Research ArticleSteady Flow of Couple-Stress Fluid in ConstrictedTapered Artery Effects of Transverse Magnetic FieldMoving Catheter and Slip Velocity

Hamzah Bakhti1 and Lahcen Azrar12

1MMC Team Department of Mathematics Faculty of Sciences amp Techniques AEU BP 416 Tangier Morocco2MMC Team Department of Mechanical Engineering Faculty of Engineering KAU Jeddah Saudi Arabia

Correspondence should be addressed to Hamzah Bakhti bakhti hamzahoutlookcom

Received 9 July 2015 Revised 22 August 2015 Accepted 24 August 2015

Academic Editor Taha Sochi

Copyright copy 2016 H Bakhti and L Azrar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Steady flow of a couple-stress fluid in constricted tapered artery has been studied under the effects of transverse magnetic fieldmoving catheter and slip velocity With the help of Besselrsquos functions analytic expressions for axial velocity flow rate impedanceand wall shear stress have been obtained It is of interest to note that these solutions can be used for different types of fluid flow intubes and not only the case of bloodThe effects of various geometric parameters the parameters arising out of the fluid consideredand the magnetic field are discussed by considering the slip velocity the catheter velocity and tapering angle The study of theabove model is very important as it has direct applications in the treatment of cardiovascular diseases

1 Introduction

Catheters are semirigid thin tubes made frommedical gradematerials serving a broad range of functions and can beinserted in the body to detect and identify diseases orperform a surgical procedure inside the heart brain armslegs or lungs [1] But the problem is how we can knowthat the catheter is going the right way or where is it Mosthospitals use X-ray machine mounted above the couch tocheck the catheterrsquos progress Low-dose X-rays are used tomonitor the progress of the catheter tip But if we wantto navigate the catheterrsquos passage in real time some ten totwenty X-ray images are made every second Even thoughthe radiation dose involved is very low no radiation at all ismuch healthier for both the patient and themedical staff whorun the risk of being exposed to the radiation on daily basisResearchers developed a magnetic navigation system formedical instruments such as catheters and guide wires Thecatheters with magnetic tips are steered within the patientwithout the need for an electrophysiologist to maneuverthe catheter or guide wire placement manually A magneticsensor on the tip of the medical instrument measures an

external magnetic field and reports exactly where the tip ofthe instrument is located [2]

Each year heart disease is at the top of the list of thecountryrsquos most serious health problems Statistics show thatcardiovascular disease in some countries like America isthe first health problem and the leading cause of death(recent statistics released by the American Heart Associ-ation) Catheters are often used to treat stenosis (partialocclusion of the blood vessel) aneurysm (dilation of theblood vessel resulting in stretching of the vessel wall) andembolism (complete occlusion of a blood vessel by a bloodclot or some other particle) Figure 1 The catheters maycontain a balloon that can be used to stretch the bloodvessel or a rinsing mechanism or laser device to removeembolism They are also used to measure blood pressurein situ Sometimes a catheter is used to deliberately createembolism for example to stop the flow of blood to a tumor inorder tomortify it An aneurysm can also be treated bymeansof embolisation [3]

The study of blood flow through different types of arteriesis of considerable importance in many cardiovascular dis-eases one of them is atherosclerosis The blood flow through

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 9289684 11 pageshttpdxdoiorg10115520169289684

2 Journal of Applied Mathematics

StenosisAneurysm

Embolism

Guide wireCatheter

Figure 1 Catheters are used to treat stenosis (partial occlusion ofthe blood vessel) aneurysm (dilation of the blood vessel resulting instretching of the vessel wall) and embolism (complete occlusion of ablood vessel by a blood clot or some other particle) (a reproductionfrom Delft Outlook 2004)

an artery has drawn the attention of researchers for a longtime up to now due to its great importance in medicalsciences Under normal conditions blood flow in the humancirculatory system depends upon the pumping action of theheart and this produces a pressure gradient throughout thearterial network

Many researchers have studied blood flow in the arteryby considering blood as either Newtonian or non-Newtonianfluids since blood is a suspension of red cells in plasma itbehaves as a non-Newtonian fluid at low shear rate In thestenosed condition substantial reduction in the lumen of anartery results in size effects (ratio of haematocrit to vesseldiameter) which influences flow characteristics significantlyTo study the size effect in the fluid flow Stokes [4] Eringen[5] and Cowin [6] have proposed continuum model Micro-continuum structure of the fluid is also referred to as couple-stress fluid which is proposed by Stokes The foregroundof couple-stress fluid is to introduce size dependent effectsas mentioned above that is not present in other classicalviscous fluids Because of its mathematical simplicity it hasbeen widely used by number of researchers Chaturani andUpadhya [7] have studied the pulsatile flow of couple-stressfluid through circular tubes The Poiseuille flow of couple-stress fluid has been examined by Chaturani and Rathod [8]

The application of Magnetohydrodynamics in physiolog-ical problems is of growing interest and it is very importantfrom both theoretical and practical points of viewThe appli-cation of Magnetohydrodynamics in physiological problemsis of growing interest The flow of blood can be controlledby applying appropriate quantity of magnetic field Kollin [9]has coined the idea of electromagnetic field in the medicalresearch for the first time in the year 1936 Korchevskii andMarochnik [10] have discussed the possibility of regulatingthe blood movement in human system by applying magneticfield Rao and Deshikachar [11] have investigated the effectof transverse magnetic field in physiological type of flowthrough a uniform circular pipe Vardanyan [12] showedthat the application of magnetic field reduces the speed ofblood flow A mathematical model for two-layer pulsatileflow of blood with microorganism in a uniform tube underlow Reynolds number and magnetic effect has been studiedby Rathod and Gayatri [13] Thus all these researchers havereported that the effect of magnetic field reduces the velocityof blood

There are many treatments available for diagnosing andtreating constricted vessels Catheterization (thin flexibletube) is one of them in which balloon angioplasty is a spe-cialized form of catheterization These procedures are widelyused in the medical field for treating the atherosclerosisInsertion of the catheter in a tube creates an annular regionbetween inner wall of the artery and outer wall of the catheterwhich influences the flow field such as pressure distributionand shear stress at the wall In view of its immense impor-tance the effect of the catheter on physiological parameterswas discussed by the researchers [14 15]

The shapes of the stenosis in the above aforesaid studieshave been considered to be radially symmetric or asymmet-ric But while stenosis is maturing it may grow up in seriesmanner overlapping with each other and it would appearlike x-shape Riahi et al [16] observed the pressure gradientforce flow velocity impedance and wall shear stress in theoverlapping stenotic zone at the critical height and at thethroats of such stenosis Here they considered steady natureof flow Srivastava andMishra [17] explored the arterial bloodflow through an overlapping stenosis by treating the bloodas a Casson fluid They figured impedance and shear stressfor different stenosis heights Chakravarty and Mandal [18]discussed the effects of the overlapping stenosis under lowshear rate flow

The presence of red cell slip at the vessel wall wasrecommended theoretically by Vand [19] experimentally byBennett [20] and Nubar [21] Chaturani and Biswas [22] andso forth They used slip velocity at the wall in their analysisLately Ponalagusamy [23 25] has developed mathematicalmodels for blood flow through stenosed arterial segmentby taking a slip velocity condition at the constricted wallThus it is very appropriate to consider slip velocity at the wallof the stenosed artery in blood flow modeling The catheteris actually moving into the body so it is recommended toconsider a nonnull velocity at the catheter wall in this paperwe consider that the catheter is moving in the 119911-axis with afixed velocity

In the present analysis a mathematical model for thesteady blood flow through tapered stenosed artery under theinfluence of a moving catheter slip velocity and a magneticfield is presented by considering blood as a couple-stress fluidin a circular tube It is assumed that the magnetic field alongthe radius of the pipe is present no external electric field isimposed and magnetic Reynolds number is very small Themotivation for studying this problem is to understand theblood flow in an artery under the effect of magnetic fieldalongside with the catheter inserted into the blood vesselalso when the fatty plaques of cholesterol and artery cloggingblood clots are formed in the lumen of the artery

The main aim of this work is to study these phenomenaobtain analytic expressions for axial velocity and shear stressand also study the effect ofmagnetic field (Hartmann number119867) and couple-stress parameter (120573) on the velocity and theeffects of Hartmann number on the fluid velocity Hence thepresent mathematical model gives a simple form of velocityexpression for the blood flow so that it will help not onlypeople working in the field of physiological fluid dynamicsbut also the medical practitioners

Journal of Applied Mathematics 3

2 Problem Formulation

Let us consider a two-dimensional steady flow of bloodthrough a rigid tapered stenosed tube by considering bloodas an electrically conducting incompressible couple-stressfluidThemagnetic field is acting along the radius of the tubeThe magnetic Reynolds number of the flow is assumed to besufficiently small that the inducedmagnetic and electric fieldscan be neglected [24] The catheter is assumed to be movingin the 119911-axis direction

Mathematical model of blood flow through a taperedstenosed arterial segment in the presence of a movingcatheter and a magnetic field is to be built to study theimpact of various geometric Hartman and fluid parameterson physiological parameters The geometry of the taperedstenosed artery is shown in Figures 2 and 3 and is expressedmathematically with inputs from [26] as

119877 (119911) =

(1198770+ 120577119911)(1 minus

120598119899119899(119899minus1)

(119899 minus 1) 119871119899

0

(119871119899minus1

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

119899)) 119889 le 119911 le 119889 + 119871

0

1198770+ 120577119911 otherwise

(1)

where 1198770is the radius of the annular region in case of

nontapered artery in the nonconstricted domain 119877119888is the

radius of the catheter 1198710is the stenosis length 119889 indicates the

location of stenosis 120598 is the maximum height of the stenosisinto the lumen and 120577 = tan120601 is the tapering parameter whichrepresents the slope of the tapered vessel with 120601 being thetapering angle 120601 lt 0 120601 gt 0 and 120601 = 0 are for convergingtaper diverging taper and no-taper respectively

The current density J is expressed by

J = 120590 (E + V times B) (2)

where E is the electric field intensity 120590 is the electrical con-ductivityB is themagnetic flux intensity andV is the velocityvector In themomentum equation the electromagnetic forceF119898is included and is defined as

F119898= J times B = 120590 (E + V times B) times B (3)

The conservation equations which govern the couple-stress fluid flow including a Lorentz force can be written inthe following form

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + (120582 + 120583) nablanabla sdot V + 120578Δnablanabla sdot V

minus 120578Δ2V + 120583ΔV + 120590 (V times B) times B + 120588f

+

1

2

nabla times (120588l)

(4)

where Δ is the Laplacian operator For couple-stress fluidshear stress tensor is not symmetric The force stress tensor120591 and the couple-stress tensor M that arises in the theory ofcouple-stress fluids are given by

120591 = (minus119901 + 120582nablaV) I + 120583 (nablaV + (nablaV)119879) + 12

I

times (nablaM + 120588l)

M = 119898I + 2120578 (nabla (nabla times V)) + 21205781015840 (nabla (nabla times V))119879

(5)

where 120588 is the density V is velocity vector 119901 is the pressure120582 and 120583 are the viscosity coefficients and 120578 and 1205781015840 are couple-stress coefficients f is a body force and l is a body couplemoment Further the materials constants 120583 120582 120578 and 1205781015840satisfy the following inequalities

120583 ge 0

3120582 + 2120583 ge 0

120578 ge 0

120578 ge 1205781015840

(6)

As the flow is steady and incompressible in the absenceof body force and body couple moment (4) reduce to

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + 120583ΔV minus 120578Δ2V + 120590 (V times B) times B(7)

The flow is considered to take place under the influence ofexternally applied magnetic field in axial direction 119911 Underthese assumptions the governing equations may be writtenin the cylindrical coordinates system as follows

Equation of continuity is

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0 (8)

Equation of radial momentum is

120588(119906

120597119906

120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 120583(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+

1205972119906

1205971199112) minus 120578(

1205974119906

1205971199034+ 2

1205974119906

12059711990321205971199112+

1205974119906

1205971199114+

2

119903

1205973119906

1205971199033

+

2

119903

1205973119906

1205971199031205971199112minus

3

1199032

1205972119906

1205971199032minus

2

1199032

1205972119906

1205971199112+

3

1199033

120597119906

120597119903

minus

3

1199034119906)

(9)

4 Journal of Applied Mathematics

L

120598

R0

Rc

n = 2n = 5n = 15

L0

d

Figure 2 2D view of a catheterized stenosed artery

L

R(z)

120598

120601 lt 0120601 = 0

120601 gt 0

Figure 3 2D view of a tapered stenosed artery

Equation of axial momentum is

120588(119906

120597119908

120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ 120583(

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+

1205972119908

1205971199112) minus 120578(

1205974119908

1205971199034+ 2

1205974119908

12059711990321205971199112+

1205974119908

1205971199114+

2

119903

1205973119908

1205971199033

+

2

119903

1205973119908

1205971199031205971199112minus

1

1199032

1205972119906

1205971199032+

1

1199033

120597119908

120597119903

) minus 1205901198612

0119908

(10)

where 1198610is the external transverse magnetic field The

nondimensional parameters are obtained as follows

119911lowast=

119911

1198710

119903lowast=

119903

1198770

119908lowast=

119908

1199080

120598lowast=

120598

1198770

120577lowast=

1205771198710

1198770

119906lowast=

1198710119906

11990801198770

119901lowast=

1199032

0119901

11990801198710120583

(11)

and then the dimensionless form of geometry (1) afterdropping the stars is obtained as follows

119877 (119911) =

(1 + 120577119911) (1 minus

120577119899119899(119899minus1)

(119899 minus 1)

((119911 minus 120574) minus (119911 minus 120574)119899)) 120574 le 119911 le 120574 + 1

1 + 120577119911 otherwise(12)

where 120574 = 11987101198711and 119908

0is a typical axial velocity

Also (8)ndash(10) on dropping the stars become

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0

Re1205853 (119906120597119906120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 1205852(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+ 1205852 1205972119906

1205971199112) minus

1205852

1205732[

1205974119906

1205971199034+

2

119903

1205973119906

1205971199033minus

3

1199032

1205972119906

1205971199032+

3

1199033

120597119906

120597119903

minus

3

1199034119906

+ 1205852(

2

119903

1205973119906

1205971199031205971199112+ 2

1205974119906

12059711990321205971199112+ 1205852 1205974119906

1205971199114minus

2

1199032

1205972119906

1205971199112)]

Re120585 (119906120597119908120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+ 1205852 1205972119908

1205971199112) minus

1

1205732[

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032

+

1

1199033

120597119908

120597119903

+ 1205854 1205974119908

1205971199114+ 21205852(

1205974119908

12059711990321205971199112+

1

119903

1205973119908

1205971199031205971199112)]

minus 1198672119908

(13)

Journal of Applied Mathematics 5

where 120585 = 11987701198710 Re = 120588119906

01198770120583 is the Reynolds number

1205732= 1198772

0120583120578 is the couple-stress fluid parameter and 1198672 =

1198612

01198772

0120590120583 is the Hartmann number

Under the assumption of mild stenosis that is 1205981198770le

1 and further assuming that 120585 = 11987701198710le 1 (13) get

transformed into120597119901

120597119903

= 0 (14)

120597119901

120597119911

= (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

)

minus

1

1205732(

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032+

1

1199033

120597119908

120597119903

)

minus 1198672119908

(15)

Equation (15) can be written as

120597119901

120597119911

= Δ119908 minus

1

1205732Δ2119908 minus119867

2119908 (16)

where

Δ =

1205972

1205971199032+

1

119903

120597

120597119903

(17)

It can be seen that the pressure variation depends only onthe axial variableThe pressure gradient 120597119901120597119911 is produced bythe pumping action of the heart

The corresponding nondimensional boundary conditionsare as shown below

119908 = V at 119903 = 119877 (119911)

119908 = 119892 at 119903 = 119877119888

1205972119908

1205971199032minus

120596

119903

120597119908

120597119903

= 0 at 119903 = 119877 (119911) 119903 = 119877119888

(18)

where V represents the slip velocity at the artery wall 119892is the velocity of the moving boundary (catheter) and120596 = 120578

1015840120578 is the parameter associated with the couple-stress

fluid No couple-stress effects will be present if 1205781015840 = 120578which is equivalent to saying that the couple-stress tensor issymmetric

3 Solution of the Problem

As a solution of (16) with the pressure gradient being taken asa constant take

minus

120597119901

120597119911

= 119896119904 (19)

Let 11989921+ 1198992

2= 1205732 and 1198992

11198992

2= 12057321198672 which are given by

1198992

1=

1205732+ radic1205734minus 412057321198672

2

(20)

1198992

2=

1205732minus radic1205734minus 412057321198672

2

(21)

then (16) is simplified to the form

(Δ minus 1198992

1) (Δ minus 119899

2

2)119908 = 120573

2119896119904 (22)

The solution of the above equation is obtained as

119908 = 1198881(119911) 1198680(1198991119903) + 1198882(119911)1198700(1198991119903) + 1198883(119911) 1198680(1198992119903)

+ 1198884 (119911)1198700 (1198992119903) +

1

1198672119896119904

(23)

where 1198680and 119870

0are modified Bessel function of order zero

first and second kinds respectively 119888119894for 119894 = 1 2 3 4

are calculated numerically by solving the algebraic systemobtained from the boundary conditions

Volumetric flow rate 119876 in steady flow through a tubeis obtained by integrating the velocity profile over a crosssection of the tubeThe nondimensional volumetric flow rate119876 across the radial distance is expressed as

119876 = int

119877(119911)

119877119888

2119903119908119889119903 (24)

which is obtained in the form 119876 = minus2119896119904119865(119877119888 119877(119911)) where

119865 (119877119888 119877 (119911)) =

1198891

1198991

(1198771198681(1198991119877) minus 119877

1198881198681(1198991119877119888))

+

1198892

1198991

(1198771198881198701(1198991119877119888) minus 119877119870

1(1198991119877))

+

1198893

1198992

(1198771198681(1198992119877) minus 119877

1198881198681(1198992119877119888))

+

1198894

1198992

(1198771198881198701(1198992119877119888) minus 119877119870

1(1198992119877))

+

1198772minus 1198772

119888

21198672

(25)

where 119889119894= minus119888119894119896119904 for 119894 = 1 2 3 4 The pressure drop Δ119901

across the stenosis between the sections 119911 = 0 and 119911 = 119871 isobtained by

Δ119901 =

119876

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911 (26)

The resistance to the flow (impedance) is obtained from

119877119891=

Δ119901

119876

=

1

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911

=

1

2

(int

119889

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

119889+1198710

119889

1

119865 (119877119888 119877 (119911))

119889119911

+ int

119871

119889+1198710

1

119865 (119877119888 119877 (119911))

119889119911)

(27)

The dimensionless form of (20) is

119877119891=

Γ

2

(int

120574

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

120574+1

120574

1

119865 (119877119888 119877 (119911))

119889119911

+ int

1120574

120574+1

1

119865 (119877119888 119877 (119911))

119889119911)

(28)

where Γ = 1198710119889

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

StenosisAneurysm

Embolism

Guide wireCatheter

Figure 1 Catheters are used to treat stenosis (partial occlusion ofthe blood vessel) aneurysm (dilation of the blood vessel resulting instretching of the vessel wall) and embolism (complete occlusion of ablood vessel by a blood clot or some other particle) (a reproductionfrom Delft Outlook 2004)

an artery has drawn the attention of researchers for a longtime up to now due to its great importance in medicalsciences Under normal conditions blood flow in the humancirculatory system depends upon the pumping action of theheart and this produces a pressure gradient throughout thearterial network

Many researchers have studied blood flow in the arteryby considering blood as either Newtonian or non-Newtonianfluids since blood is a suspension of red cells in plasma itbehaves as a non-Newtonian fluid at low shear rate In thestenosed condition substantial reduction in the lumen of anartery results in size effects (ratio of haematocrit to vesseldiameter) which influences flow characteristics significantlyTo study the size effect in the fluid flow Stokes [4] Eringen[5] and Cowin [6] have proposed continuum model Micro-continuum structure of the fluid is also referred to as couple-stress fluid which is proposed by Stokes The foregroundof couple-stress fluid is to introduce size dependent effectsas mentioned above that is not present in other classicalviscous fluids Because of its mathematical simplicity it hasbeen widely used by number of researchers Chaturani andUpadhya [7] have studied the pulsatile flow of couple-stressfluid through circular tubes The Poiseuille flow of couple-stress fluid has been examined by Chaturani and Rathod [8]

The application of Magnetohydrodynamics in physiolog-ical problems is of growing interest and it is very importantfrom both theoretical and practical points of viewThe appli-cation of Magnetohydrodynamics in physiological problemsis of growing interest The flow of blood can be controlledby applying appropriate quantity of magnetic field Kollin [9]has coined the idea of electromagnetic field in the medicalresearch for the first time in the year 1936 Korchevskii andMarochnik [10] have discussed the possibility of regulatingthe blood movement in human system by applying magneticfield Rao and Deshikachar [11] have investigated the effectof transverse magnetic field in physiological type of flowthrough a uniform circular pipe Vardanyan [12] showedthat the application of magnetic field reduces the speed ofblood flow A mathematical model for two-layer pulsatileflow of blood with microorganism in a uniform tube underlow Reynolds number and magnetic effect has been studiedby Rathod and Gayatri [13] Thus all these researchers havereported that the effect of magnetic field reduces the velocityof blood

There are many treatments available for diagnosing andtreating constricted vessels Catheterization (thin flexibletube) is one of them in which balloon angioplasty is a spe-cialized form of catheterization These procedures are widelyused in the medical field for treating the atherosclerosisInsertion of the catheter in a tube creates an annular regionbetween inner wall of the artery and outer wall of the catheterwhich influences the flow field such as pressure distributionand shear stress at the wall In view of its immense impor-tance the effect of the catheter on physiological parameterswas discussed by the researchers [14 15]

The shapes of the stenosis in the above aforesaid studieshave been considered to be radially symmetric or asymmet-ric But while stenosis is maturing it may grow up in seriesmanner overlapping with each other and it would appearlike x-shape Riahi et al [16] observed the pressure gradientforce flow velocity impedance and wall shear stress in theoverlapping stenotic zone at the critical height and at thethroats of such stenosis Here they considered steady natureof flow Srivastava andMishra [17] explored the arterial bloodflow through an overlapping stenosis by treating the bloodas a Casson fluid They figured impedance and shear stressfor different stenosis heights Chakravarty and Mandal [18]discussed the effects of the overlapping stenosis under lowshear rate flow

The presence of red cell slip at the vessel wall wasrecommended theoretically by Vand [19] experimentally byBennett [20] and Nubar [21] Chaturani and Biswas [22] andso forth They used slip velocity at the wall in their analysisLately Ponalagusamy [23 25] has developed mathematicalmodels for blood flow through stenosed arterial segmentby taking a slip velocity condition at the constricted wallThus it is very appropriate to consider slip velocity at the wallof the stenosed artery in blood flow modeling The catheteris actually moving into the body so it is recommended toconsider a nonnull velocity at the catheter wall in this paperwe consider that the catheter is moving in the 119911-axis with afixed velocity

In the present analysis a mathematical model for thesteady blood flow through tapered stenosed artery under theinfluence of a moving catheter slip velocity and a magneticfield is presented by considering blood as a couple-stress fluidin a circular tube It is assumed that the magnetic field alongthe radius of the pipe is present no external electric field isimposed and magnetic Reynolds number is very small Themotivation for studying this problem is to understand theblood flow in an artery under the effect of magnetic fieldalongside with the catheter inserted into the blood vesselalso when the fatty plaques of cholesterol and artery cloggingblood clots are formed in the lumen of the artery

The main aim of this work is to study these phenomenaobtain analytic expressions for axial velocity and shear stressand also study the effect ofmagnetic field (Hartmann number119867) and couple-stress parameter (120573) on the velocity and theeffects of Hartmann number on the fluid velocity Hence thepresent mathematical model gives a simple form of velocityexpression for the blood flow so that it will help not onlypeople working in the field of physiological fluid dynamicsbut also the medical practitioners

Journal of Applied Mathematics 3

2 Problem Formulation

Let us consider a two-dimensional steady flow of bloodthrough a rigid tapered stenosed tube by considering bloodas an electrically conducting incompressible couple-stressfluidThemagnetic field is acting along the radius of the tubeThe magnetic Reynolds number of the flow is assumed to besufficiently small that the inducedmagnetic and electric fieldscan be neglected [24] The catheter is assumed to be movingin the 119911-axis direction

Mathematical model of blood flow through a taperedstenosed arterial segment in the presence of a movingcatheter and a magnetic field is to be built to study theimpact of various geometric Hartman and fluid parameterson physiological parameters The geometry of the taperedstenosed artery is shown in Figures 2 and 3 and is expressedmathematically with inputs from [26] as

119877 (119911) =

(1198770+ 120577119911)(1 minus

120598119899119899(119899minus1)

(119899 minus 1) 119871119899

0

(119871119899minus1

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

119899)) 119889 le 119911 le 119889 + 119871

0

1198770+ 120577119911 otherwise

(1)

where 1198770is the radius of the annular region in case of

nontapered artery in the nonconstricted domain 119877119888is the

radius of the catheter 1198710is the stenosis length 119889 indicates the

location of stenosis 120598 is the maximum height of the stenosisinto the lumen and 120577 = tan120601 is the tapering parameter whichrepresents the slope of the tapered vessel with 120601 being thetapering angle 120601 lt 0 120601 gt 0 and 120601 = 0 are for convergingtaper diverging taper and no-taper respectively

The current density J is expressed by

J = 120590 (E + V times B) (2)

where E is the electric field intensity 120590 is the electrical con-ductivityB is themagnetic flux intensity andV is the velocityvector In themomentum equation the electromagnetic forceF119898is included and is defined as

F119898= J times B = 120590 (E + V times B) times B (3)

The conservation equations which govern the couple-stress fluid flow including a Lorentz force can be written inthe following form

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + (120582 + 120583) nablanabla sdot V + 120578Δnablanabla sdot V

minus 120578Δ2V + 120583ΔV + 120590 (V times B) times B + 120588f

+

1

2

nabla times (120588l)

(4)

where Δ is the Laplacian operator For couple-stress fluidshear stress tensor is not symmetric The force stress tensor120591 and the couple-stress tensor M that arises in the theory ofcouple-stress fluids are given by

120591 = (minus119901 + 120582nablaV) I + 120583 (nablaV + (nablaV)119879) + 12

I

times (nablaM + 120588l)

M = 119898I + 2120578 (nabla (nabla times V)) + 21205781015840 (nabla (nabla times V))119879

(5)

where 120588 is the density V is velocity vector 119901 is the pressure120582 and 120583 are the viscosity coefficients and 120578 and 1205781015840 are couple-stress coefficients f is a body force and l is a body couplemoment Further the materials constants 120583 120582 120578 and 1205781015840satisfy the following inequalities

120583 ge 0

3120582 + 2120583 ge 0

120578 ge 0

120578 ge 1205781015840

(6)

As the flow is steady and incompressible in the absenceof body force and body couple moment (4) reduce to

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + 120583ΔV minus 120578Δ2V + 120590 (V times B) times B(7)

The flow is considered to take place under the influence ofexternally applied magnetic field in axial direction 119911 Underthese assumptions the governing equations may be writtenin the cylindrical coordinates system as follows

Equation of continuity is

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0 (8)

Equation of radial momentum is

120588(119906

120597119906

120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 120583(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+

1205972119906

1205971199112) minus 120578(

1205974119906

1205971199034+ 2

1205974119906

12059711990321205971199112+

1205974119906

1205971199114+

2

119903

1205973119906

1205971199033

+

2

119903

1205973119906

1205971199031205971199112minus

3

1199032

1205972119906

1205971199032minus

2

1199032

1205972119906

1205971199112+

3

1199033

120597119906

120597119903

minus

3

1199034119906)

(9)

4 Journal of Applied Mathematics

L

120598

R0

Rc

n = 2n = 5n = 15

L0

d

Figure 2 2D view of a catheterized stenosed artery

L

R(z)

120598

120601 lt 0120601 = 0

120601 gt 0

Figure 3 2D view of a tapered stenosed artery

Equation of axial momentum is

120588(119906

120597119908

120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ 120583(

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+

1205972119908

1205971199112) minus 120578(

1205974119908

1205971199034+ 2

1205974119908

12059711990321205971199112+

1205974119908

1205971199114+

2

119903

1205973119908

1205971199033

+

2

119903

1205973119908

1205971199031205971199112minus

1

1199032

1205972119906

1205971199032+

1

1199033

120597119908

120597119903

) minus 1205901198612

0119908

(10)

where 1198610is the external transverse magnetic field The

nondimensional parameters are obtained as follows

119911lowast=

119911

1198710

119903lowast=

119903

1198770

119908lowast=

119908

1199080

120598lowast=

120598

1198770

120577lowast=

1205771198710

1198770

119906lowast=

1198710119906

11990801198770

119901lowast=

1199032

0119901

11990801198710120583

(11)

and then the dimensionless form of geometry (1) afterdropping the stars is obtained as follows

119877 (119911) =

(1 + 120577119911) (1 minus

120577119899119899(119899minus1)

(119899 minus 1)

((119911 minus 120574) minus (119911 minus 120574)119899)) 120574 le 119911 le 120574 + 1

1 + 120577119911 otherwise(12)

where 120574 = 11987101198711and 119908

0is a typical axial velocity

Also (8)ndash(10) on dropping the stars become

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0

Re1205853 (119906120597119906120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 1205852(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+ 1205852 1205972119906

1205971199112) minus

1205852

1205732[

1205974119906

1205971199034+

2

119903

1205973119906

1205971199033minus

3

1199032

1205972119906

1205971199032+

3

1199033

120597119906

120597119903

minus

3

1199034119906

+ 1205852(

2

119903

1205973119906

1205971199031205971199112+ 2

1205974119906

12059711990321205971199112+ 1205852 1205974119906

1205971199114minus

2

1199032

1205972119906

1205971199112)]

Re120585 (119906120597119908120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+ 1205852 1205972119908

1205971199112) minus

1

1205732[

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032

+

1

1199033

120597119908

120597119903

+ 1205854 1205974119908

1205971199114+ 21205852(

1205974119908

12059711990321205971199112+

1

119903

1205973119908

1205971199031205971199112)]

minus 1198672119908

(13)

Journal of Applied Mathematics 5

where 120585 = 11987701198710 Re = 120588119906

01198770120583 is the Reynolds number

1205732= 1198772

0120583120578 is the couple-stress fluid parameter and 1198672 =

1198612

01198772

0120590120583 is the Hartmann number

Under the assumption of mild stenosis that is 1205981198770le

1 and further assuming that 120585 = 11987701198710le 1 (13) get

transformed into120597119901

120597119903

= 0 (14)

120597119901

120597119911

= (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

)

minus

1

1205732(

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032+

1

1199033

120597119908

120597119903

)

minus 1198672119908

(15)

Equation (15) can be written as

120597119901

120597119911

= Δ119908 minus

1

1205732Δ2119908 minus119867

2119908 (16)

where

Δ =

1205972

1205971199032+

1

119903

120597

120597119903

(17)

It can be seen that the pressure variation depends only onthe axial variableThe pressure gradient 120597119901120597119911 is produced bythe pumping action of the heart

The corresponding nondimensional boundary conditionsare as shown below

119908 = V at 119903 = 119877 (119911)

119908 = 119892 at 119903 = 119877119888

1205972119908

1205971199032minus

120596

119903

120597119908

120597119903

= 0 at 119903 = 119877 (119911) 119903 = 119877119888

(18)

where V represents the slip velocity at the artery wall 119892is the velocity of the moving boundary (catheter) and120596 = 120578

1015840120578 is the parameter associated with the couple-stress

fluid No couple-stress effects will be present if 1205781015840 = 120578which is equivalent to saying that the couple-stress tensor issymmetric

3 Solution of the Problem

As a solution of (16) with the pressure gradient being taken asa constant take

minus

120597119901

120597119911

= 119896119904 (19)

Let 11989921+ 1198992

2= 1205732 and 1198992

11198992

2= 12057321198672 which are given by

1198992

1=

1205732+ radic1205734minus 412057321198672

2

(20)

1198992

2=

1205732minus radic1205734minus 412057321198672

2

(21)

then (16) is simplified to the form

(Δ minus 1198992

1) (Δ minus 119899

2

2)119908 = 120573

2119896119904 (22)

The solution of the above equation is obtained as

119908 = 1198881(119911) 1198680(1198991119903) + 1198882(119911)1198700(1198991119903) + 1198883(119911) 1198680(1198992119903)

+ 1198884 (119911)1198700 (1198992119903) +

1

1198672119896119904

(23)

where 1198680and 119870

0are modified Bessel function of order zero

first and second kinds respectively 119888119894for 119894 = 1 2 3 4

are calculated numerically by solving the algebraic systemobtained from the boundary conditions

Volumetric flow rate 119876 in steady flow through a tubeis obtained by integrating the velocity profile over a crosssection of the tubeThe nondimensional volumetric flow rate119876 across the radial distance is expressed as

119876 = int

119877(119911)

119877119888

2119903119908119889119903 (24)

which is obtained in the form 119876 = minus2119896119904119865(119877119888 119877(119911)) where

119865 (119877119888 119877 (119911)) =

1198891

1198991

(1198771198681(1198991119877) minus 119877

1198881198681(1198991119877119888))

+

1198892

1198991

(1198771198881198701(1198991119877119888) minus 119877119870

1(1198991119877))

+

1198893

1198992

(1198771198681(1198992119877) minus 119877

1198881198681(1198992119877119888))

+

1198894

1198992

(1198771198881198701(1198992119877119888) minus 119877119870

1(1198992119877))

+

1198772minus 1198772

119888

21198672

(25)

where 119889119894= minus119888119894119896119904 for 119894 = 1 2 3 4 The pressure drop Δ119901

across the stenosis between the sections 119911 = 0 and 119911 = 119871 isobtained by

Δ119901 =

119876

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911 (26)

The resistance to the flow (impedance) is obtained from

119877119891=

Δ119901

119876

=

1

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911

=

1

2

(int

119889

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

119889+1198710

119889

1

119865 (119877119888 119877 (119911))

119889119911

+ int

119871

119889+1198710

1

119865 (119877119888 119877 (119911))

119889119911)

(27)

The dimensionless form of (20) is

119877119891=

Γ

2

(int

120574

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

120574+1

120574

1

119865 (119877119888 119877 (119911))

119889119911

+ int

1120574

120574+1

1

119865 (119877119888 119877 (119911))

119889119911)

(28)

where Γ = 1198710119889

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

2 Problem Formulation

Let us consider a two-dimensional steady flow of bloodthrough a rigid tapered stenosed tube by considering bloodas an electrically conducting incompressible couple-stressfluidThemagnetic field is acting along the radius of the tubeThe magnetic Reynolds number of the flow is assumed to besufficiently small that the inducedmagnetic and electric fieldscan be neglected [24] The catheter is assumed to be movingin the 119911-axis direction

Mathematical model of blood flow through a taperedstenosed arterial segment in the presence of a movingcatheter and a magnetic field is to be built to study theimpact of various geometric Hartman and fluid parameterson physiological parameters The geometry of the taperedstenosed artery is shown in Figures 2 and 3 and is expressedmathematically with inputs from [26] as

119877 (119911) =

(1198770+ 120577119911)(1 minus

120598119899119899(119899minus1)

(119899 minus 1) 119871119899

0

(119871119899minus1

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

119899)) 119889 le 119911 le 119889 + 119871

0

1198770+ 120577119911 otherwise

(1)

where 1198770is the radius of the annular region in case of

nontapered artery in the nonconstricted domain 119877119888is the

radius of the catheter 1198710is the stenosis length 119889 indicates the

location of stenosis 120598 is the maximum height of the stenosisinto the lumen and 120577 = tan120601 is the tapering parameter whichrepresents the slope of the tapered vessel with 120601 being thetapering angle 120601 lt 0 120601 gt 0 and 120601 = 0 are for convergingtaper diverging taper and no-taper respectively

The current density J is expressed by

J = 120590 (E + V times B) (2)

where E is the electric field intensity 120590 is the electrical con-ductivityB is themagnetic flux intensity andV is the velocityvector In themomentum equation the electromagnetic forceF119898is included and is defined as

F119898= J times B = 120590 (E + V times B) times B (3)

The conservation equations which govern the couple-stress fluid flow including a Lorentz force can be written inthe following form

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + (120582 + 120583) nablanabla sdot V + 120578Δnablanabla sdot V

minus 120578Δ2V + 120583ΔV + 120590 (V times B) times B + 120588f

+

1

2

nabla times (120588l)

(4)

where Δ is the Laplacian operator For couple-stress fluidshear stress tensor is not symmetric The force stress tensor120591 and the couple-stress tensor M that arises in the theory ofcouple-stress fluids are given by

120591 = (minus119901 + 120582nablaV) I + 120583 (nablaV + (nablaV)119879) + 12

I

times (nablaM + 120588l)

M = 119898I + 2120578 (nabla (nabla times V)) + 21205781015840 (nabla (nabla times V))119879

(5)

where 120588 is the density V is velocity vector 119901 is the pressure120582 and 120583 are the viscosity coefficients and 120578 and 1205781015840 are couple-stress coefficients f is a body force and l is a body couplemoment Further the materials constants 120583 120582 120578 and 1205781015840satisfy the following inequalities

120583 ge 0

3120582 + 2120583 ge 0

120578 ge 0

120578 ge 1205781015840

(6)

As the flow is steady and incompressible in the absenceof body force and body couple moment (4) reduce to

nabla sdot V = 0

120588 (V sdot nabla)V = minusnabla119901 + 120583ΔV minus 120578Δ2V + 120590 (V times B) times B(7)

The flow is considered to take place under the influence ofexternally applied magnetic field in axial direction 119911 Underthese assumptions the governing equations may be writtenin the cylindrical coordinates system as follows

Equation of continuity is

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0 (8)

Equation of radial momentum is

120588(119906

120597119906

120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 120583(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+

1205972119906

1205971199112) minus 120578(

1205974119906

1205971199034+ 2

1205974119906

12059711990321205971199112+

1205974119906

1205971199114+

2

119903

1205973119906

1205971199033

+

2

119903

1205973119906

1205971199031205971199112minus

3

1199032

1205972119906

1205971199032minus

2

1199032

1205972119906

1205971199112+

3

1199033

120597119906

120597119903

minus

3

1199034119906)

(9)

4 Journal of Applied Mathematics

L

120598

R0

Rc

n = 2n = 5n = 15

L0

d

Figure 2 2D view of a catheterized stenosed artery

L

R(z)

120598

120601 lt 0120601 = 0

120601 gt 0

Figure 3 2D view of a tapered stenosed artery

Equation of axial momentum is

120588(119906

120597119908

120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ 120583(

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+

1205972119908

1205971199112) minus 120578(

1205974119908

1205971199034+ 2

1205974119908

12059711990321205971199112+

1205974119908

1205971199114+

2

119903

1205973119908

1205971199033

+

2

119903

1205973119908

1205971199031205971199112minus

1

1199032

1205972119906

1205971199032+

1

1199033

120597119908

120597119903

) minus 1205901198612

0119908

(10)

where 1198610is the external transverse magnetic field The

nondimensional parameters are obtained as follows

119911lowast=

119911

1198710

119903lowast=

119903

1198770

119908lowast=

119908

1199080

120598lowast=

120598

1198770

120577lowast=

1205771198710

1198770

119906lowast=

1198710119906

11990801198770

119901lowast=

1199032

0119901

11990801198710120583

(11)

and then the dimensionless form of geometry (1) afterdropping the stars is obtained as follows

119877 (119911) =

(1 + 120577119911) (1 minus

120577119899119899(119899minus1)

(119899 minus 1)

((119911 minus 120574) minus (119911 minus 120574)119899)) 120574 le 119911 le 120574 + 1

1 + 120577119911 otherwise(12)

where 120574 = 11987101198711and 119908

0is a typical axial velocity

Also (8)ndash(10) on dropping the stars become

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0

Re1205853 (119906120597119906120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 1205852(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+ 1205852 1205972119906

1205971199112) minus

1205852

1205732[

1205974119906

1205971199034+

2

119903

1205973119906

1205971199033minus

3

1199032

1205972119906

1205971199032+

3

1199033

120597119906

120597119903

minus

3

1199034119906

+ 1205852(

2

119903

1205973119906

1205971199031205971199112+ 2

1205974119906

12059711990321205971199112+ 1205852 1205974119906

1205971199114minus

2

1199032

1205972119906

1205971199112)]

Re120585 (119906120597119908120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+ 1205852 1205972119908

1205971199112) minus

1

1205732[

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032

+

1

1199033

120597119908

120597119903

+ 1205854 1205974119908

1205971199114+ 21205852(

1205974119908

12059711990321205971199112+

1

119903

1205973119908

1205971199031205971199112)]

minus 1198672119908

(13)

Journal of Applied Mathematics 5

where 120585 = 11987701198710 Re = 120588119906

01198770120583 is the Reynolds number

1205732= 1198772

0120583120578 is the couple-stress fluid parameter and 1198672 =

1198612

01198772

0120590120583 is the Hartmann number

Under the assumption of mild stenosis that is 1205981198770le

1 and further assuming that 120585 = 11987701198710le 1 (13) get

transformed into120597119901

120597119903

= 0 (14)

120597119901

120597119911

= (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

)

minus

1

1205732(

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032+

1

1199033

120597119908

120597119903

)

minus 1198672119908

(15)

Equation (15) can be written as

120597119901

120597119911

= Δ119908 minus

1

1205732Δ2119908 minus119867

2119908 (16)

where

Δ =

1205972

1205971199032+

1

119903

120597

120597119903

(17)

It can be seen that the pressure variation depends only onthe axial variableThe pressure gradient 120597119901120597119911 is produced bythe pumping action of the heart

The corresponding nondimensional boundary conditionsare as shown below

119908 = V at 119903 = 119877 (119911)

119908 = 119892 at 119903 = 119877119888

1205972119908

1205971199032minus

120596

119903

120597119908

120597119903

= 0 at 119903 = 119877 (119911) 119903 = 119877119888

(18)

where V represents the slip velocity at the artery wall 119892is the velocity of the moving boundary (catheter) and120596 = 120578

1015840120578 is the parameter associated with the couple-stress

fluid No couple-stress effects will be present if 1205781015840 = 120578which is equivalent to saying that the couple-stress tensor issymmetric

3 Solution of the Problem

As a solution of (16) with the pressure gradient being taken asa constant take

minus

120597119901

120597119911

= 119896119904 (19)

Let 11989921+ 1198992

2= 1205732 and 1198992

11198992

2= 12057321198672 which are given by

1198992

1=

1205732+ radic1205734minus 412057321198672

2

(20)

1198992

2=

1205732minus radic1205734minus 412057321198672

2

(21)

then (16) is simplified to the form

(Δ minus 1198992

1) (Δ minus 119899

2

2)119908 = 120573

2119896119904 (22)

The solution of the above equation is obtained as

119908 = 1198881(119911) 1198680(1198991119903) + 1198882(119911)1198700(1198991119903) + 1198883(119911) 1198680(1198992119903)

+ 1198884 (119911)1198700 (1198992119903) +

1

1198672119896119904

(23)

where 1198680and 119870

0are modified Bessel function of order zero

first and second kinds respectively 119888119894for 119894 = 1 2 3 4

are calculated numerically by solving the algebraic systemobtained from the boundary conditions

Volumetric flow rate 119876 in steady flow through a tubeis obtained by integrating the velocity profile over a crosssection of the tubeThe nondimensional volumetric flow rate119876 across the radial distance is expressed as

119876 = int

119877(119911)

119877119888

2119903119908119889119903 (24)

which is obtained in the form 119876 = minus2119896119904119865(119877119888 119877(119911)) where

119865 (119877119888 119877 (119911)) =

1198891

1198991

(1198771198681(1198991119877) minus 119877

1198881198681(1198991119877119888))

+

1198892

1198991

(1198771198881198701(1198991119877119888) minus 119877119870

1(1198991119877))

+

1198893

1198992

(1198771198681(1198992119877) minus 119877

1198881198681(1198992119877119888))

+

1198894

1198992

(1198771198881198701(1198992119877119888) minus 119877119870

1(1198992119877))

+

1198772minus 1198772

119888

21198672

(25)

where 119889119894= minus119888119894119896119904 for 119894 = 1 2 3 4 The pressure drop Δ119901

across the stenosis between the sections 119911 = 0 and 119911 = 119871 isobtained by

Δ119901 =

119876

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911 (26)

The resistance to the flow (impedance) is obtained from

119877119891=

Δ119901

119876

=

1

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911

=

1

2

(int

119889

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

119889+1198710

119889

1

119865 (119877119888 119877 (119911))

119889119911

+ int

119871

119889+1198710

1

119865 (119877119888 119877 (119911))

119889119911)

(27)

The dimensionless form of (20) is

119877119891=

Γ

2

(int

120574

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

120574+1

120574

1

119865 (119877119888 119877 (119911))

119889119911

+ int

1120574

120574+1

1

119865 (119877119888 119877 (119911))

119889119911)

(28)

where Γ = 1198710119889

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

L

120598

R0

Rc

n = 2n = 5n = 15

L0

d

Figure 2 2D view of a catheterized stenosed artery

L

R(z)

120598

120601 lt 0120601 = 0

120601 gt 0

Figure 3 2D view of a tapered stenosed artery

Equation of axial momentum is

120588(119906

120597119908

120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ 120583(

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+

1205972119908

1205971199112) minus 120578(

1205974119908

1205971199034+ 2

1205974119908

12059711990321205971199112+

1205974119908

1205971199114+

2

119903

1205973119908

1205971199033

+

2

119903

1205973119908

1205971199031205971199112minus

1

1199032

1205972119906

1205971199032+

1

1199033

120597119908

120597119903

) minus 1205901198612

0119908

(10)

where 1198610is the external transverse magnetic field The

nondimensional parameters are obtained as follows

119911lowast=

119911

1198710

119903lowast=

119903

1198770

119908lowast=

119908

1199080

120598lowast=

120598

1198770

120577lowast=

1205771198710

1198770

119906lowast=

1198710119906

11990801198770

119901lowast=

1199032

0119901

11990801198710120583

(11)

and then the dimensionless form of geometry (1) afterdropping the stars is obtained as follows

119877 (119911) =

(1 + 120577119911) (1 minus

120577119899119899(119899minus1)

(119899 minus 1)

((119911 minus 120574) minus (119911 minus 120574)119899)) 120574 le 119911 le 120574 + 1

1 + 120577119911 otherwise(12)

where 120574 = 11987101198711and 119908

0is a typical axial velocity

Also (8)ndash(10) on dropping the stars become

120597119906

120597119903

+

119906

119903

+

120597119908

120597119911

= 0

Re1205853 (119906120597119906120597119903

+ 119908

120597119906

120597119911

) = minus

120597119901

120597119903

+ 1205852(

1205972119906

1205971199032+

1

119903

120597119906

120597119903

minus

119906

1199032

+ 1205852 1205972119906

1205971199112) minus

1205852

1205732[

1205974119906

1205971199034+

2

119903

1205973119906

1205971199033minus

3

1199032

1205972119906

1205971199032+

3

1199033

120597119906

120597119903

minus

3

1199034119906

+ 1205852(

2

119903

1205973119906

1205971199031205971199112+ 2

1205974119906

12059711990321205971199112+ 1205852 1205974119906

1205971199114minus

2

1199032

1205972119906

1205971199112)]

Re120585 (119906120597119908120597119903

+ 119908

120597119908

120597119911

) = minus

120597119901

120597119911

+ (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

+ 1205852 1205972119908

1205971199112) minus

1

1205732[

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032

+

1

1199033

120597119908

120597119903

+ 1205854 1205974119908

1205971199114+ 21205852(

1205974119908

12059711990321205971199112+

1

119903

1205973119908

1205971199031205971199112)]

minus 1198672119908

(13)

Journal of Applied Mathematics 5

where 120585 = 11987701198710 Re = 120588119906

01198770120583 is the Reynolds number

1205732= 1198772

0120583120578 is the couple-stress fluid parameter and 1198672 =

1198612

01198772

0120590120583 is the Hartmann number

Under the assumption of mild stenosis that is 1205981198770le

1 and further assuming that 120585 = 11987701198710le 1 (13) get

transformed into120597119901

120597119903

= 0 (14)

120597119901

120597119911

= (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

)

minus

1

1205732(

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032+

1

1199033

120597119908

120597119903

)

minus 1198672119908

(15)

Equation (15) can be written as

120597119901

120597119911

= Δ119908 minus

1

1205732Δ2119908 minus119867

2119908 (16)

where

Δ =

1205972

1205971199032+

1

119903

120597

120597119903

(17)

It can be seen that the pressure variation depends only onthe axial variableThe pressure gradient 120597119901120597119911 is produced bythe pumping action of the heart

The corresponding nondimensional boundary conditionsare as shown below

119908 = V at 119903 = 119877 (119911)

119908 = 119892 at 119903 = 119877119888

1205972119908

1205971199032minus

120596

119903

120597119908

120597119903

= 0 at 119903 = 119877 (119911) 119903 = 119877119888

(18)

where V represents the slip velocity at the artery wall 119892is the velocity of the moving boundary (catheter) and120596 = 120578

1015840120578 is the parameter associated with the couple-stress

fluid No couple-stress effects will be present if 1205781015840 = 120578which is equivalent to saying that the couple-stress tensor issymmetric

3 Solution of the Problem

As a solution of (16) with the pressure gradient being taken asa constant take

minus

120597119901

120597119911

= 119896119904 (19)

Let 11989921+ 1198992

2= 1205732 and 1198992

11198992

2= 12057321198672 which are given by

1198992

1=

1205732+ radic1205734minus 412057321198672

2

(20)

1198992

2=

1205732minus radic1205734minus 412057321198672

2

(21)

then (16) is simplified to the form

(Δ minus 1198992

1) (Δ minus 119899

2

2)119908 = 120573

2119896119904 (22)

The solution of the above equation is obtained as

119908 = 1198881(119911) 1198680(1198991119903) + 1198882(119911)1198700(1198991119903) + 1198883(119911) 1198680(1198992119903)

+ 1198884 (119911)1198700 (1198992119903) +

1

1198672119896119904

(23)

where 1198680and 119870

0are modified Bessel function of order zero

first and second kinds respectively 119888119894for 119894 = 1 2 3 4

are calculated numerically by solving the algebraic systemobtained from the boundary conditions

Volumetric flow rate 119876 in steady flow through a tubeis obtained by integrating the velocity profile over a crosssection of the tubeThe nondimensional volumetric flow rate119876 across the radial distance is expressed as

119876 = int

119877(119911)

119877119888

2119903119908119889119903 (24)

which is obtained in the form 119876 = minus2119896119904119865(119877119888 119877(119911)) where

119865 (119877119888 119877 (119911)) =

1198891

1198991

(1198771198681(1198991119877) minus 119877

1198881198681(1198991119877119888))

+

1198892

1198991

(1198771198881198701(1198991119877119888) minus 119877119870

1(1198991119877))

+

1198893

1198992

(1198771198681(1198992119877) minus 119877

1198881198681(1198992119877119888))

+

1198894

1198992

(1198771198881198701(1198992119877119888) minus 119877119870

1(1198992119877))

+

1198772minus 1198772

119888

21198672

(25)

where 119889119894= minus119888119894119896119904 for 119894 = 1 2 3 4 The pressure drop Δ119901

across the stenosis between the sections 119911 = 0 and 119911 = 119871 isobtained by

Δ119901 =

119876

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911 (26)

The resistance to the flow (impedance) is obtained from

119877119891=

Δ119901

119876

=

1

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911

=

1

2

(int

119889

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

119889+1198710

119889

1

119865 (119877119888 119877 (119911))

119889119911

+ int

119871

119889+1198710

1

119865 (119877119888 119877 (119911))

119889119911)

(27)

The dimensionless form of (20) is

119877119891=

Γ

2

(int

120574

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

120574+1

120574

1

119865 (119877119888 119877 (119911))

119889119911

+ int

1120574

120574+1

1

119865 (119877119888 119877 (119911))

119889119911)

(28)

where Γ = 1198710119889

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

where 120585 = 11987701198710 Re = 120588119906

01198770120583 is the Reynolds number

1205732= 1198772

0120583120578 is the couple-stress fluid parameter and 1198672 =

1198612

01198772

0120590120583 is the Hartmann number

Under the assumption of mild stenosis that is 1205981198770le

1 and further assuming that 120585 = 11987701198710le 1 (13) get

transformed into120597119901

120597119903

= 0 (14)

120597119901

120597119911

= (

1205972119908

1205971199032+

1

119903

120597119908

120597119903

)

minus

1

1205732(

1205974119908

1205971199034+

2

119903

1205973119908

1205971199033minus

1

1199032

1205972119908

1205971199032+

1

1199033

120597119908

120597119903

)

minus 1198672119908

(15)

Equation (15) can be written as

120597119901

120597119911

= Δ119908 minus

1

1205732Δ2119908 minus119867

2119908 (16)

where

Δ =

1205972

1205971199032+

1

119903

120597

120597119903

(17)

It can be seen that the pressure variation depends only onthe axial variableThe pressure gradient 120597119901120597119911 is produced bythe pumping action of the heart

The corresponding nondimensional boundary conditionsare as shown below

119908 = V at 119903 = 119877 (119911)

119908 = 119892 at 119903 = 119877119888

1205972119908

1205971199032minus

120596

119903

120597119908

120597119903

= 0 at 119903 = 119877 (119911) 119903 = 119877119888

(18)

where V represents the slip velocity at the artery wall 119892is the velocity of the moving boundary (catheter) and120596 = 120578

1015840120578 is the parameter associated with the couple-stress

fluid No couple-stress effects will be present if 1205781015840 = 120578which is equivalent to saying that the couple-stress tensor issymmetric

3 Solution of the Problem

As a solution of (16) with the pressure gradient being taken asa constant take

minus

120597119901

120597119911

= 119896119904 (19)

Let 11989921+ 1198992

2= 1205732 and 1198992

11198992

2= 12057321198672 which are given by

1198992

1=

1205732+ radic1205734minus 412057321198672

2

(20)

1198992

2=

1205732minus radic1205734minus 412057321198672

2

(21)

then (16) is simplified to the form

(Δ minus 1198992

1) (Δ minus 119899

2

2)119908 = 120573

2119896119904 (22)

The solution of the above equation is obtained as

119908 = 1198881(119911) 1198680(1198991119903) + 1198882(119911)1198700(1198991119903) + 1198883(119911) 1198680(1198992119903)

+ 1198884 (119911)1198700 (1198992119903) +

1

1198672119896119904

(23)

where 1198680and 119870

0are modified Bessel function of order zero

first and second kinds respectively 119888119894for 119894 = 1 2 3 4

are calculated numerically by solving the algebraic systemobtained from the boundary conditions

Volumetric flow rate 119876 in steady flow through a tubeis obtained by integrating the velocity profile over a crosssection of the tubeThe nondimensional volumetric flow rate119876 across the radial distance is expressed as

119876 = int

119877(119911)

119877119888

2119903119908119889119903 (24)

which is obtained in the form 119876 = minus2119896119904119865(119877119888 119877(119911)) where

119865 (119877119888 119877 (119911)) =

1198891

1198991

(1198771198681(1198991119877) minus 119877

1198881198681(1198991119877119888))

+

1198892

1198991

(1198771198881198701(1198991119877119888) minus 119877119870

1(1198991119877))

+

1198893

1198992

(1198771198681(1198992119877) minus 119877

1198881198681(1198992119877119888))

+

1198894

1198992

(1198771198881198701(1198992119877119888) minus 119877119870

1(1198992119877))

+

1198772minus 1198772

119888

21198672

(25)

where 119889119894= minus119888119894119896119904 for 119894 = 1 2 3 4 The pressure drop Δ119901

across the stenosis between the sections 119911 = 0 and 119911 = 119871 isobtained by

Δ119901 =

119876

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911 (26)

The resistance to the flow (impedance) is obtained from

119877119891=

Δ119901

119876

=

1

2

int

119871

0

1

119865 (119877119888 119877 (119911))

119889119911

=

1

2

(int

119889

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

119889+1198710

119889

1

119865 (119877119888 119877 (119911))

119889119911

+ int

119871

119889+1198710

1

119865 (119877119888 119877 (119911))

119889119911)

(27)

The dimensionless form of (20) is

119877119891=

Γ

2

(int

120574

0

1

119865 (119877119888 119877 (119911))

119889119911 + int

120574+1

120574

1

119865 (119877119888 119877 (119911))

119889119911

+ int

1120574

120574+1

1

119865 (119877119888 119877 (119911))

119889119911)

(28)

where Γ = 1198710119889

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

The shear stress 120591119903119911is calculated using the expression

120591119903119911= 120591119878

119903119911+ 120591119860

119903119911 (29)

where the symmetric and skew-symmetric parts of the stressare given by

120591119878

119903119911= 120583

120597119908

120597119903

(30)

120591119860

119903119911= (

120597119898119903120579

120597119903

+

119898119903120579+ 119898120579119903

119903

) (31)

respectively Here

119898119903120579=

1

4

(

1205781015840

119903

120597119908

120597119903

minus 120578

1205972119908

1205971199032)

119898120579119903=

1

4

(

120578

119903

120597119908

120597119903

minus 1205781015840 1205972119908

1205971199032)

(32)

Hence the dimensionless shear stress for the artery isgiven by

120591119903119911

= 1198991(11988811198681(1198991119903) minus 11988821198701(1198991119903)) (1 minus

1198992

1

4 (1198992

1+ 1198992

2)

)

+ 1198992(11988831198681(1198992119903) minus 11988841198701(1198992119903)) (1 minus

1198992

2

4 (1198992

1+ 1198992

2)

)

(33)

Thus the shear stress at the wall can be computed from(33) by taking 119903 = 119877(119911)

4 Results and Discussion

The study of blood flow through catheterized stenosedtapered artery with the presence of a transverse magneticfield involves the integration of various geometric and fluidvariables which influences the physiological parameters suchas the fluid velocity rate flow and wall shear stress Closedform solutions are obtained in terms of modified BesselrsquosfunctionsThe physiological dimensionless quantities such asthe fluid velocity in the stenosis region and the wall shearstress at the maximum height of the stenosis are computednumerically for various values of the fluid and geometricparameters using the program MATLAB The parametersconsidered are 119899 (shape parameter) 120577 (tapered parameter)119877119888(catheter radius) Γ (stenosis length) 120573 (couple-stress fluid

parameter)119867 (the Hartmann number) V (slip velocity) and119892 (catheter wall velocity) The obtained results are analyzedgraphically The results obtained in this study are in goodagreement with those reported in the literature

Figures 4ndash6 illustrate the variation of axial velocity profilefor different values of 119899 120577 and 120573 It is observed from Figure 4that the velocity profile increases for increasing the shapeparameter and that actually this occurred due to the changeof the stenosis height It is observed from Figure 5 that thevelocity increases by the increase in the tapered parameter

w

0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

n = 2n = 5n = 15

Figure 4 Variation of axial velocity119908with respect to 119899when 120577 = 0120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

001 0015 002 0025 003 0035 004 0045 005

r

01

02

03

04

05

06

07

08

120577 = minus005120577 = 000120577 = 005

Figure 5 Variation of axial velocity119908 with respect to 120577 when 119899 = 5120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

120577 and the same goes for the couple-stress fluid parameter 120573Figure 6 It is to be noted as120573 rarr infin the properties of couple-stress in the fluid vanish and hence behave like a Newtonianfluid Hence it is understood that the velocity is low in couple-stress fluid when compared to that of Newtonian fluid

In general from Figure 7 it can be observed that theaxial velocity is decreasing The effect of the Hartmannnumber 119867 on the axial velocity is shown in Figure 8 Byobserving Figure 8 the fluid velocity is decreasing as theHartmann number is increasing Further as 119867 rarr 0 theHartmann number loses its properties and behaves like anormal blood flow without any magnetic field applied to itWe have concluded that the application of magnetic fieldreduces the speed of blood flow and that meets the results ofPonalagusamy and Tamil Selvi [25] also indicating that the

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

w

001 0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

120573 = 5120573 = 6120573 = 7

Figure 6 Variation of axial velocity119908with respect to 120573when 119899 = 5120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

w

002 003 004 005 006

r

01

02

03

04

05

06

07

08

09

1

120598 = 00

120598 = 01

120598 = 02

Figure 7 Variation of axial velocity119908 with respect to 120598 when 119899 = 5120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

wall shear stress increases when we increase the Hartmannnumber The influence of catheter radius 119877

119888on the axial

velocity is directly proportional as shown in Figure 9 Fromthe above result we understand that as the catheter radiusincreases the annular region gets narrowed which leads tothe rise in the obstruction to the flow

The axial velocity of the blood is high in case of high slipvelocity V or catheter velocity 119892 it is observed that the slipvelocity at the boundary facilitates the fluid flow and the samegoes for the moving catheter Figures 10 and 11

The shear stress at the wall is a significant physiologicalparameter to be considered in the blood flow study Precisepredictions of the distribution of the shear stress at the wallare particularly useful in assimilating the effect of blood flowin arteries in general The shear stress at the wall is calculated

w

001 0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

H = 10H = 15H = 20

Figure 8 Variation of axial velocity119908with respect to119867when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

w

0015 002 0025 003 0035 004 0045

r

01

02

03

04

05

06

07

08

Rc = 01Rc = 02Rc = 03

Figure 9 Variation of axial velocity119908with respect to119877119888when 119899 = 5

120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

at the maximum height of the stenosis When we increasethe height of the stenosis 120598 the wall shear stress increasesFigure 15 Furthermore it is observed that the shear stressalong the wall is reaching the maximum at the throat of thestenosis

The effect of the Hartmann number 119867 on the shearstress is shown in Figure 16 As observed from Figure 12the shear stress at the wall is independent of the shapeparameter 119899 as pointed out by Ponalagusamy and Tamil Selvi[25] The variation of the shear stress at the wall as thetapered parameter increases is depicted in Figure 13Here theshear stress at the wall is more significant in the untaperedartery compared with the tapered artery which is divergingFurther the converging tapered artery possesses higher wall

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

= 001 = 002 = 003

Figure 10 Variation of axial velocity119908with respect to Vwhen 119899 = 5120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

w

0015 002 0025 003 0035 004 0045 005 0055

r

01

02

03

04

05

06

07

08

g = 003

g = 004g = 005

Figure 11 Variation of axial velocity119908with respect to 119892when 119899 = 5120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

shear stress than the untapered and tapered artery which isdiverging

The slip velocity at the wall of the stenosed artery andthe moving catheter significantly influence the shear stressat the wall which is noticed from Figures 18 and 19 Here itis observed that gain in slip velocity (resp moving catheter)reduces the shear stress at the wall The influence of catheterradius on shear stress at the wall is shown in Figure 17 Theincreasing catheter radius narrows down the lumen of theartery thus resulting in the higher values of shear stress atthe wall As couple-stress fluid parameter 120573 increases theshear stress at the wall decreases This behavior is shown inFigure 14 It is observed that the theoretical distribution ofshear stress along the wall reaches amaximum at throat of thestenosis and then rapidly decreases in the diverging section

z

0 05 1 15 2

120591

n = 2n = 5n = 15

minus0155

minus015

minus016

minus017

minus0175

minus0165

minus018

minus0185

minus019

minus0195

Figure 12 Variation of wall shear stress 119908 with respect to 119899 when120577 = 0 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

120591

120577 = 000

120577 = 005

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

minus014

120577 = minus005

Figure 13 Variation of wall shear stress 119908 with respect to 120577 when119899 = 5 120598 = 02 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

This behavior is the same in terms of the change in both theradius of the catheter119877

119888and the couple-stress fluid parameter

120573whose behavior is depicted graphically in Figures 17 and 14respectively

5 Conclusion

A mathematical model has been built to discuss the flow ofblood through a catheterized asymmetric tapered stenosedartery with slip velocity at the stenosed wall and a movingcatheter Closed form solution is obtained and the effects ofvarious geometric fluid parameters andmagnetic field on theaxial velocity of the blood and the shear stress at the wall arestudied

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

z

0 05 1 15 2

120591

120573 = 5

120573 = 6

120573 = 7

minus015

minus016

minus017

minus018

minus019

minus02

minus021

minus022

Figure 14 Variation of wall shear stress 119908 with respect to 120573 when119899 = 5 120598 = 02 120577 = 0119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

Wall shear stress

120598 = 01120598 = 02120598 = 03

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

Figure 15 Variation of wall shear stress 119908 with respect to 120598 when119899 = 5 120577 = 0 120573 = 5119867 = 1 V = 001 119892 = 003 and 119877

119888= 01

There is special importance of couple-stress fluids com-pared to the Newtonian fluids because of their wide existencesuch as oil blood and polymeric solutions In view of whatis mentioned above an analytic approach was followed tosolve themathematicalmodel of blood flow through stenosedtapered artery under the assumption of mild stenosis Theresultant observations are summarized as follows

(i) As the height and the stenosis length are increasingthe obstruction to the flow of blood is increasing

(ii) Converging tapered artery hasmore shear stress at thewall than the nontapered and the diverging taperedartery

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

H = 10H = 15H = 20

Figure 16 Variation of wall shear stress 119908 with respect to 119867 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and 119877

119888= 01

z

0 05 1 15 2

minus019

120591minus015

minus016

minus017

minus018

minus014

minus013

minus012

Rc = 01

Rc = 02Rc = 03

Figure 17 Variation of wall shear stress 119908 with respect to 119877119888when

119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119892 = 003 and119867 = 1

(iii) Diverging tapered artery has least shear stress at thewall

(iv) The increment in theHartmannnumber enhances theblood velocity and wall shear stress

(v) The axial velocity is decreasingwhile the couple-stressfluid parameters 120573 and the height of the stenosis 120598 areincreasing

(vi) Three different values of the slip velocity and thevelocity of the catheter at the arterial boundary andthe catheter wall are considered they are showing asignificant influence on the shear stress at the wall andthe axial velocity

The modeling and simulation of the above phenomenaare very realistic and are expected to be very useful in

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

z

0 05 1 15 2

minus019

120591

minus015

minus016

minus017

minus018

minus014

minus013

= 001

= 002

= 003

Figure 18 Variation of wall shear stress 119908 with respect to V when119899 = 5 120577 = 0 120573 = 5 120598 = 02 119877

119888= 01 119892 = 003 and119867 = 1

z

0 05 1 15 2

120591

minus019

minus02

minus015

minus016

minus017

minus018

g = 003

g = 004

g = 005

Figure 19 Variation of wall shear stress 119908 with respect to 119892 when119899 = 5 120577 = 0 120573 = 5 120598 = 02 V = 001 119877

119888= 01 and119867 = 1

predicting the behavior of physiological parameters in thediagnosis of various arterial diseases

Symbols

B Magnetic flux intensity1198610 External transverse

magnetic field119888119894 119894 = 1 4 Constants of integration119889 Location of the stenosis119889119894= minus119888119894119896119904 119894 = 1 4 Defined constants

E Electric field intensityf Body forceF119898 Electromagnetic force

119892 Velocity of the movingcatheter

119867 = 119861radic1198772

0120590120583 Hartmann number

1198680 Modified Bessel function of

the first kind and orderzero

J Current density1198700 Modified Bessel function of

the second kind and orderzero

l Body couple moment1198710 Stenosis length

M Couple-stress tensor

1198991= radic(120573

2+ radic1205734minus 412057321198672)2 Defined constant

1198992= radic(120573

2minus radic1205734minus 412057321198672)2 Defined constant

119901 Pressure119876 Volumetric flow rate119877119888 Radius of the

nonconstricted regionRe = 120588119906

01198770120583 Reynolds number

119877(119911) Radius of the tube119906 Radial velocityV Velocity vectorV Slip velocity119908 Axial velocity1199080 Typical axial velocity

120573 = 1198770radic120583120578 Couple-stress fluid

parameterΔ = 120597

21205971199032+ (1119903)(120597120597119903) Laplacian operator

120598 Maximum height of thestenosis

120578 1205781015840 Couple-stress coefficients

Γ = 1198710119889 Defined constant

120574 = 11987101198711 Defined constant

119877119891 Resistance to the flow

120582 120583 Viscosity coefficients120596 = 120578

1015840120578 Parameter associated with

the couple-stress fluid120601 Tapering angle120588 Density of the fluid120590 Electrical conductivity120591 Shear stress tensor120591119903119911 Shear stress

120591119860

119903119911 Skew-symmetric part of

shear stress120591119878

119903119911 Symmetric part of shear

stress120585 = 11987701198710 Defined constant

120577 Tapering parameter

Conflict of Interests

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

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

References

[1] R C Diggery andD T GrintCatheters Types Applications andPotential Complications Nova Science 2012

[2] D Filgueiras-Rama A Estrada J Shachar et al ldquoRemotemagnetic navigation for accurate real-time catheter positioningand ablation in cardiac electrophysiology proceduresrdquo Journalof Visualized Experiments no 74 Article ID e3658 2013

[3] B Mols ldquoMagnetic-field navigation for cathetersrdquo TU DelftOutlook University Magazine 2004

[4] V K StokesTheories of Fluids withMicrostructure An Introduc-tion Springer Berlin Germany 1984

[5] A C Eringen Theory of Micropolar Fluids no RR-27 Pur-due University School of Aeronautics and Astronautics WestLafayette Ind USA 1965

[6] S C Cowin ldquoThe theory of polar fluidsrdquo Advances in AppliedMechanics vol 14 pp 82ndash279 1974

[7] P Chaturani andV S Upadhya ldquoPulsatile flow of a couple stressfluid through circular tubes with applications to blood flowrdquoBiorheology vol 15 no 3-4 pp 193ndash201 1978

[8] P Chaturani and V P Rathod ldquoA critical study of Poiseuilleflow of couple stress fluid with applications to blood flowrdquoBiorheology vol 18 no 2 pp 235ndash244 1981

[9] AKollin ldquoElectromagnetic flowmeter Principle ofmethod andits application to blood flow measurementrdquo Proceedings of theSociety for Experimental Biology and Medicine vol 35 pp 235ndash244 1936

[10] E M Korchevskii and L S Marochnik ldquoMagnetohydrody-namic version of movement of bloodrdquo Biophysics vol 10 no2 pp 411ndash413 1965

[11] A R Rao and K S Deshikachar ldquoPhysiological type flow ina circular pipe in the presence of a transverse magnetic fieldrdquoJournal of the Indian Institute of Science vol 68 pp 247ndash2601988

[12] V A Vardanyan ldquoEffect of a magnetic field on blood flowrdquoBiofizika vol 18 no 3 pp 491ndash496 1973

[13] V P Rathod and Gayatri ldquoEffect of magnetic field on two layermodel of pulsatile blood flow with micro-organismsrdquo Bulletinof Pure and Applied Sciences E Mathematics and Statistics vol19 no 1 pp 1ndash13 2000

[14] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through catheterized arteriesmdasha mathematicalmodelrdquoAppliedMathematicalModelling vol 31 no 8 pp 1497ndash1517 2007

[15] D Srinivasacharya and D Srikanth ldquoEffect of couple stresseson the pulsatile flow through a constricted annulusrdquo ComptesRendus Mecanique vol 336 no 11-12 pp 820ndash827 2008

[16] D N Riahi R Roy and S Cavazos ldquoOn arterial blood flowin the presence of an overlapping stenosisrdquo Mathematical andComputer Modelling vol 54 no 11-12 pp 2999ndash3006 2011

[17] V P Srivastava and S Mishra ldquoNon-Newtonian arterial bloodflow through an overlapping stenosisrdquoApplications and AppliedMathematics vol 5 no 1 pp 225ndash238 2010

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

[19] V Vand ldquoViscosity of solutions and suspensions ITheoryrdquoTheJournal of Physical Chemistry vol 52 no 2 pp 277ndash299 1948

[20] L Bennett ldquoRed cell slip at a wall in vitrordquo Science vol 155 no3769 pp 1554ndash1556 1967

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

[22] P Chaturani and D Biswas ldquoA comparative study of poiseuilleflow of a polar fluid under various boundary conditions withapplications to blood flowrdquo Rheologica Acta vol 23 no 4 pp435ndash445 1984

[23] 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

[24] BKMishra and S S Shekhawat ldquoMagnetic effect on bloodflowthrough a tapered artery with stenosisrdquoActa Ciencia Indica vol1029 pp 295ndash301 2007

[25] R Ponalagusamy and R Tamil Selvi ldquoBlood flow in stenosedarteries with radially variable viscosity peripheral plasma layerthickness and magnetic fieldrdquo Meccanica vol 48 no 10 pp2427ndash2438 2013

[26] D Biswas and U S Chakraborty ldquoPulsatile blood flow througha catheterized artery with an axially nonsymmetrical stenosisrdquoApplied Mathematical Sciences vol 4 pp 2865ndash2880 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of