119909= 0

V (119909 119911 119905) = 0

at 119909 = 0

119906 (119909 119911 119905) = 0 = V (119909 119911 119905) at 119909 = 1

(13)


120597119906

120597119911=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119911

1205972119906

1205971199112=

(119906)119899

119894+1119895minus 2 (119906)

119899

119894119895+ (119906)119899

119894minus1119895

Δ1199112

120597119906

120597119909=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119909

120597119906

120597119905=

(119906)119899+1

119894119895minus (119906)119899

119894119895

Δ119905

(14)

where

(119906)119899

119894119895= 119906 (119909

119895 119911119894 119905119899)

119911119894= (119894 minus 1) Δ119911 119894 = 1 2 119872 + 1

119909119895= (119895 minus 1) Δ119909 119895 = 1 2 119873 + 1

119905119899= (119899 minus 1) Δ119905 119899 = 1 2

(15)


119899



119899with the help


(119906)1

119894119895= 0

(V)1119894119895= 0

(119906)119899

1198941= (119906)119899

1198942

(V)1198991198941= 0

(119906)119899

119894119873+1= (V)119899119894119873+1

= 0

(16)


0 02 04 06 08 10

002

004

006

008

01

012

x

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(a)

0

01

02

03

04

05

06

07

08

x

1

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

0 02 04 06 08

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(b)





119876 = 2120587int

ℎ

0

119903119906 119889119903 = 2120587ℎ (119911)2int

1

0

119909119906 119889119909 (17)


120591 = minus120583119889119906

119889119903

10038161003816100381610038161003816100381610038161003816119903=ℎ(119911)

= minus120583

ℎ (119911)

119889119906

119889119909(18)


(119876)119899

119894= 2120587 (ℎ

119899

119894)2int

1

0

119909119895(119906)119899

119894119895119889119909119895

120591119899

119894= minus

120583

ℎ119899

119894

(119906119894119873+1

minus 119906119894119873

Δ119909)

(19)



0= 01 and 119901

1= 02 times 119901

0 The results








0 1 2 3 4 5 6 7 8 9

1

z

0

01

02

03

04

05

06

07

09

08

u a

xial

vel

ocity

forx

=085

timet

=1000

Re = 800

Re = 600

Re = 400

(a)

0

01

02

03

04

05

minus01

minus02

minus03

minus04

minus051 2 3 4 5 6 7 8 9

z

Re = 800

Re = 600

Re = 400

r

adia

l velo

city

at ti

met=10000000

(b)


minus01

minus02

minus03

minus04

minus045

minus035

minus025

minus015

minus005

0

x

Re = 800

Re = 600

Re = 400

r

adia

l vel

ocity

atz=3898305

10 02 04 06 08







x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)









v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119903 119911 119905) = 0

V (119903 119911 119905) = 0

at 119905 = 0

(7)


120597119906 (119903 119911 119905)

120597119903= 0

V (119903 119911 119905) = 0

at 119903 = 0

119906 (119903 119911 119905) = 0 = V (119903 119911 119905) at 119903 = ℎ (119911)

(8)



119909 =119903

ℎ (119911) (9)


120597119906

120597119911+

V119909ℎ (119911)

+1

ℎ (119911)

120597V120597119909

minus119909

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911= 0 (10)

120597V120597119905

= minus(V

ℎ (119911)

120597V120597119909

+ 119906120597V120597119911

minus119909119906

ℎ (119911)

120597V120597119909

119889ℎ

119889119911)

+1

Re

1

ℎ2(119911)

(1205972V1205971199092+1

119909

120597V120597119909

minusV1199092) +

1205972V1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972V

120597119909120597119911+

119909

ℎ (119911)

120597V120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597V120597119909

+ 1199092 1205972V1205971199092)

(11)

120597119906

120597119905= minus(

Vℎ (119911)

120597119906

120597119909+ 119906

120597119906

120597119911minus

119909119906

ℎ (119911)

120597119906

120597119909

119889ℎ

119889119911) minus

120597119901

120597119911

+1

Re

1

ℎ2(119911)

(1205972119906

1205971199092+1

119909

120597119906

120597119909) +

1205972119906

1205971199112

minus1

Re

2119909

ℎ (119911)

119889ℎ (119911)

119889119911

1205972119906

120597119909120597119911+

119909

ℎ (119911)

120597119906

120597119909

1198892ℎ

1198891199112

minus (119889ℎ119889119911

ℎ (119911))

2

(2119909120597119906

120597119909+ 1199092 1205972119906

1205971199092) + 119865ext

(12)


119906 (119909 119911 119905) = 0

V (119909 119911 119905) = 0

at 119905 = 0

120597119906 (119909 119911 119905)

120597119909= 0

V (119909 119911 119905) = 0

at 119909 = 0

119906 (119909 119911 119905) = 0 = V (119909 119911 119905) at 119909 = 1

(13)


120597119906

120597119911=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119911

1205972119906

1205971199112=

(119906)119899

119894+1119895minus 2 (119906)

119899

119894119895+ (119906)119899

119894minus1119895

Δ1199112

120597119906

120597119909=

(119906)119899

119894+1119895minus (119906)119899

119894minus1119895

2Δ119909

120597119906

120597119905=

(119906)119899+1

119894119895minus (119906)119899

119894119895

Δ119905

(14)

where

(119906)119899

119894119895= 119906 (119909

119895 119911119894 119905119899)

119911119894= (119894 minus 1) Δ119911 119894 = 1 2 119872 + 1

119909119895= (119895 minus 1) Δ119909 119895 = 1 2 119873 + 1

119905119899= (119899 minus 1) Δ119905 119899 = 1 2

(15)


119899



119899with the help


(119906)1

119894119895= 0

(V)1119894119895= 0

(119906)119899

1198941= (119906)119899

1198942

(V)1198991198941= 0

(119906)119899

119894119873+1= (V)119899119894119873+1

= 0

(16)


0 02 04 06 08 10

002

004

006

008

01

012

x

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(a)

0

01

02

03

04

05

06

07

08

x

1

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

0 02 04 06 08

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(b)





119876 = 2120587int

ℎ

0

119903119906 119889119903 = 2120587ℎ (119911)2int

1

0

119909119906 119889119909 (17)


120591 = minus120583119889119906

119889119903

10038161003816100381610038161003816100381610038161003816119903=ℎ(119911)

= minus120583

ℎ (119911)

119889119906

119889119909(18)


(119876)119899

119894= 2120587 (ℎ

119899

119894)2int

1

0

119909119895(119906)119899

119894119895119889119909119895

120591119899

119894= minus

120583

ℎ119899

119894

(119906119894119873+1

minus 119906119894119873

Δ119909)

(19)



0= 01 and 119901

1= 02 times 119901

0 The results








0 1 2 3 4 5 6 7 8 9

1

z

0

01

02

03

04

05

06

07

09

08

u a

xial

vel

ocity

forx

=085

timet

=1000

Re = 800

Re = 600

Re = 400

(a)

0

01

02

03

04

05

minus01

minus02

minus03

minus04

minus051 2 3 4 5 6 7 8 9

z

Re = 800

Re = 600

Re = 400

r

adia

l velo

city

at ti

met=10000000

(b)


minus01

minus02

minus03

minus04

minus045

minus035

minus025

minus015

minus005

0

x

Re = 800

Re = 600

Re = 400

r

adia

l vel

ocity

atz=3898305

10 02 04 06 08







x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)









v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

002

004

006

008

01

012

x

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(a)

0

01

02

03

04

05

06

07

08

x

1

u a

xial

vel

ocity

forR

e=400

at ti

met

=10000000

0 02 04 06 08

z = 8125

z = 65625

z = 5

z = 34375

z = 1875

(b)





119876 = 2120587int

ℎ

0

119903119906 119889119903 = 2120587ℎ (119911)2int

1

0

119909119906 119889119909 (17)


120591 = minus120583119889119906

119889119903

10038161003816100381610038161003816100381610038161003816119903=ℎ(119911)

= minus120583

ℎ (119911)

119889119906

119889119909(18)


(119876)119899

119894= 2120587 (ℎ

119899

119894)2int

1

0

119909119895(119906)119899

119894119895119889119909119895

120591119899

119894= minus

120583

ℎ119899

119894

(119906119894119873+1

minus 119906119894119873

Δ119909)

(19)



0= 01 and 119901

1= 02 times 119901

0 The results








0 1 2 3 4 5 6 7 8 9

1

z

0

01

02

03

04

05

06

07

09

08

u a

xial

vel

ocity

forx

=085

timet

=1000

Re = 800

Re = 600

Re = 400

(a)

0

01

02

03

04

05

minus01

minus02

minus03

minus04

minus051 2 3 4 5 6 7 8 9

z

Re = 800

Re = 600

Re = 400

r

adia

l velo

city

at ti

met=10000000

(b)


minus01

minus02

minus03

minus04

minus045

minus035

minus025

minus015

minus005

0

x

Re = 800

Re = 600

Re = 400

r

adia

l vel

ocity

atz=3898305

10 02 04 06 08







x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)









v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9

1

z

0

01

02

03

04

05

06

07

09

08

u a

xial

vel

ocity

forx

=085

timet

=1000

Re = 800

Re = 600

Re = 400

(a)

0

01

02

03

04

05

minus01

minus02

minus03

minus04

minus051 2 3 4 5 6 7 8 9

z

Re = 800

Re = 600

Re = 400

r

adia

l velo

city

at ti

met=10000000

(b)


minus01

minus02

minus03

minus04

minus045

minus035

minus025

minus015

minus005

0

x

Re = 800

Re = 600

Re = 400

r

adia

l vel

ocity

atz=3898305

10 02 04 06 08







x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)









v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(a)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(b)

x

u-velocity

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

06

05

04

03

02

01

0

z

(c)

u-velocity

0

002

004

006

008

01

012

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)









v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










v-velocity

minus2

minus1

0

1

2

3

4

5

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(a)

minus1

1

08

06

04

02

0

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

minus02

minus04

minus08

minus06

(b)

minus3

minus2

minus1

0

1

2

3

4

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(c)

minus1

minus05

0

05

1

v-velocity

x

0 2 4 6 8 10

1

09

08

07

06

05

04

03

02

01

0

z

(d)








5 Conclusions



020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










020

4060

020

40600

005

01

015

02

zx

u a

xial

vel

ocity

(a)0

2040

60

020

4060minus2

minus1

0

1

2

zx

v r

adia

l vel

ocity

(b)

020

4060

020

40600

02

04

06

08

zx

u a

xial

vel

ocity

(c)0

2040

60

020

4060minus1

minus05

0

05

1

15

zx

v r

adia

l vel

ocity

(d)


0 2 4 6 8 10

minus1

0

05

minus05

minus15

1

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

001

002

003

004

005

006

007

008

z

Q w

all s

hear

stre

ss at

t=1000

Re = 800

Re = 600

Re = 400

(b)



0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 100

05

1

15

2

25

3

35

4

45

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 400

(a)

0 2 4 6 8 100

005

01

015

02

025

03

035

04

z

Q v

olum

etric

flow

rate

att=10000000

Re = 800

Re = 600

Re = 300


0 10 20 30 40 50 600

10

20

30

40

50

60

(a)0 10 20 30 40 50 60

0

10

20

30

40

50

60


0 10 20 30 40 50 60minus60

minus40

minus20

0

20

40




0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

01

02

03

04

05

06

07

08

x

Axi

al v

eloc

ity u

Re = 800

Re = 600

Re = 300






References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Date post:	07-Mar-2021
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Research Article Numerical Simulation of Nonlinear Pulsatile Newtonian Blood Flow ... · 2019. 7....

Documents