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
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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 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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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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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
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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
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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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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal 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
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