Hindawi Publishing CorporationISRN Biomedical EngineeringVolume 2013 Article ID 925876 10 pageshttpdxdoiorg1011552013925876

Research ArticleSlip Effects on Pulsatile Flow of Blood througha Stenosed Arterial Segment under Periodic Body Acceleration

A Sinha G C Shit and P K Kundu

Department of Mathematics Jadavpur University Kolkata 700032 India

Correspondence should be addressed to G C Shit gopal iitkgpyahoocoin

Received 17 June 2013 Accepted 7 July 2013

Academic Editors A Cappozzo and D S Naidu

Copyright copy 2013 A Sinha et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A theoretical investigation concerning the influence of externally imposed periodic body acceleration on the flow of blood througha time-dependent stenosed arterial segment by taking into account the slip velocity at the wall of the artery has been carried out Amathematical model is developed by treating blood as a non-Newtonian fluid obeying the Casson fluid model The pulsatile flowis analyzed by considering a periodic pressure gradient and the inertial effects as negligibly small A suitable generalized geometryfor time-dependent stenosis is taken into account Perturbation method is used to solve the coupled implicit system of nonlineardifferential equations that govern the flow of blood Analytical expressions for the velocity profile volumetric flow rate and wallshear stress are obtained A thorough quantitative analysis has beenmade through numerical computations of the variables involvedin the analysis that are of special interest in this study The computational results are presented graphically The results for differentvalues of the parameters involved in the problem under consideration presented here show that the flow is appreciably influencedby slip velocity in the presence of periodic body acceleration

1 Introduction

There are number of evidences available in the scientificliteratures that vascular fluid dynamics plays a major role inthe development and progression of arterial diseases Localnarrowing in the lumen of an arterial segment is commonlyreferred to as stenosis This occurs due to deposition of vari-ous substances like cholesterol on the endothelium of arterialwall When an obstruction is developed in an artery one ofthemost serious consequences is the increased resistance andthe associated reduction of the blood flow to the particularvascular bed supplied by the artery

Thus the presence of a stenosis leads to stroke heartattack and serious circulatory disorders Different studies onthe flow of blood through arterial segments with obstructionhave been carried out experimentally and theoretically byseveral investigators [1ndash7] The assumption of Newtonianbehavior of blood is acceptable for high shear rate flowthrough larger arteries [4] But blood being a suspensionof cells in plasma exhibits non-Newtonian behavior at lowshear rate ( 120574 lt 10119904) in small diameter arteries (002ndash01mm) [8] Several studies were performed to analyze the

steady flow of blood treating it as a Newtonian fluid [9 10]It is well known that blood flow in the human circulatorysystem is caused by the pumping action of the heart whichin turn produces a pulsatile pressure gradient throughout thesystem [11] Human heart is a muscular pump and due tocontraction and expansion of heart muscles there producesa pressure difference in its systolic and diastolic conditionspopularly known as pressure pulse which physicians check atthe wrist The cyclic nature of heart pump creates pulsatileconditions in all arteries The ejects and fills with blood inalternating cycles are called systolic and diastolic Blood ispumped out of the heart during systolic whereas the heartrests during diastole and no blood is ejected Pressure andflow rate are characteristic in pulsatile shapes that vary indifferent parts of the arterial systemThus several researchershave studied pulsatile flowof blood treating it as aNewtonianfluid [12ndash14] Long et al [8] numerically investigated thepulsatile flow behaviour of blood in the poststenotic regionby considering inlet diameter as 8mm prestenotic length48mm poststenosis domain 180mm and stenosis length1607mm Clark [1] performed the experimental studies ofthe pulsatile flow in a model of aortic stenosis taking the

2 ISRN Biomedical Engineering

Reynolds number 740 and 2000 Nagarani and Sarojamma[15] developed a mathematical model of pulsatile flow ofCasson fluid for blood flow through stenosed narrow arteriesThey used perturbation technique to solve their problemSiddiqui et al [16] mathematically analyzed the flow of bloodthrough narrow arteries by considering Herschel-Bulkleyfluid model as well as Casson fluid model

In our daily life we often face some external bodyacceleration such as traveling in high velocity vehicles andaircrafts In various sports during the performance a highaccelerationvibration suddenly takes place These types ofsituations undoubtedly affect the normal flow of blood whichlead to headache vomiting tendency loss of vision abnor-mality in pulse rate and so forth Therefore it is necessary tomaintain such type of body accelerations to avoid these typesof health hazards Due to physiological importance of bodyacceleration many theoretical investigations are developedfor the flow of blood under the influence of body accelerationwith and with out stenosis Sud and Sekhon [17] made ananalysis on blood flow under the time-dependent acceler-ation They pointed out that the high blood velocity andhigh shear rate are capable of harming the circulation whichis produced under the influence of such time-dependentacceleration Sud and Sekhon [18] also analyzed the bloodflow through a model of the human arterial system underthe influence of periodic accelerationThey observed that thebody acceleration has an enhancing effect on the flow rateChaturani and Palanisamy [19] investigated the pulsatile flowof blood under the influence of periodic body accelerationby treating blood as a Power-law fluid Majhi and Nair [20]studied the pulsatile flow of blood under the influence ofbody acceleration by assuming blood as a third grade fluidShit and Roy [21] examined the effect of externally imposedbody acceleration and magnetic field on pulsatile flow ofblood through an arterial segment having stenosis with no-slip velocity condition

Misra et al [22] conducted a theoretical study concerningblood flow through a stenosed arterial segment wherein theyconsidered no-slip condition at the vessel wall There arehowever numerous situations where there may be a partialslip between the fluid and the boundary For many fluids themotion of the particulate fluid is still governed by the Navier-Stokes equations but the usual no-slip condition at theboundary should be replaced by the slip condition [2] Misraand Shit [23] carried out the role of slip velocity in blood flowthrough stenosed arteries Several authors [24 25] suggestedthe presence of a red blood cell occurring in slip condition atthe vessel wall Recently Ponalagusamy [26] and Biswas andChakraborty [27 28] have developed mathematical modelsfor blood flow through stenosed arterial segment by taking avelocity slip condition at the constricted wall Thus it seemsthat consideration of a velocity slip at the stenosed vessel wallwill be quite rational in blood flow modeling However theeffect of body acceleration on pulsatile flow of blood throughan arterial segment having time-dependent stenosis in thepresence velocity-slip has not been considered so far to thebest of our knowledge

Themotivation of this paper is to study the unsteady flowof blood through an arterial segment with time-dependent

r998400

d998400

r9984000

R998400(z

998400)

R0

z998400

120576998400

l9984000

u998400= 120573

998400 120575u998400

120575r998400

o

Figure 1 Schematic diagram of the model geometry

stenosis in the presence of velocity slip The analysis iscarried out by employing appropriate analytical methods andsome important predictions have been made on the basisof the present study In order to illustrate the applicabilityof the theoretical analysis the derived analytical expressionshave been computed for a specific situation with an aim toobserve the variation of some quantities of special interestThe computed values are reported graphically

2 Formulation of the Problem

Let us consider an axially symmetric incompressible laminarpulsatile and fully developed flow of blood through anarterial segment with time-dependent stenosis as shown inFigure 1 Although the modelling of blood flow in arteriesmay appear to be more intuitive by considering tube flow itis worthwhile mentioning that the flow in a tube resemblesthe flow behavior in a channel in many situations [29 30]With this rationale the present problem is formulated byconsidering flow through a channel Body acceleration andslip effect are taken into account for the present problemBlood flowing in arteries is considered here as a suspensionof erythrocytes (red blood cells) in plasma It is assumedthat the fluid is uniformly dense throughout Here blood isrepresented by Casson fluid model The length of the arteryis assumed to be large enough as compared to its radius sothat at the entrance and exit sections special wall effects canbe neglected It has been reported that the radial velocity isnegligibly small for a low Reynolds number flow in a narrowartery with stenosis [31]

With the above considerations the equations that governthe flow of blood may be put in the form

1205881205971199061015840

1205971199051015840= minus

1205971199011015840

1205971199111015840minus

1

1199031015840120597

1205971199031015840(11990310158401205911015840) + 1198661015840(1199051015840) (1)

0 =1205971199011015840

1205971199031015840 (2)

where 1199061015840 is the axial component of blood velocity 1199011015840 is thepressure 120588 is the density of blood 1205911015840 is the shear stress and

ISRN Biomedical Engineering 3

1198661015840(1199051015840) is the body acceleration The constitutive equation ofCasson fluid (which represents the blood) is given by

minus1205971199061015840

1205971199031015840=

1

120583(120591101584012

minus 12059112

119910)2

1199031015840

0le 1199031015840le 1198771015840 (3)

1205971199061015840

1205971199031015840= 0 0 le 119903

1015840le 1199031015840

0 (4)

where 120591119910is the yield stress and 120583 is the coefficient of viscosity

Let us consider a generalized geometry of time-depend-ent single stenosis (cf Figure 1) as

1198771015840(1199111015840 1199051015840) = 119877

0

[[[

[

1 minus1205981015840 (1 minus 119890minus119905

10158401198791015840

)

119877011989710158401198990

119899119899(119899minus1)

119899 minus 1

times (1198971015840119899minus1

0(1199111015840minus 1198891015840) minus (119911

1015840minus 1198891015840)119899

)]]]

]

1198891015840le 1199111015840le 1198971015840

0+ 1198891015840

(5)

in which 1198771015840(1199111015840 1199051015840) is the radius of the arterial segment in thestenotic region at an axial distance 1199111015840 at a time 1199051015840 119877

0is the

radius of a normal portion of the artery 11989710158400is the length of

the stenosis 1198891015840 indicates the location 119899 is a parameter thatdetermines the shape of the stenosis and 1205981015840 is the maximumheight of the stenosis located at

1199111015840= 1198891015840+

11989710158400

(1198991(119899minus1)) (6)

The ratio of the height of the stenosis to the radius of thenormal portion of the artery is considered to be much lessthan unity

The boundary conditions for the present problemmay beput mathematically in the form

1199061015840= 1205731015840 1205971199061015840

1205971199031015840at 1199031015840= 1198771015840(1199111015840)

1205911015840 is finite at 119903

1015840= 0

(7)

where 1205731015840 denotes the slip length

The periodic body acceleration in axial direction is givenby

1198661015840(1199051015840) = 1198860cos (1205961015840

11199051015840+ 120601) (8)

where 1198860is the amplitude 1205961015840

1= 21205871198911015840

1 11989110158401is the frequency

in Hz is assumed to be small so that the wave effect can beneglected and 120601 is the lead angle of 119866

1015840(1199051015840)with respect to theheart action

Since the pressure gradient is a function of 1199111015840 and 1199051015840 wecan take

minus1205971199011015840 (1199111015840 1199051015840)

1205971199111015840= 1198600+ 1198601cos (1205961015840

21199051015840) (9)

where1198600and119860

1 respectively are the steady component and

the amplitude of the fluctuating component of the pressuregradient and 120596

1015840

2= 21205871198911015840

2 11989110158402is the pulse frequency in Hz

Let us introduce the following nondimensional variables

119911 =1199111015840

1198770

119877 =1198771015840

1198770

119903 =1199031015840

1198770

119889 =1198891015840

1198770

1198970=

11989710158400

1198770

1199030=

11990310158400

1198770

120598 =1205981015840

1198770

120591 =21205911015840

11986001198770

120579 =2120591119910

11986001198770

120596 =12059610158401

12059610158402

119905 = 1205961015840

21199051015840 119879 = 120596

1015840

21198791015840 119861 =

1198860

1198600

119890 =1198601

1198600

119906 =41199061015840120583

119860011987720

1205722=

1198772012059610158402120588

120583 120573 =

1205731015840

1198770

(10)

Using the nondimensional variables defined in (10) (1) (3)and (4) respectively reduces to

1205722 120597119906

120597119905= 4 (1 + 119890 cos 119905) + 4119861 cos (120596119905 + 120601) minus

2

119903

120597 (119903120591)

120597119903 (11)

120597119906

120597119903= 0 0 le 119903 le 119903

0 (12)

12059112

= 12057912

+1

radic2(minus

120597119906

120597119903)

12

1199030le 119903 le 119877 (13)

Similarly the boundary conditions (7) are also transformedto

119906 = 120573120597119906

120597119903at 119903 = 119877 (119911) (14)

120591 is finite at 119903 = 0 (15)

where

119877 (119911 119905) = 1 minus120598 (1 minus 119890

minus119905119879)

1198971198990

119899119899(119899minus1)

119899 minus 1

times (119897119899minus1

0(119911 minus 119889) minus (119911 minus 119889)

119899) 119889 le 119911 le 119897

0+ 119889

(16)

3 Analytical Solution

Considering the Womersley parameter to be small (1205722 ≪ 1)the axial velocity component119906 shear stress 120591 plug core radius1199030 and plug core velocity 119906

119901are expressed in the following

form

119906 = 1199060+ 12057221199061+ sdot sdot sdot (17)

120591 = 1205910+ 12057221205911+ sdot sdot sdot (18)

1199030= 11990300

+ 120572211990310

+ sdot sdot sdot (19)

119906119901= 1199060119901

+ 12057221199061119901

+ sdot sdot sdot (20)

4 ISRN Biomedical Engineering

Using (17) and (18) in (11) we get

120597 (1199031205910)

120597119903= 2119903119892 (119905) (21)

1205971199060

120597119905= minus

2

119903

120597 (1199031205911)

120597119903 (22)

where 119892(119905) = (1 + 119890 cos 119905) + 119861 cos(120596119905 + 120601)Integrating (21) and using the boundary condition (15)

we obtain

1205910= 1199032119892 (119905) (23)

Using (17) and (18) (12) can be written as

minus1205971199060

120597119903= 2 (120579 + 120591

0minus 2radic120579120591

0) (24)

minus1205971199061

120597119903= 21205911(1 minus radic

120579

1205910

) (25)

Substituting (17) in (14) we obtain

1199060= 120573

1205971199060

120597119903at 119903 = 119877 (26)

1199061= 120573

1205971199061

120597119903at 119903 = 119877 (27)

Using (23) and the boundary condition (26) the solution of(24) yields

1199060= 119892 (119877

2minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(28)

Plug velocity can be obtained from (28) by putting 119903 = 11990300as

1199060119901

= 119892 (1198772minus 1199032

00) minus

8radic120579119892

3(11987732

minus 11990332

00)

+ 2120579 (119877 minus 11990300) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(29)

Using the relation (28) and boundary condition (15) in (22)we obtain the expression for 120591

1as

1205911=

11989210158401198773

8[2 (

119903

119877) minus (

119903

119877)3

minus3

8radic

120579

119892119877

times(7 (119903

119877) minus 4(

119903

119877)52

)]

minus1198921015840120573119877

2(1 minus radic

120579

119892119877) 119903

(30)

Using (23) and (30) and the boundary condition (27) from(25) we obtain the expression for 119906

1as

1199061=

11989210158401198774

8[(

119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(31)

The expression for 1199061119901can be obtained from (31) when 119903 = 119903

10

as

1199061119901

=11989210158401198774

8[(

11990310

119877)4

minus 4(11990310

119877)2

+ 3 + radic120579

119892119877

times (16

3(11990310

119877)2

minus424

127(11990310

119877)72

+16

3(11990310

119877)32

minus1144

147) +

120579

119892119877

times (128

63(11990310

119877)3

minus64

9(11990310

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032

10) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

10)]

(32)

ISRN Biomedical Engineering 5

The total velocity distribution can be written as

119906 = 119892 (1198772minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579 +120572211989210158401198774

8

times[[

[

(119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]]

]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(33)

The nondimensional volumetric flow rate 119876 is given by

119876 (119905) = 4int119877

0

119906119903 119889119903 = 1198921198774minus

16

7radic

119892120579

1198771198774

+1

31198773120579 +

1

12119877612057221198921015840minus

15

77radic

120579

119892119877119877612057221198921015840

+4120579119877612057221198921015840

35119892119877+ 119892120573119877

3(1 minus radic

120579

119892119877)

2

+ 12057311989210158401198775

times [(radic120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877) minus

120573

119877(1 minus radic

120579

119892119877))

minus1

4(1 + radic

120579

119892119877) +

2

7radic

120579

119892119877(1 minus radic

120579

119892119877)]

(34)

where 119876 = 1198761015840(1205871198600119877402120583) 1198761015840 is the volumetric flow rate

In dimensionless form resistance to the flow is given by

120582 (119905) =1198750minus 119875119871

119876=

119871 (1 + 119890 cos (119905 + 120601))

119876 (35)

where 119875 = 1198750at 119911 = 0 and 119875 = 119875

119871at 119911 = 119871

4 Results and Discussion

With a view to illustrate the applicability of the mathematicalmodel developed and analyzed in the preceding sectionsthe analytical expressions for the axial velocity profile wallshear stress and volumetric flow rate are presented by takinginto account the velocity-slip condition at the arterial wallIn order to get a proper insight into the flow behaviour ofblood through a time-dependent stenosed arterial segmentunder body acceleration the variations of 119906 119876 120591 and 120582

have been estimated and the computed results are presentedin graphical form The flowing blood is modeled as Cassonfluid model The governing equations of the flow are solvedusing perturbation analysis with the assumption that theWomersley frequency parameter is small which is valid forphysiological situations in small blood vessels In our analysisthe value of the shape parameter (119899) of the stenosis is taken tobe 2Themaximumheight of the stenosis is generally taken as03 and only to pronounce its effect we have taken the rangefrom 00 to 06The values of the nondimensional yield stress120579 for the blood of the normal subject are between 001 and003 and in diseased state it is quite high and in such a casethe value of the yield stress is taken to lie between 01 and 04The velocity slip parameter is taken between minus20 and minus05The pressure gradient parameter (119890) is taken in the range 01ndash04 For this problem the value of pulsatile Reynolds numberis taken as 005 To discuss the effects of the body accelerationparameter 119861 on the various flow quantities its value is takenin the range between 00 and 08

Figures 2ndash6 give an idea of the axial velocity distributionin the case of blood flow in the vicinity of the stenotic portionof the arterial segment under the purview of the presentstudyThe presence of velocity slip at the wall alters the bloodvelocity significantly as shown in Figure 2 We observe fromthis figure that blood velocity decreases as the slip lengthincreases It is interesting to note that at 119911 = 10 (ie at thethroat the stenosis) and at a particular time 119905 = 1205874 for largevalues of slip length (120573 = minus10 minus12) axial velocity increaseswith the radial distance whereas the reverse trend is observedfor 120573 = minus05 minus08 Figure 3 represents the axial velocitydistribution at a particular time 119905 = 1205874 with the radialdistance for different values of body acceleration parameter119861It is seen that at the throat of the stenosis (ie 119911 = 10) axialvelocity monotonically decreases with the radial distanceFrom the same figure we observed that the body accelerationparameter 119861 brings quantitative as well as qualitative changesin velocity profiles It reveals that the velocity increases asthe the body acceleration parameter increases This resultsupports the phenomenon that body acceleration reduces theflow resistance and so the velocity of blood flow increaseswith the increase in body acceleration Figure 4 gives thedistribution of the axial velocity at the onset of the stenosis for

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

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2 ISRN Biomedical Engineering

Reynolds number 740 and 2000 Nagarani and Sarojamma[15] developed a mathematical model of pulsatile flow ofCasson fluid for blood flow through stenosed narrow arteriesThey used perturbation technique to solve their problemSiddiqui et al [16] mathematically analyzed the flow of bloodthrough narrow arteries by considering Herschel-Bulkleyfluid model as well as Casson fluid model

In our daily life we often face some external bodyacceleration such as traveling in high velocity vehicles andaircrafts In various sports during the performance a highaccelerationvibration suddenly takes place These types ofsituations undoubtedly affect the normal flow of blood whichlead to headache vomiting tendency loss of vision abnor-mality in pulse rate and so forth Therefore it is necessary tomaintain such type of body accelerations to avoid these typesof health hazards Due to physiological importance of bodyacceleration many theoretical investigations are developedfor the flow of blood under the influence of body accelerationwith and with out stenosis Sud and Sekhon [17] made ananalysis on blood flow under the time-dependent acceler-ation They pointed out that the high blood velocity andhigh shear rate are capable of harming the circulation whichis produced under the influence of such time-dependentacceleration Sud and Sekhon [18] also analyzed the bloodflow through a model of the human arterial system underthe influence of periodic accelerationThey observed that thebody acceleration has an enhancing effect on the flow rateChaturani and Palanisamy [19] investigated the pulsatile flowof blood under the influence of periodic body accelerationby treating blood as a Power-law fluid Majhi and Nair [20]studied the pulsatile flow of blood under the influence ofbody acceleration by assuming blood as a third grade fluidShit and Roy [21] examined the effect of externally imposedbody acceleration and magnetic field on pulsatile flow ofblood through an arterial segment having stenosis with no-slip velocity condition

Misra et al [22] conducted a theoretical study concerningblood flow through a stenosed arterial segment wherein theyconsidered no-slip condition at the vessel wall There arehowever numerous situations where there may be a partialslip between the fluid and the boundary For many fluids themotion of the particulate fluid is still governed by the Navier-Stokes equations but the usual no-slip condition at theboundary should be replaced by the slip condition [2] Misraand Shit [23] carried out the role of slip velocity in blood flowthrough stenosed arteries Several authors [24 25] suggestedthe presence of a red blood cell occurring in slip condition atthe vessel wall Recently Ponalagusamy [26] and Biswas andChakraborty [27 28] have developed mathematical modelsfor blood flow through stenosed arterial segment by taking avelocity slip condition at the constricted wall Thus it seemsthat consideration of a velocity slip at the stenosed vessel wallwill be quite rational in blood flow modeling However theeffect of body acceleration on pulsatile flow of blood throughan arterial segment having time-dependent stenosis in thepresence velocity-slip has not been considered so far to thebest of our knowledge

Themotivation of this paper is to study the unsteady flowof blood through an arterial segment with time-dependent

r998400

d998400

r9984000

R998400(z

998400)

R0

z998400

120576998400

l9984000

u998400= 120573

998400 120575u998400

120575r998400

o

Figure 1 Schematic diagram of the model geometry

stenosis in the presence of velocity slip The analysis iscarried out by employing appropriate analytical methods andsome important predictions have been made on the basisof the present study In order to illustrate the applicabilityof the theoretical analysis the derived analytical expressionshave been computed for a specific situation with an aim toobserve the variation of some quantities of special interestThe computed values are reported graphically

2 Formulation of the Problem

Let us consider an axially symmetric incompressible laminarpulsatile and fully developed flow of blood through anarterial segment with time-dependent stenosis as shown inFigure 1 Although the modelling of blood flow in arteriesmay appear to be more intuitive by considering tube flow itis worthwhile mentioning that the flow in a tube resemblesthe flow behavior in a channel in many situations [29 30]With this rationale the present problem is formulated byconsidering flow through a channel Body acceleration andslip effect are taken into account for the present problemBlood flowing in arteries is considered here as a suspensionof erythrocytes (red blood cells) in plasma It is assumedthat the fluid is uniformly dense throughout Here blood isrepresented by Casson fluid model The length of the arteryis assumed to be large enough as compared to its radius sothat at the entrance and exit sections special wall effects canbe neglected It has been reported that the radial velocity isnegligibly small for a low Reynolds number flow in a narrowartery with stenosis [31]

With the above considerations the equations that governthe flow of blood may be put in the form

1205881205971199061015840

1205971199051015840= minus

1205971199011015840

1205971199111015840minus

1

1199031015840120597

1205971199031015840(11990310158401205911015840) + 1198661015840(1199051015840) (1)

0 =1205971199011015840

1205971199031015840 (2)

where 1199061015840 is the axial component of blood velocity 1199011015840 is thepressure 120588 is the density of blood 1205911015840 is the shear stress and

ISRN Biomedical Engineering 3

1198661015840(1199051015840) is the body acceleration The constitutive equation ofCasson fluid (which represents the blood) is given by

minus1205971199061015840

1205971199031015840=

1

120583(120591101584012

minus 12059112

119910)2

1199031015840

0le 1199031015840le 1198771015840 (3)

1205971199061015840

1205971199031015840= 0 0 le 119903

1015840le 1199031015840

0 (4)

where 120591119910is the yield stress and 120583 is the coefficient of viscosity

Let us consider a generalized geometry of time-depend-ent single stenosis (cf Figure 1) as

1198771015840(1199111015840 1199051015840) = 119877

0

[[[

[

1 minus1205981015840 (1 minus 119890minus119905

10158401198791015840

)

119877011989710158401198990

119899119899(119899minus1)

119899 minus 1

times (1198971015840119899minus1

0(1199111015840minus 1198891015840) minus (119911

1015840minus 1198891015840)119899

)]]]

]

1198891015840le 1199111015840le 1198971015840

0+ 1198891015840

(5)

in which 1198771015840(1199111015840 1199051015840) is the radius of the arterial segment in thestenotic region at an axial distance 1199111015840 at a time 1199051015840 119877

0is the

radius of a normal portion of the artery 11989710158400is the length of

the stenosis 1198891015840 indicates the location 119899 is a parameter thatdetermines the shape of the stenosis and 1205981015840 is the maximumheight of the stenosis located at

1199111015840= 1198891015840+

11989710158400

(1198991(119899minus1)) (6)

The ratio of the height of the stenosis to the radius of thenormal portion of the artery is considered to be much lessthan unity

The boundary conditions for the present problemmay beput mathematically in the form

1199061015840= 1205731015840 1205971199061015840

1205971199031015840at 1199031015840= 1198771015840(1199111015840)

1205911015840 is finite at 119903

1015840= 0

(7)

where 1205731015840 denotes the slip length

The periodic body acceleration in axial direction is givenby

1198661015840(1199051015840) = 1198860cos (1205961015840

11199051015840+ 120601) (8)

where 1198860is the amplitude 1205961015840

1= 21205871198911015840

1 11989110158401is the frequency

in Hz is assumed to be small so that the wave effect can beneglected and 120601 is the lead angle of 119866

1015840(1199051015840)with respect to theheart action

Since the pressure gradient is a function of 1199111015840 and 1199051015840 wecan take

minus1205971199011015840 (1199111015840 1199051015840)

1205971199111015840= 1198600+ 1198601cos (1205961015840

21199051015840) (9)

where1198600and119860

1 respectively are the steady component and

the amplitude of the fluctuating component of the pressuregradient and 120596

1015840

2= 21205871198911015840

2 11989110158402is the pulse frequency in Hz

Let us introduce the following nondimensional variables

119911 =1199111015840

1198770

119877 =1198771015840

1198770

119903 =1199031015840

1198770

119889 =1198891015840

1198770

1198970=

11989710158400

1198770

1199030=

11990310158400

1198770

120598 =1205981015840

1198770

120591 =21205911015840

11986001198770

120579 =2120591119910

11986001198770

120596 =12059610158401

12059610158402

119905 = 1205961015840

21199051015840 119879 = 120596

1015840

21198791015840 119861 =

1198860

1198600

119890 =1198601

1198600

119906 =41199061015840120583

119860011987720

1205722=

1198772012059610158402120588

120583 120573 =

1205731015840

1198770

(10)

Using the nondimensional variables defined in (10) (1) (3)and (4) respectively reduces to

1205722 120597119906

120597119905= 4 (1 + 119890 cos 119905) + 4119861 cos (120596119905 + 120601) minus

2

119903

120597 (119903120591)

120597119903 (11)

120597119906

120597119903= 0 0 le 119903 le 119903

0 (12)

12059112

= 12057912

+1

radic2(minus

120597119906

120597119903)

12

1199030le 119903 le 119877 (13)

Similarly the boundary conditions (7) are also transformedto

119906 = 120573120597119906

120597119903at 119903 = 119877 (119911) (14)

120591 is finite at 119903 = 0 (15)

where

119877 (119911 119905) = 1 minus120598 (1 minus 119890

minus119905119879)

1198971198990

119899119899(119899minus1)

119899 minus 1

times (119897119899minus1

0(119911 minus 119889) minus (119911 minus 119889)

119899) 119889 le 119911 le 119897

0+ 119889

(16)

3 Analytical Solution

Considering the Womersley parameter to be small (1205722 ≪ 1)the axial velocity component119906 shear stress 120591 plug core radius1199030 and plug core velocity 119906

119901are expressed in the following

form

119906 = 1199060+ 12057221199061+ sdot sdot sdot (17)

120591 = 1205910+ 12057221205911+ sdot sdot sdot (18)

1199030= 11990300

+ 120572211990310

+ sdot sdot sdot (19)

119906119901= 1199060119901

+ 12057221199061119901

+ sdot sdot sdot (20)

4 ISRN Biomedical Engineering

Using (17) and (18) in (11) we get

120597 (1199031205910)

120597119903= 2119903119892 (119905) (21)

1205971199060

120597119905= minus

2

119903

120597 (1199031205911)

120597119903 (22)

where 119892(119905) = (1 + 119890 cos 119905) + 119861 cos(120596119905 + 120601)Integrating (21) and using the boundary condition (15)

we obtain

1205910= 1199032119892 (119905) (23)

Using (17) and (18) (12) can be written as

minus1205971199060

120597119903= 2 (120579 + 120591

0minus 2radic120579120591

0) (24)

minus1205971199061

120597119903= 21205911(1 minus radic

120579

1205910

) (25)

Substituting (17) in (14) we obtain

1199060= 120573

1205971199060

120597119903at 119903 = 119877 (26)

1199061= 120573

1205971199061

120597119903at 119903 = 119877 (27)

Using (23) and the boundary condition (26) the solution of(24) yields

1199060= 119892 (119877

2minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(28)

Plug velocity can be obtained from (28) by putting 119903 = 11990300as

1199060119901

= 119892 (1198772minus 1199032

00) minus

8radic120579119892

3(11987732

minus 11990332

00)

+ 2120579 (119877 minus 11990300) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(29)

Using the relation (28) and boundary condition (15) in (22)we obtain the expression for 120591

1as

1205911=

11989210158401198773

8[2 (

119903

119877) minus (

119903

119877)3

minus3

8radic

120579

119892119877

times(7 (119903

119877) minus 4(

119903

119877)52

)]

minus1198921015840120573119877

2(1 minus radic

120579

119892119877) 119903

(30)

Using (23) and (30) and the boundary condition (27) from(25) we obtain the expression for 119906

1as

1199061=

11989210158401198774

8[(

119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(31)

The expression for 1199061119901can be obtained from (31) when 119903 = 119903

10

as

1199061119901

=11989210158401198774

8[(

11990310

119877)4

minus 4(11990310

119877)2

+ 3 + radic120579

119892119877

times (16

3(11990310

119877)2

minus424

127(11990310

119877)72

+16

3(11990310

119877)32

minus1144

147) +

120579

119892119877

times (128

63(11990310

119877)3

minus64

9(11990310

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032

10) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

10)]

(32)

ISRN Biomedical Engineering 5

The total velocity distribution can be written as

119906 = 119892 (1198772minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579 +120572211989210158401198774

8

times[[

[

(119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]]

]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(33)

The nondimensional volumetric flow rate 119876 is given by

119876 (119905) = 4int119877

0

119906119903 119889119903 = 1198921198774minus

16

7radic

119892120579

1198771198774

+1

31198773120579 +

1

12119877612057221198921015840minus

15

77radic

120579

119892119877119877612057221198921015840

+4120579119877612057221198921015840

35119892119877+ 119892120573119877

3(1 minus radic

120579

119892119877)

2

+ 12057311989210158401198775

times [(radic120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877) minus

120573

119877(1 minus radic

120579

119892119877))

minus1

4(1 + radic

120579

119892119877) +

2

7radic

120579

119892119877(1 minus radic

120579

119892119877)]

(34)

where 119876 = 1198761015840(1205871198600119877402120583) 1198761015840 is the volumetric flow rate

In dimensionless form resistance to the flow is given by

120582 (119905) =1198750minus 119875119871

119876=

119871 (1 + 119890 cos (119905 + 120601))

119876 (35)

where 119875 = 1198750at 119911 = 0 and 119875 = 119875

119871at 119911 = 119871

4 Results and Discussion

With a view to illustrate the applicability of the mathematicalmodel developed and analyzed in the preceding sectionsthe analytical expressions for the axial velocity profile wallshear stress and volumetric flow rate are presented by takinginto account the velocity-slip condition at the arterial wallIn order to get a proper insight into the flow behaviour ofblood through a time-dependent stenosed arterial segmentunder body acceleration the variations of 119906 119876 120591 and 120582

have been estimated and the computed results are presentedin graphical form The flowing blood is modeled as Cassonfluid model The governing equations of the flow are solvedusing perturbation analysis with the assumption that theWomersley frequency parameter is small which is valid forphysiological situations in small blood vessels In our analysisthe value of the shape parameter (119899) of the stenosis is taken tobe 2Themaximumheight of the stenosis is generally taken as03 and only to pronounce its effect we have taken the rangefrom 00 to 06The values of the nondimensional yield stress120579 for the blood of the normal subject are between 001 and003 and in diseased state it is quite high and in such a casethe value of the yield stress is taken to lie between 01 and 04The velocity slip parameter is taken between minus20 and minus05The pressure gradient parameter (119890) is taken in the range 01ndash04 For this problem the value of pulsatile Reynolds numberis taken as 005 To discuss the effects of the body accelerationparameter 119861 on the various flow quantities its value is takenin the range between 00 and 08

Figures 2ndash6 give an idea of the axial velocity distributionin the case of blood flow in the vicinity of the stenotic portionof the arterial segment under the purview of the presentstudyThe presence of velocity slip at the wall alters the bloodvelocity significantly as shown in Figure 2 We observe fromthis figure that blood velocity decreases as the slip lengthincreases It is interesting to note that at 119911 = 10 (ie at thethroat the stenosis) and at a particular time 119905 = 1205874 for largevalues of slip length (120573 = minus10 minus12) axial velocity increaseswith the radial distance whereas the reverse trend is observedfor 120573 = minus05 minus08 Figure 3 represents the axial velocitydistribution at a particular time 119905 = 1205874 with the radialdistance for different values of body acceleration parameter119861It is seen that at the throat of the stenosis (ie 119911 = 10) axialvelocity monotonically decreases with the radial distanceFrom the same figure we observed that the body accelerationparameter 119861 brings quantitative as well as qualitative changesin velocity profiles It reveals that the velocity increases asthe the body acceleration parameter increases This resultsupports the phenomenon that body acceleration reduces theflow resistance and so the velocity of blood flow increaseswith the increase in body acceleration Figure 4 gives thedistribution of the axial velocity at the onset of the stenosis for

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

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International Journal of

ISRN Biomedical Engineering 3

1198661015840(1199051015840) is the body acceleration The constitutive equation ofCasson fluid (which represents the blood) is given by

minus1205971199061015840

1205971199031015840=

1

120583(120591101584012

minus 12059112

119910)2

1199031015840

0le 1199031015840le 1198771015840 (3)

1205971199061015840

1205971199031015840= 0 0 le 119903

1015840le 1199031015840

0 (4)

where 120591119910is the yield stress and 120583 is the coefficient of viscosity

Let us consider a generalized geometry of time-depend-ent single stenosis (cf Figure 1) as

1198771015840(1199111015840 1199051015840) = 119877

0

[[[

[

1 minus1205981015840 (1 minus 119890minus119905

10158401198791015840

)

119877011989710158401198990

119899119899(119899minus1)

119899 minus 1

times (1198971015840119899minus1

0(1199111015840minus 1198891015840) minus (119911

1015840minus 1198891015840)119899

)]]]

]

1198891015840le 1199111015840le 1198971015840

0+ 1198891015840

(5)

in which 1198771015840(1199111015840 1199051015840) is the radius of the arterial segment in thestenotic region at an axial distance 1199111015840 at a time 1199051015840 119877

0is the

radius of a normal portion of the artery 11989710158400is the length of

the stenosis 1198891015840 indicates the location 119899 is a parameter thatdetermines the shape of the stenosis and 1205981015840 is the maximumheight of the stenosis located at

1199111015840= 1198891015840+

11989710158400

(1198991(119899minus1)) (6)

The ratio of the height of the stenosis to the radius of thenormal portion of the artery is considered to be much lessthan unity

The boundary conditions for the present problemmay beput mathematically in the form

1199061015840= 1205731015840 1205971199061015840

1205971199031015840at 1199031015840= 1198771015840(1199111015840)

1205911015840 is finite at 119903

1015840= 0

(7)

where 1205731015840 denotes the slip length

The periodic body acceleration in axial direction is givenby

1198661015840(1199051015840) = 1198860cos (1205961015840

11199051015840+ 120601) (8)

where 1198860is the amplitude 1205961015840

1= 21205871198911015840

1 11989110158401is the frequency

in Hz is assumed to be small so that the wave effect can beneglected and 120601 is the lead angle of 119866

1015840(1199051015840)with respect to theheart action

Since the pressure gradient is a function of 1199111015840 and 1199051015840 wecan take

minus1205971199011015840 (1199111015840 1199051015840)

1205971199111015840= 1198600+ 1198601cos (1205961015840

21199051015840) (9)

where1198600and119860

1 respectively are the steady component and

the amplitude of the fluctuating component of the pressuregradient and 120596

1015840

2= 21205871198911015840

2 11989110158402is the pulse frequency in Hz

Let us introduce the following nondimensional variables

119911 =1199111015840

1198770

119877 =1198771015840

1198770

119903 =1199031015840

1198770

119889 =1198891015840

1198770

1198970=

11989710158400

1198770

1199030=

11990310158400

1198770

120598 =1205981015840

1198770

120591 =21205911015840

11986001198770

120579 =2120591119910

11986001198770

120596 =12059610158401

12059610158402

119905 = 1205961015840

21199051015840 119879 = 120596

1015840

21198791015840 119861 =

1198860

1198600

119890 =1198601

1198600

119906 =41199061015840120583

119860011987720

1205722=

1198772012059610158402120588

120583 120573 =

1205731015840

1198770

(10)

Using the nondimensional variables defined in (10) (1) (3)and (4) respectively reduces to

1205722 120597119906

120597119905= 4 (1 + 119890 cos 119905) + 4119861 cos (120596119905 + 120601) minus

2

119903

120597 (119903120591)

120597119903 (11)

120597119906

120597119903= 0 0 le 119903 le 119903

0 (12)

12059112

= 12057912

+1

radic2(minus

120597119906

120597119903)

12

1199030le 119903 le 119877 (13)

Similarly the boundary conditions (7) are also transformedto

119906 = 120573120597119906

120597119903at 119903 = 119877 (119911) (14)

120591 is finite at 119903 = 0 (15)

where

119877 (119911 119905) = 1 minus120598 (1 minus 119890

minus119905119879)

1198971198990

119899119899(119899minus1)

119899 minus 1

times (119897119899minus1

0(119911 minus 119889) minus (119911 minus 119889)

119899) 119889 le 119911 le 119897

0+ 119889

(16)

3 Analytical Solution

Considering the Womersley parameter to be small (1205722 ≪ 1)the axial velocity component119906 shear stress 120591 plug core radius1199030 and plug core velocity 119906

119901are expressed in the following

form

119906 = 1199060+ 12057221199061+ sdot sdot sdot (17)

120591 = 1205910+ 12057221205911+ sdot sdot sdot (18)

1199030= 11990300

+ 120572211990310

+ sdot sdot sdot (19)

119906119901= 1199060119901

+ 12057221199061119901

+ sdot sdot sdot (20)

4 ISRN Biomedical Engineering

Using (17) and (18) in (11) we get

120597 (1199031205910)

120597119903= 2119903119892 (119905) (21)

1205971199060

120597119905= minus

2

119903

120597 (1199031205911)

120597119903 (22)

where 119892(119905) = (1 + 119890 cos 119905) + 119861 cos(120596119905 + 120601)Integrating (21) and using the boundary condition (15)

we obtain

1205910= 1199032119892 (119905) (23)

Using (17) and (18) (12) can be written as

minus1205971199060

120597119903= 2 (120579 + 120591

0minus 2radic120579120591

0) (24)

minus1205971199061

120597119903= 21205911(1 minus radic

120579

1205910

) (25)

Substituting (17) in (14) we obtain

1199060= 120573

1205971199060

120597119903at 119903 = 119877 (26)

1199061= 120573

1205971199061

120597119903at 119903 = 119877 (27)

Using (23) and the boundary condition (26) the solution of(24) yields

1199060= 119892 (119877

2minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(28)

Plug velocity can be obtained from (28) by putting 119903 = 11990300as

1199060119901

= 119892 (1198772minus 1199032

00) minus

8radic120579119892

3(11987732

minus 11990332

00)

+ 2120579 (119877 minus 11990300) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(29)

Using the relation (28) and boundary condition (15) in (22)we obtain the expression for 120591

1as

1205911=

11989210158401198773

8[2 (

119903

119877) minus (

119903

119877)3

minus3

8radic

120579

119892119877

times(7 (119903

119877) minus 4(

119903

119877)52

)]

minus1198921015840120573119877

2(1 minus radic

120579

119892119877) 119903

(30)

Using (23) and (30) and the boundary condition (27) from(25) we obtain the expression for 119906

1as

1199061=

11989210158401198774

8[(

119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(31)

The expression for 1199061119901can be obtained from (31) when 119903 = 119903

10

as

1199061119901

=11989210158401198774

8[(

11990310

119877)4

minus 4(11990310

119877)2

+ 3 + radic120579

119892119877

times (16

3(11990310

119877)2

minus424

127(11990310

119877)72

+16

3(11990310

119877)32

minus1144

147) +

120579

119892119877

times (128

63(11990310

119877)3

minus64

9(11990310

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032

10) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

10)]

(32)

ISRN Biomedical Engineering 5

The total velocity distribution can be written as

119906 = 119892 (1198772minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579 +120572211989210158401198774

8

times[[

[

(119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]]

]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(33)

The nondimensional volumetric flow rate 119876 is given by

119876 (119905) = 4int119877

0

119906119903 119889119903 = 1198921198774minus

16

7radic

119892120579

1198771198774

+1

31198773120579 +

1

12119877612057221198921015840minus

15

77radic

120579

119892119877119877612057221198921015840

+4120579119877612057221198921015840

35119892119877+ 119892120573119877

3(1 minus radic

120579

119892119877)

2

+ 12057311989210158401198775

times [(radic120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877) minus

120573

119877(1 minus radic

120579

119892119877))

minus1

4(1 + radic

120579

119892119877) +

2

7radic

120579

119892119877(1 minus radic

120579

119892119877)]

(34)

where 119876 = 1198761015840(1205871198600119877402120583) 1198761015840 is the volumetric flow rate

In dimensionless form resistance to the flow is given by

120582 (119905) =1198750minus 119875119871

119876=

119871 (1 + 119890 cos (119905 + 120601))

119876 (35)

where 119875 = 1198750at 119911 = 0 and 119875 = 119875

119871at 119911 = 119871

4 Results and Discussion

With a view to illustrate the applicability of the mathematicalmodel developed and analyzed in the preceding sectionsthe analytical expressions for the axial velocity profile wallshear stress and volumetric flow rate are presented by takinginto account the velocity-slip condition at the arterial wallIn order to get a proper insight into the flow behaviour ofblood through a time-dependent stenosed arterial segmentunder body acceleration the variations of 119906 119876 120591 and 120582

have been estimated and the computed results are presentedin graphical form The flowing blood is modeled as Cassonfluid model The governing equations of the flow are solvedusing perturbation analysis with the assumption that theWomersley frequency parameter is small which is valid forphysiological situations in small blood vessels In our analysisthe value of the shape parameter (119899) of the stenosis is taken tobe 2Themaximumheight of the stenosis is generally taken as03 and only to pronounce its effect we have taken the rangefrom 00 to 06The values of the nondimensional yield stress120579 for the blood of the normal subject are between 001 and003 and in diseased state it is quite high and in such a casethe value of the yield stress is taken to lie between 01 and 04The velocity slip parameter is taken between minus20 and minus05The pressure gradient parameter (119890) is taken in the range 01ndash04 For this problem the value of pulsatile Reynolds numberis taken as 005 To discuss the effects of the body accelerationparameter 119861 on the various flow quantities its value is takenin the range between 00 and 08

Figures 2ndash6 give an idea of the axial velocity distributionin the case of blood flow in the vicinity of the stenotic portionof the arterial segment under the purview of the presentstudyThe presence of velocity slip at the wall alters the bloodvelocity significantly as shown in Figure 2 We observe fromthis figure that blood velocity decreases as the slip lengthincreases It is interesting to note that at 119911 = 10 (ie at thethroat the stenosis) and at a particular time 119905 = 1205874 for largevalues of slip length (120573 = minus10 minus12) axial velocity increaseswith the radial distance whereas the reverse trend is observedfor 120573 = minus05 minus08 Figure 3 represents the axial velocitydistribution at a particular time 119905 = 1205874 with the radialdistance for different values of body acceleration parameter119861It is seen that at the throat of the stenosis (ie 119911 = 10) axialvelocity monotonically decreases with the radial distanceFrom the same figure we observed that the body accelerationparameter 119861 brings quantitative as well as qualitative changesin velocity profiles It reveals that the velocity increases asthe the body acceleration parameter increases This resultsupports the phenomenon that body acceleration reduces theflow resistance and so the velocity of blood flow increaseswith the increase in body acceleration Figure 4 gives thedistribution of the axial velocity at the onset of the stenosis for

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

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International Journal of

4 ISRN Biomedical Engineering

Using (17) and (18) in (11) we get

120597 (1199031205910)

120597119903= 2119903119892 (119905) (21)

1205971199060

120597119905= minus

2

119903

120597 (1199031205911)

120597119903 (22)

where 119892(119905) = (1 + 119890 cos 119905) + 119861 cos(120596119905 + 120601)Integrating (21) and using the boundary condition (15)

we obtain

1205910= 1199032119892 (119905) (23)

Using (17) and (18) (12) can be written as

minus1205971199060

120597119903= 2 (120579 + 120591

0minus 2radic120579120591

0) (24)

minus1205971199061

120597119903= 21205911(1 minus radic

120579

1205910

) (25)

Substituting (17) in (14) we obtain

1199060= 120573

1205971199060

120597119903at 119903 = 119877 (26)

1199061= 120573

1205971199061

120597119903at 119903 = 119877 (27)

Using (23) and the boundary condition (26) the solution of(24) yields

1199060= 119892 (119877

2minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(28)

Plug velocity can be obtained from (28) by putting 119903 = 11990300as

1199060119901

= 119892 (1198772minus 1199032

00) minus

8radic120579119892

3(11987732

minus 11990332

00)

+ 2120579 (119877 minus 11990300) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579

(29)

Using the relation (28) and boundary condition (15) in (22)we obtain the expression for 120591

1as

1205911=

11989210158401198773

8[2 (

119903

119877) minus (

119903

119877)3

minus3

8radic

120579

119892119877

times(7 (119903

119877) minus 4(

119903

119877)52

)]

minus1198921015840120573119877

2(1 minus radic

120579

119892119877) 119903

(30)

Using (23) and (30) and the boundary condition (27) from(25) we obtain the expression for 119906

1as

1199061=

11989210158401198774

8[(

119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(31)

The expression for 1199061119901can be obtained from (31) when 119903 = 119903

10

as

1199061119901

=11989210158401198774

8[(

11990310

119877)4

minus 4(11990310

119877)2

+ 3 + radic120579

119892119877

times (16

3(11990310

119877)2

minus424

127(11990310

119877)72

+16

3(11990310

119877)32

minus1144

147) +

120579

119892119877

times (128

63(11990310

119877)3

minus64

9(11990310

119877)32

+320

63)]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032

10) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

10)]

(32)

ISRN Biomedical Engineering 5

The total velocity distribution can be written as

119906 = 119892 (1198772minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579 +120572211989210158401198774

8

times[[

[

(119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]]

]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(33)

The nondimensional volumetric flow rate 119876 is given by

119876 (119905) = 4int119877

0

119906119903 119889119903 = 1198921198774minus

16

7radic

119892120579

1198771198774

+1

31198773120579 +

1

12119877612057221198921015840minus

15

77radic

120579

119892119877119877612057221198921015840

+4120579119877612057221198921015840

35119892119877+ 119892120573119877

3(1 minus radic

120579

119892119877)

2

+ 12057311989210158401198775

times [(radic120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877) minus

120573

119877(1 minus radic

120579

119892119877))

minus1

4(1 + radic

120579

119892119877) +

2

7radic

120579

119892119877(1 minus radic

120579

119892119877)]

(34)

where 119876 = 1198761015840(1205871198600119877402120583) 1198761015840 is the volumetric flow rate

In dimensionless form resistance to the flow is given by

120582 (119905) =1198750minus 119875119871

119876=

119871 (1 + 119890 cos (119905 + 120601))

119876 (35)

where 119875 = 1198750at 119911 = 0 and 119875 = 119875

119871at 119911 = 119871

4 Results and Discussion

With a view to illustrate the applicability of the mathematicalmodel developed and analyzed in the preceding sectionsthe analytical expressions for the axial velocity profile wallshear stress and volumetric flow rate are presented by takinginto account the velocity-slip condition at the arterial wallIn order to get a proper insight into the flow behaviour ofblood through a time-dependent stenosed arterial segmentunder body acceleration the variations of 119906 119876 120591 and 120582

have been estimated and the computed results are presentedin graphical form The flowing blood is modeled as Cassonfluid model The governing equations of the flow are solvedusing perturbation analysis with the assumption that theWomersley frequency parameter is small which is valid forphysiological situations in small blood vessels In our analysisthe value of the shape parameter (119899) of the stenosis is taken tobe 2Themaximumheight of the stenosis is generally taken as03 and only to pronounce its effect we have taken the rangefrom 00 to 06The values of the nondimensional yield stress120579 for the blood of the normal subject are between 001 and003 and in diseased state it is quite high and in such a casethe value of the yield stress is taken to lie between 01 and 04The velocity slip parameter is taken between minus20 and minus05The pressure gradient parameter (119890) is taken in the range 01ndash04 For this problem the value of pulsatile Reynolds numberis taken as 005 To discuss the effects of the body accelerationparameter 119861 on the various flow quantities its value is takenin the range between 00 and 08

Figures 2ndash6 give an idea of the axial velocity distributionin the case of blood flow in the vicinity of the stenotic portionof the arterial segment under the purview of the presentstudyThe presence of velocity slip at the wall alters the bloodvelocity significantly as shown in Figure 2 We observe fromthis figure that blood velocity decreases as the slip lengthincreases It is interesting to note that at 119911 = 10 (ie at thethroat the stenosis) and at a particular time 119905 = 1205874 for largevalues of slip length (120573 = minus10 minus12) axial velocity increaseswith the radial distance whereas the reverse trend is observedfor 120573 = minus05 minus08 Figure 3 represents the axial velocitydistribution at a particular time 119905 = 1205874 with the radialdistance for different values of body acceleration parameter119861It is seen that at the throat of the stenosis (ie 119911 = 10) axialvelocity monotonically decreases with the radial distanceFrom the same figure we observed that the body accelerationparameter 119861 brings quantitative as well as qualitative changesin velocity profiles It reveals that the velocity increases asthe the body acceleration parameter increases This resultsupports the phenomenon that body acceleration reduces theflow resistance and so the velocity of blood flow increaseswith the increase in body acceleration Figure 4 gives thedistribution of the axial velocity at the onset of the stenosis for

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

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DistributedSensor Networks

International Journal of

ISRN Biomedical Engineering 5

The total velocity distribution can be written as

119906 = 119892 (1198772minus 1199032) minus

8radic120579119892

3(11987732

minus 11990332

)

+ 2120579 (119877 minus 119903) minus 2120573119892119877 + 4radic119877119892120579 minus 2120573120579 +120572211989210158401198774

8

times[[

[

(119903

119877)4

minus 4(119903

119877)2

+ 3 + radic120579

119892119877

times (16

3(

119903

119877)2

minus424

127(

119903

119877)72

+16

3(

119903

119877)32

minus1144

147)

+120579

119892119877(128

63(

119903

119877)3

minus64

9(

119903

119877)32

+320

63)]]

]

+1198921015840119877120573

2[2119877(radic

120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877)119877 minus 120573(1 minus radic

120579

119892119877))

minus (1 minus radic120579

119892119877) (1198772minus 1199032) +

4

3radic

120579

119892

times(1 minus radic120579

119892119877) (11987732

minus 11990332

)]

(33)

The nondimensional volumetric flow rate 119876 is given by

119876 (119905) = 4int119877

0

119906119903 119889119903 = 1198921198774minus

16

7radic

119892120579

1198771198774

+1

31198773120579 +

1

12119877612057221198921015840minus

15

77radic

120579

119892119877119877612057221198921015840

+4120579119877612057221198921015840

35119892119877+ 119892120573119877

3(1 minus radic

120579

119892119877)

2

+ 12057311989210158401198775

times [(radic120579

119892119877minus 1)

times ((1 minus8

7radic

120579

119892119877) minus

120573

119877(1 minus radic

120579

119892119877))

minus1

4(1 + radic

120579

119892119877) +

2

7radic

120579

119892119877(1 minus radic

120579

119892119877)]

(34)

where 119876 = 1198761015840(1205871198600119877402120583) 1198761015840 is the volumetric flow rate

In dimensionless form resistance to the flow is given by

120582 (119905) =1198750minus 119875119871

119876=

119871 (1 + 119890 cos (119905 + 120601))

119876 (35)

where 119875 = 1198750at 119911 = 0 and 119875 = 119875

119871at 119911 = 119871

4 Results and Discussion

With a view to illustrate the applicability of the mathematicalmodel developed and analyzed in the preceding sectionsthe analytical expressions for the axial velocity profile wallshear stress and volumetric flow rate are presented by takinginto account the velocity-slip condition at the arterial wallIn order to get a proper insight into the flow behaviour ofblood through a time-dependent stenosed arterial segmentunder body acceleration the variations of 119906 119876 120591 and 120582

have been estimated and the computed results are presentedin graphical form The flowing blood is modeled as Cassonfluid model The governing equations of the flow are solvedusing perturbation analysis with the assumption that theWomersley frequency parameter is small which is valid forphysiological situations in small blood vessels In our analysisthe value of the shape parameter (119899) of the stenosis is taken tobe 2Themaximumheight of the stenosis is generally taken as03 and only to pronounce its effect we have taken the rangefrom 00 to 06The values of the nondimensional yield stress120579 for the blood of the normal subject are between 001 and003 and in diseased state it is quite high and in such a casethe value of the yield stress is taken to lie between 01 and 04The velocity slip parameter is taken between minus20 and minus05The pressure gradient parameter (119890) is taken in the range 01ndash04 For this problem the value of pulsatile Reynolds numberis taken as 005 To discuss the effects of the body accelerationparameter 119861 on the various flow quantities its value is takenin the range between 00 and 08

Figures 2ndash6 give an idea of the axial velocity distributionin the case of blood flow in the vicinity of the stenotic portionof the arterial segment under the purview of the presentstudyThe presence of velocity slip at the wall alters the bloodvelocity significantly as shown in Figure 2 We observe fromthis figure that blood velocity decreases as the slip lengthincreases It is interesting to note that at 119911 = 10 (ie at thethroat the stenosis) and at a particular time 119905 = 1205874 for largevalues of slip length (120573 = minus10 minus12) axial velocity increaseswith the radial distance whereas the reverse trend is observedfor 120573 = minus05 minus08 Figure 3 represents the axial velocitydistribution at a particular time 119905 = 1205874 with the radialdistance for different values of body acceleration parameter119861It is seen that at the throat of the stenosis (ie 119911 = 10) axialvelocity monotonically decreases with the radial distanceFrom the same figure we observed that the body accelerationparameter 119861 brings quantitative as well as qualitative changesin velocity profiles It reveals that the velocity increases asthe the body acceleration parameter increases This resultsupports the phenomenon that body acceleration reduces theflow resistance and so the velocity of blood flow increaseswith the increase in body acceleration Figure 4 gives thedistribution of the axial velocity at the onset of the stenosis for

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 ISRN Biomedical Engineering

0

05

1

15

2

25

0 01 02 03 04 05 06 07 08 09

u

r

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 2 Velocity distribution for different values of 120573 when 119899 =

2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 =

01 and 119905 = 1205874

u

r

0

02

04

06

08

1

12

14

16

18

2

0 01 02 03 04 05 06 07 08

B = 00

B = 02

B = 04

B = 08

Figure 3 Velocity distribution for different values of 119861 when 119899 = 2119890 = 02 120573 = 00 1205722 = 005 120598 = 03 120596 = 20 120601 = 00 120579 = 01 and119905 = 1205874

different values of yield stressThis figure depicts that velocitydecreases with increasing the yield stress It is also notedthat velocity profile oscillates with time Figure 5 depicts thedistribution of the velocity at the throat of the stenosis withtime for different values of the height of the stenosis It hasbeen observed from this figure that for any value of 120598 themaximum value attains on the axis of symmetry Figure 6illustrates the velocity distribution at 119911 = 15 for differentvalues of the pressure gradient parameter 119890This figure revealsthat velocity increases as the pressure gradient parameter 119890

increases when 119905 isin [00 095] [41 74] and so on whereasthe reverse trend is observed in rest of the time intervals

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12 14 16 18 20t

120579 = 00

120579 = 005

120579 = 01

120579 = 02

u

Figure 4 Variation of central line velocity with time 119905 for differentvalues of 120579 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 and 120573 = minus05

005

115

225

335

445

555

0 2 4 6 8 10 12 14 16 18 20t

u

120598 = 01

120598 = 03

120598 = 05

120598 = 07

Figure 5 Variation of central line velocity with time 119905 for differentvalues of 120598 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120573 = minus05120596 = 20 120601 = 00 and 120579 = 01

The distributions of flow rate for different values ofvelocity slip yield stress body acceleration parameter andpressure gradient parameter have also been computed Theresults are plotted and presented graphically through Figures7ndash9 Figure 7 reveals that the volumetric flow rate increasesas the values of the velocity slip increases It can also be notedthat in the stenotic region initially the velocity increasesalong with the axis of the artery attaining its maximum andthen it decreases monotonically Onemay note from Figure 8that for any values of yield stress 120579 flow rate oscillates withtime It has been seen from the same figure that for eachoscillation troughcrest is maximum for minimum value ofyield stress Figure 9 gives the variation of volumetric flow

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

ISRN Biomedical Engineering 7

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

u

e = 01

e = 02

e = 03

e = 04

Figure 6 Variation of central line axial velocity with time 119905 fordifferent values of 119890 when 119899 = 2 120573 = minus05 119861 = 04 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

0

5

10

15

20

25

30

06 08 1 12 14

Q

z

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 7 Variation of volumetric flow rate for different values of 120573when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 =

00 120579 = 01 and 119905 = 1205874

rate with pressure gradient parameter 119890 for different valuesof body acceleration parameter 119861 This figure shows that flowrate decreases with pressure gradient parameter 119890 whereasthe flow rate increases as the body acceleration parameterincreases Therefore the pressure gradient parameter 119890 has alinear relation with flow rate 119876 with decreasing trend

The computational results for the wall shear stress com-puted on the basis of the present study are presented inFigures 10ndash12 for different values of the parameters 120573 119861and 120579 respectively Figure 10 shows that for any values of 120573the wall shear stress oscillates with time Also it has beenobserved from this figure that the wall shear stress decreasesas the velocity-slip increases when 119905 isin [15 32] [45 61] and

120579 = 00

120579 = 005

120579 = 01

120579 = 02

minus6

minus4

minus2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20t

Q

Figure 8 Variation of volumetric flow rate for different values of 120579when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874

05

1

15

2

3

25

0 01 02 03 04 05 06 07 08 09 1e

B = 00

B = 02

B = 04

B = 08

Q

Figure 9 Variation of volumetric flow ratewith 119890 for different valuesof 119861 (119899 = 2 1205722 = 005 120579 = 01 120598 = 03 120596 = 20 120601 = 00 120573 = minus05and 119905 = 1205874)

so on while the trend is reversed in rest of the time intervalsFigure 11 depicts the distribution of wall shear stress for dif-ferent values of body acceleration parameter119861 Onemay notefrom this figure that the period of oscillation increases as thebody acceleration parameter decreases Figure 12 representsthe wall shear stress distribution in the stenotic region fordifferent values of yield stress This figure indicates that thewall shear stress decreases as 120579 increases We also observedthat the maximum wall shear stress occurs at the throat ofthe stenosis

The variation of flow resistance with the height of stenosisis shown in Figure 13 for different values of slip velocityparameter 120573 The results presented in this figure reveal

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 ISRN Biomedical Engineering

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18 20t

120591

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 10Distribution ofwall shear stress (120591) at 119911 = 10 for differentvalues of 120573 with 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

B = 00

B = 02

B = 04

B = 08

0 2 4 6 8 10 12 14 16 18 20t

0

1

2

3

4

5

6

7

9

8

120591

Figure 11 Variation of the wall shear stress (120591) with time at 119911 = 15

for different values of 119861 with 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005120598 = 03 120596 = 20 120601 = 00 and 120579 = 01

that the flow resistance increases with the increase in thestenosis height and that as the velocity slip increases theflow resistance decreases Figure 14 gives the variation offlow resistance with the maximum height of the stenosis fordifferent values of the body acceleration parameter 119861 Thisfigure shows that the body acceleration has a reducing effecton the flow resistance

5 Concluding Remarks

The present analysis deals with a theoretical investigation ofblood flow characteristics through a narrow and time-dependent stenosed artery in the presence of body

06 08 1 12 14z

120579 = 00

120579 = 005

120579 = 01

120579 = 02

4

45

5

55

6

65

7

75

8

120591

Figure 12 Variation of the wall shear stress (120591) for different valuesof 120579 (119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120598 = 03 120596 = 20 120601 = 00120573 = minus05 and 119905 = 1205874)

0 01 02 03 04 05 061

2

3

4

5

6

7

120582

120576

120573 = minus05

120573 = minus08

120573 = minus10

120573 = minus12

Figure 13 Variation of the resistance to the flow with 120598 for differentvalues of 120573 when 119899 = 2 119890 = 02 119861 = 04 1205722 = 005 120596 = 20 120601 = 00120579 = 01 and 119905 = 1205874

acceleration and velocity slip by treating it as a Cassonfluid model Using the appropriate boundary conditionsanalytical expressions for the velocity wall shear stress andflow resistance have been estimated The computationalresults were presented graphically for different values of theparameters involved in the present problem under considera-tion Figure 15 shows the comparison of our results with thoseof computer generated results [32] in order to validate thepresent mathematical model For the purpose of comparisonboth the studies have been naturally brought to the sameplatform by disregarding the slip effects and by considering119879

being very small for the present study while for the previousstudy the inclination angle has been considered to be zero

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

ISRN Biomedical Engineering 9

B = 00

B = 02

B = 04

B = 08

0 01 02 03 04 05 06120576

0

5

10

15

20

25

30

120582

Figure 14 Variation of the resistance to the flow with 120598 for differentvalues of 119861 when 119899 = 2 119890 = 02 120573 = minus05 1205722 = 005 120596 = 20120601 = 00 120579 = 01 and 119905 = 1205874

Present StudyResult of Neeraja and Vidya [32]

0

05

1

15

2

25

u

0 01 02 03 04 05 06 07 08r

Figure 15 Comparison of axial velocity with those of Neeraja andVidya [32] when 120573 = 00 119879 = 01 120598 = 02 119861 = 10 120596 = 201205722= 001 120579 = 001 and 119899 = 2

The present study bears the potential to examine thecomplex flow behavior of blood under the simultaneousinfluence of different factors like the size of the stenosisbody acceleration parameter pressure gradient parameterand the velocity slip The observations made on the basisof the present study are quite significant The investigationshows that the blood velocity increases as body accelerationparameter increases This fact is harmful for the heart

Caro et al [33] made a conjecture that the arterial diseaseatheroma develops in the regions where the mean wall shearstress is relatively low Experimental studies also revealedthat low velocity regions are prone to the development ofatherosclerosisfurther deposition

Themain objective in our present study has been to assessthe role of velocity slip in blood flow through arteries and todetermine those regions where the velocity is low and also theregionswhere thewall shear stress is lowThus the study bearsthe potential to further explore the causes and developmentof arterial diseases like atherosclerosis and atheroma

An increase in the size of stenoses causes enhancement ofthe resistance to blood flow through the arteries in the brainheart and other organs of the body This may lead to strokeheart attack and various other cardiovascular diseases

Nomenclature

1199061015840 Velocity of blood1199011015840 Pressure1205911015840 Shear stress120591119910 Yield stress

119905 Time1198891015840 The distance of the onset of stenosis from

entrance1198661015840(1199051015840) Body acceleration11989710158400 Length of stenosis

11990310158400 Radius of plug

1198770 Radius of the pericardial surface of normal

portion of the arterial segment1198771015840 Radius of the endothelium of the stenosed

portion119899 (ge 2) Shape parameter of stenosis1205981015840 Maximum height of the stenosis120583 Coefficient of viscosity120588 Density of blood1205731015840 Slip coefficient12059610158401 Angular frequency

1198860 Amplitude of acceleration

120601 Phase difference1198912 Pulse frequency

1205722 Womersley parameter119890 Pressure gradient parameter120579 Nondimensional yield stress119861 Body acceleration parameter120573 Slip length

Acknowledgments

One of the authors (A Sinha) is grateful to the NBHM DAEMumbai India for the financial support of this investigationThis work is also partly supported by DST New DelhiGovernment of India throughprojectGrant no SRFTPMS-0422011

References

[1] C Clark ldquoTurbulent velocitymeasurements in amodel of aorticstenosisrdquo Journal of Biomechanics vol 9 no 11 pp 677ndash6871976

[2] G S Beavers and D D Joseph ldquoBoundary conditions at anaturally permeable wallrdquo Journal of Fluid Mechanics vol 30no 1 pp 197ndash207 1967

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 ISRN Biomedical Engineering

[3] L M Srivastava ldquoFlow of couple stress fluid through stenoticblood vesselsrdquo Journal of Biomechanics vol 18 no 7 pp 479ndash485 1985

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

[5] S Chakravarty ldquoEffects of stenosis on the flow-behaviour ofblood in an arteryrdquo International Journal of Engineering Sciencevol 25 no 8 pp 1003ndash1016 1987

[6] J C Misra and G C Shit ldquoBlood flow through arteries in apathological state a theoretical studyrdquo International Journal ofEngineering Science vol 44 no 10 pp 662ndash671 2006

[7] J B Shukla R S Parihar and S P Gupta ldquoEffects of peripherallayer viscosity on blood flow through the artery with mildstenosisrdquo Bulletin of Mathematical Biology vol 42 no 6 pp797ndash805 1980

[8] Q Long X Y Xu K V Ramnarine and P Hoskins ldquoNumericalinvestigation of physiologically realistic pulsatile flow througharterial stenosisrdquo Journal of Biomechanics vol 34 no 10 pp1229ndash1242 2001

[9] M Texon ldquoThe hemodynamic concept of atherosclerosisrdquoNational Library of Medicine vol 36 no 4 pp 263ndash273 1960

[10] J C Misra A Sinha and G C Shit ldquoMathematical modelingof blood flow in a porous vessel having double stenoses in thepresence of an external magnetic fieldrdquo International Journal ofBiomathematics vol 4 no 2 pp 207ndash225 2011

[11] J C Misra S D Adhikary and G G Shit ldquoMathematicalanalysis of blood flow through an arterial segment with time-dependent stenosisrdquoMathematical Modelling and Analysis vol13 no 3 pp 401ndash412 2008

[12] A Sarkar and G Jayaraman ldquoCorrection to flow ratemdashpressuredrop relation in coronary angioplasty steady streaming effectrdquoJournal of Biomechanics vol 31 no 9 pp 781ndash791 1998

[13] M El-Shahed ldquoPulsatile flow of blood through a stenosedporous medium under periodic body accelerationrdquo AppliedMathematics and Computation vol 138 no 2-3 pp 479ndash4882003

[14] E F Elshehawey E M E Elbarbary N A S Afifi and MEl-Shahed ldquoPulsatile flow of blood through a porous mediumunder periodic body accelerationrdquo International Journal ofTheoretical Physics vol 39 no 1 pp 183ndash188 2000

[15] P Nagarani and G Sarojamma ldquoEffect of body acceleration onpulsatile flow of casson fluid through a mild stenosed arteryrdquoKorea Australia Rheology Journal vol 20 no 4 pp 189ndash1962008

[16] S U Siddiqui N K Verma S Mishra and R S GuptaldquoMathematical modelling of pulsatile flow of Cassonrsquos fluid inarterial stenosisrdquo Applied Mathematics and Computation vol210 no 1 pp 1ndash10 2009

[17] V K Sud and G S Sekhon ldquoArterial flow under periodic bodyaccelerationrdquo Bulletin of Mathematical Biology vol 47 no 1 pp35ndash52 1985

[18] V K Sud and G S Sekhon ldquoAnalysis of blood flow througha model of the human arterial system under periodic bodyaccelerationrdquo Journal of Biomechanics vol 19 no 11 pp 929ndash941 1986

[19] P Chaturani and V Palanisamy ldquoPulsatile flow of power-lawfluid model for blood flow under periodic body accelerationrdquoBiorheology vol 27 no 5 pp 747ndash758 1990

[20] S N Majhi and V R Nair ldquoPulsatile flow of third grade fluidsunder body acceleration-modelling blood flowrdquo InternationalJournal of Engineering Science vol 32 no 5 pp 839ndash846 1994

[21] G C Shit and M Roy ldquoPulsatile flow and heat transfer ofa magneto-micropolar fluid through a stenosed artery underthe influence of body accelerationrdquo Journal of Mechanics inMedicine and Biology vol 11 no 3 pp 643ndash661 2011

[22] J C Misra M K Patra and S C Misra ldquoA non-Newtonianfluid model for blood flow through arteries under stenoticconditionserdquo Journal of Biomechanics vol 26 no 9 pp 1129ndash1141 1993

[23] J C Misra and G C Shit ldquoRole of slip velocity in blood flowthrough stenosed arteries a non-Newtonian modelrdquo Journal ofMechanics in Medicine and Biology vol 7 no 3 pp 337ndash3532007

[24] P Brunn ldquoThe velocity slip of polar fluidsrdquo Rheologica Acta vol14 no 12 pp 1039ndash1054 1975

[25] Y Nubar ldquoBlood flow slip and viscometryrdquo Biophysical Journalvol 11 no 3 pp 252ndash264 1971

[26] R Ponalagusamy ldquoBlood flow through an artery with mildstenosis a two-layered model different shapes of stenoses andslip velocity at the wallrdquo Journal of Applied Sciences vol 7 no 7pp 1071ndash1077 2007

[27] D Biswas and U S Chakraborty ldquoPulsatile flow of blood ina constricted artery with body accelerationrdquo Applications andApplied Mathematics vol 4 pp 329ndash342 2009

[28] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoAppliedMathematical Sciences vol 4 no 57ndash60 pp 2865ndash28802010

[29] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wavelength at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

[30] S Takabatake K Ayukawa and A Mori ldquoPeristaltic pumpingin circular cylindrical tubes a numerical study of fluid transportand its efficiencyrdquo Journal of Fluid Mechanics vol 193 pp 267ndash283 1988

[31] P Chaturani and R P Samy ldquoPulsatile flow of Cassonrsquos fluidthrough stenosed arteries with applications to blood flowrdquoBiorheology vol 23 no 5 pp 499ndash511 1986

[32] G Neeraja and K Vidya ldquoEffect of body acceleration on pul-satile flow of Herschel-Bulkley fluid through an inclined mildstenosed arteryrdquo International Journal of Engineering Researchamp Technology vol 1 pp 1ndash10 2012

[33] C G Caro J M Fitz-Gerald and R C Schroter ldquoAtheromaand arterial wall shear stress observation correlation andproposal of a shear dependent mass transfer mechanism foratherogenesisrdquo Proceedings of the Royal Society of London B vol177 no 46 pp 109ndash159 1971

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of