Research Article Mathematical Analysis of Casson Fluid...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 583809 11 pageshttpdxdoiorg1011552013583809

Research ArticleMathematical Analysis of Casson Fluid Model forBlood Rheology in Stenosed Narrow Arteries

J Venkatesan1 D S Sankar2 K Hemalatha3 and Yazariah Yatim4

1 Department of Mathematics Rajalakshmi Engineering College Thandalam Chennai 602 105 India2Division of Mathematics School of Advanced Sciences VIT University Chennai Campus Chennai 600 127 India3 Department of Mathematics Anna University Chennai 600 025 India4 School of Mathematical Sciences Universiti Sains Malaysia 11800 Penang Malaysia

Correspondence should be addressed to D S Sankar sankar dsyahoocoin

Received 1 April 2013 Accepted 14 July 2013

Academic Editor Mohamed Fathy El-Amin

Copyright copy 2013 J Venkatesan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Theflowof blood through a narrow arterywith bell-shaped stenosis is investigated treating blood asCasson fluid Present results arecompared with the results of the Herschel-Bulkley fluidmodel obtained byMisra and Shit (2006) for the same geometry Resistanceto flow and skin friction are normalized in two different ways such as (i) with respect to the same non-Newtonian fluid in a normalartery which gives the effect of a stenosis and (ii) with respect to the Newtonian fluid in the stenosed artery which spells out thenon-Newtonian effects of the fluid It is found that the resistance to flow and skin friction increase with the increase of maximumdepth of the stenosis but these flow quantities (when normalized with non-Newtonian fluid in normal artery) decrease with theincrease of the yield stress as obtained by Misra and Shit (2006) It is also noticed that the resistance to flow and skin frictionincrease (when normalized with Newtonian fluid in stenosed artery) with the increase of the yield stress

1 Introduction

The study of the fluid dynamical aspects of blood flowthrough a stenosed artery is useful for the fundamentalunderstanding of circulatory disorders Stenosis in an arteryis the narrowing of the blood flow area in the artery by thedevelopment of arteriosclerosis plaques due to the deposits offats cholesterol and so forth on the inner wall of the arteryThis leads to an increase in the resistance to flow and asso-ciated reduction in blood supply in the downstream whichleads to serious cardiovascular diseases such as myocardialinfarction and cerebral strokes [1ndash3]

Blood shows Newtonian fluidrsquos character when it flowsthrough larger diameter arteries at high shear rates but itexhibits a remarkable non-Newtonian behavior when it flowsthrough small diameter arteries at low shear rates [4 5]Moreover there is an increase in viscosity of blood at low rates of shear as the red blood cells tend to aggregate into theRouleaux form [6] Rouleaux form behaves as a semi-solidalong the center forming a plug flow region In the plug flowregion we have a flattened parabolic velocity profile rather

than the parabolic velocity profile of a Newtonian fluid Thisbehavior can be modeled by the concept of yield stress Theyield stress for blood depends strongly on fibrinogen concen-tration and is also dependent on the hematocrit The yieldstress values for normal human blood is between 001 and006 dyncm2 [7]

Casson fluid model is a non-Newtonian fluid with yieldstress which is widely used formodeling blood flow in narrowarteries Many researchers have used the Casson fluid modelfor mathematical modeling of blood flow in narrow arteriesat low shear rates It has been demonstrated by Blair [8] andCopley [9] that theCassonfluidmodel is adequate for the rep-resentation of the simple shear behavior of blood in narrowarteries Casson [10] examined the validity of Casson fluidmodel in his studies pertaining to the flow characteristics ofblood and reported that at low shear rates the yield stressfor blood is nonzero It has been established by Merrill et al[11] that the Casson fluid model predicts satisfactorily theflow behaviors of blood in tubes with the diameter of 130ndash1000 120583m

2 Journal of Applied Mathematics

Charm and Kurland [12] pointed out in their experimen-tal findings that the Casson fluidmodel could be the best rep-resentative of blood when it flows through narrow arteries atlow shear rates and that it could be applied to human blood ata wide range of hematocrit and shear rates Blair and Spanner[13] reported that blood behaves like aCasson fluid in the caseof moderate shear rate flows and it is appropriate to assumeblood as a Casson fluid Aroesty and Gross [14] have devel-oped a Casson fluid theory for pulsatile blood flow throughnarrow uniform arteries Chaturani and Samy [15] analyzedthe pulsatile flow of Casson fluid through stenosed arteriesusing the perturbation method

Herschel-Bulkley fluid is also a non-Newtonian fluidwithyield stress which is more general in the sense that it containstwo parameters such as the yield stress and power law indexwhereas the Casson fluid has only one parameter which is theyield stress Herschel-Bulkley fluidrsquos constitutive equation canbe reduced to the constitutive equations of Newtonian Powerlaw and Bingham fluid models by taking appropriate valuesto the parameters Chaturani and Ponnalagar Samy [16] ana-lyzed the steady flow of Herschel-Bulkley fluid for blood flowthrough cosine-shaped stenosed arteries

Misra and Shit [17] analyzed the steady flow of Herschel-Bulkley fluid for blood flow in narrow arteries with bell-shaped mild stenosis The mathematical modeling of Cassonfluid model for steady flow of blood in narrow arteries withbell-shaped mild stenosis was not studied by anyone so faraccording to the knowledge of the authors Hence in thepresent study a mathematical model is developed to analyzethe blood flow at low shear rates in narrow arteries with mildbell-shaped stenosis treating blood as Casson fluid modelThe results of the present study are compared with the resultsof Misra and Shit [17] and Chaturani and Ponnalagar Samy[16] and some possible clinical applications to the presentstudy are also given

2 Formulation

Let us consider an axially symmetric laminar steady andfully developed flow of a non-Newtonian incompressible vis-cous fluid (blood) in the axial direction (119911) through a circularartery with bell-shaped mild stenosis The non-Newtonianbehavior of the flowing blood is characterized by Casson fluidmodel The artery wall is assumed to be rigid (due to thepresence of the stenosis) and the artery is assumed to be longenough so that the entrance and end effects can be neglectedin the arterial segment under study A cylindrical polar coor-dinate system (119903 120595 119911) is used to analyze the blood flow where119903 and 119911 are the variables taken in the radial and axial direc-tions respectively and120595 is the azimuthal angleThegeometryof the arterial segment with mild constriction is shown inFigure 1

Since the blood flow in narrow arteries is slow themagnitude of the inertial forces is negligibly small and thusthe inertial terms in the momentum equations are neglectedSince the considered flow is unidirectional and is in the axialdirection the radial component of the momentum equation

r

L0L0

0

R(z)R0

LL

120575

Z

Figure 1 Geometry of the arterial segment with stenosis

is ignored The axial component of the momentum equationis simplified to the following

minus119889119901

119889119911=1

119903

119889 (119903120591)

119889119903 (1)

where 120591 is the shear stress and 119901 is the pressure The con-stitutive equation (relationship between the shear stress andstrain rate) of Casson fluid model is defined as follows

minus119889119906

119889119903= 119891 (120591) =

1

119896(radic120591 minus radic120591119888)

2

120591 ge 120591119888

0 120591 le 120591119888

(2)

where 119906 is the velocity of blood in the axial direction 120591119888is the

yield stress and 119896 is the viscosity coefficient of Casson fluidThe geometry of the segment of the narrow artery with

mild bell-shaped stenosis ismathematicaly defined as follows

119877 (119911) = 1198770[1 minus

120575

1198770

119890minus1198982120598211991121198772

0] (3)

where 1198770is the radius of the normal artery 119877(119911) is the radius

of the artery in the stenosis region 120575 is the depth of thestenosis at its throat 119898 is a parametric constant and 120598 char-acterizes the relative length of the constriction defined as 120598 =11987701198710

Equation (3) can be rewritten as

119877 (119911)

1198770

= 1 minus 119886119890minus1198871199112

(4)

where 119886 = 1205751198770and 119887 = 119898212059821198772

0 Note that 119886 and 119887 are the

parameters in the nondimensional form corresponding to themaximum projection of the stenosis at its throat and variablelength of the stenosis in the segment of the narrow arteryunder study respectively Equation (3) spells out that the bell-shaped geometry has the advantage of having two parameterssuch as 119886 and 119887 compared to cosine-curve shaped geometrywhich has only one parameter namely the maximum depthof the stenosis [15] In the bell-shaped stenosis geometry bykeeping 119887 as variable and 119886 as constant one can generatearteries with different stenosis lengths with the same maxi-mumdepth of the stenosis 119886 and also by keeping 119886 as variableand 119887 as constant one can generate arteries with differentmaximum depths with the same length of the stenosis Thedifferent stenosis shapes obtained by varying the stenosisheight 119886 with 30 stenosis that is 119871

0= 15 are shown in

Journal of Applied Mathematics 3

04

05

06

07

08

09

1

0 05 1 15minus15 minus1 minus05

RR

0

a = 01

a = 02

a = 03

a = 04

a = 05

z

Figure 2 Stenosis geometries for different values of stenosis height119886

Figure 2The percentage of stenosis is given by 1198710119871times100 In

the present study we have taken 119871 = 5 cm and119898 = 2 as takenby Misra and Shit [17]

Figure 3(a) depicts the shapes of stenoses with differentlengths by fixing 119886 = 02 and keeping 119887 as variable (fordifferent values of stenosis the length 119871

0with fixed value of

119898 = 2) Figure 3(b) shows the shapes of stenoses with differ-ent lengths by fixing 119886 = 02 and varying the values of 119887 (fordifferent values of 119898 and with a fixed value of 119871

0= 15) It

is noticed that the width of the stenosis decreases with theincrease in the values of119898

Equations (1) and (2) have to be solved with the help ofthe following no slip boundary condition

119906 = 0 at 119903 = 119877 (119911) (5)

and the regularity condition

120591 is finite at 119903 = 0 (6)

3 Method of Solution

Integrating (1) and then using (6) we get

120591 = minus119903

2

119889119901

119889119911 (7)

From (7) the skin friction 120591119877is obtained as

120591119877= minus

119877

2

119889119901

119889119911 (8)

where 119877 = 119877(119911)The volumetric flow rate 119876 is as follows

119876 =1205871198773

1205913

119877

int

120591119877

0

1205912119891 (120591) 119889120591 (9)

where 120591 and 120591119877are given by (7) and (8) respectively

Substituting (2) into (9) we get

119876 =1205871198773

1205913

119877

int

120591119877

120591119862

1205912 1


2

119889120591 (10)

Integrating (10) and then simplifying one can get

119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) minus1

21(120591119888

120591119877

)

4

(11)

Since 120591119888120591119877≪ 1 neglecting the term involving (120591

119888120591119877)4 in

(11) we get the expression for flow rate as

119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) (12)

Using (8) in (12) we get

minus119889119901

119889119911=128120591119888

49119877+8

119877(119876119896

1205871198773minus120591119888

147) +

64

7119877

radic119896119876120591119888

1205871198773minus1205912

119888

147

(13)

Neglecting the term involving 1205912119888in (13) we get

minus119889119901

119889119911=128120591119888

49119877+8119876119896

1205871198774+64

7119877

radic119896119876120591119888

1205871198773 (14)

Integrating (14) along the length of the artery and using theconditions that 119901 = 119901

1at 119911 = minus119871 and 119901 = 119901

2at 119911 = 119871 we

obtain

1199011minus 1199012=128120591119888

491198770

int

119871

minus119871

119889119911

(1198771198770)+8119876119896

12058711987740

int

119871

minus119871

119889119911

(1198771198770)4

+64

7radic119876119896120591119888

1205871198775

0

int

119871

minus119871

119889119911

(1198771198770)52

(15)

Simplifying (15) one can obtain the following expression forpressure drop

1199011minus 1199012=256

49

120591119888

1198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119876119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic119876119896120591119888

1205871198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(16)

31 Resistance to Flow The resistance to flow 120582 is defined asfollows

120582 =1199011minus 1199012

119876 (17)



06

07

08

09

1R

R0

z

L0 = 05

L0 = 15

L0 = 1

L0 = 2

(a)


z

06

07

08

09

1

RR

0

m = 3m = 4

m = 6m = 5

m = 2m = 1

(b)

Figure 3 (a) Shapes of the arterial stenosis for different values of the stenosis length 1198710 (b) Shapes of the arterial stenosis for different values

of the stenosis shape parameter119898

The resistance to flow for Casson fluid in a stenosed artery isobtained as follows

120582 =256120591119888

491198761198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic

119896120591119888

1205871198761198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(18)

In the absence of any constriction (120575 = 0 and 119877 = 1198770) the

resistance to flow (in the normal artery) 120582119873is given by the

following

120582119873=16119871

1198770

119896

1205871198773

0

+16120591119888

49119876+8

7radic

119896120591119888

1205871198761198773

0

(19)

The expression for resistance to flow in the dimensionlessform is obtained as follows

1205821=120582

120582119873

= 1 minus1198710

119871+1

119871(

(1612059111988849119876) 119868

1+ (119896120587119877

3

0) 1198682+ (87)radic(119896120591

119888120587119876119877

3

0)1198683

(1612059111988849119876) + (119896120587119877

3

0) + (87)radic119896120591

119888120587119876119877

3

0

) (20)

where 1198681= int1198710

0(119889119911(119877119877

0)) 1198682= int1198710

0(119889119911(119877119877

0)4) and 119868

3=

int1198710

0119889119911(119877119877

0)(52)

It is noted that 1205821measures the relative resistance in a

stenosed artery compared to normal artery Substituting theexpression for 119877119877

0from (4) the integrals 119868

1 1198682 and 119868

3are

reduced to the following

1198681= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)

1198682= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)4

1198683= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)52

(21)

Using a two-point Gauss quadrature formula the integrals in(21) are evaluated as follows

1198681=1198710

2[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

+1

(1 minus 119886119890minus(1198871198712

012)(4minus2radic3))

]

1198682=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

4

+1

(1 minus 119886119890minus(1198871198712


4

]

]


1198683=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

52

+1

(1 minus 119886119890minus(1198871198712


52

]

]

(22)

If we want to compare the resistance to flow for different fluidmodels we have to normalize it with respect to resistanceto flow 120582

119873119890

of Newtonian fluid in normal artery and therespective expression for Casson fluid model is obtained asfollows

1205822=

120582

120582119873119890

= (1 +16

49(1205871205911198621198773

0

119876119896) +

8

7

radic1205871205911198621198773

0

119876119896)

times (1 minus1198710

119871) +

1

119871(16

49(1205871205911198621198773

0

119876119896) 1198681

+1198682+8

7

radic1205871205911198621198773

0

1198761198961198683)

(23)

where

120582119873119890

=16119896119871

12058711987740

(24)

which is obtained from (18) with 120575 = 0119877(119911) = 1198770 and 120591

119888= 0

32 Skin Friction From (8) and (14) the expression for theskin friction is obtained as follows

120591119877= minus

119877

2sdot119889119901

119889119911=64

49120591119888+4119876119896

1205871198773+32

7

radic119896119876120591119888

1205871198773 (25)

In the absence of any constriction when (119877(119911) = 1198770) the

expression for the skin friction becomes the following

120591119873=64

49120591119888+4119876119896

1205871198773

0

+32

7radic119896119876120591119888

1205871198773

0

(26)

The nondimensional form of skin friction with effects onstenosis is defined as the ratio between the skin friction in thestenosed artery and skin friction in the normal artery From(25) and (26) the skin friction with effects on stenosis isobtained as

1205911=120591119877

120591119873

=4119876119896 + (327) (119876119896120591

1198881205871198773

0)12

(1198771198770)32

+ (6449) 1205911198881205871198773

0(1198771198770)3

(1198771198770)3

4119876119896 + (327) (1198761198961205911198881205871198773

0)12

+ (6449) 1205911198881205871198773

0

(27)

The nondimensional form of skin friction with effects onthe non-Newtonian behavior of blood is defined as the ratiobetween the skin friction of the non-Newtonian fluid in thestenosed artery and the skin friction of theNewtonian fluid inthe same stenosed arteryThe expression for skin frictionwitheffects on the non-Newtonian behavior of blood is obtainedas follows

1205912=120591119877

120591119873119890

=1

(1198771198770)3+16

49(1205871198773

0120591119862

119876119896)

+8

7

1

(1198771198770)32

radic1205871198773

0120591119862

119876119896

(28)

where

120591119873119890

=4119896119876

1205871198773

0

(29)

4 Numerical Simulations of the Results

The objective of this study is to discuss the effects of variousparameters on the physiologically important flow quantities

such as skin friction resistance to flow and flow rate Thefollowing parameters with their ranges mentioned as 119876 0ndash1 119896 20ndash70 (CP)119899sec119899minus1 120591

119888 00ndash05 dynecm2 119898 = 2 119871 =

5 cm and 1198770= 040 are used to evaluate the expressions of

these flow quantities and get data for plotting the graphs

41 Skin-Friction

411 Effects of Stenosis on Skin-Friction The variation of skinfriction 120591

1with axial distance for different values of 119871

0119871 and

yield stress 120591119888with 119896 = 4 is shown in Figure 4 It is observed

that the skin friction increases considerably with the increasein the stenosis length for a given value of yield stress 120591

119888 but it

decreases very slightly when the yield stress increases for thefixed value of 119871

0119871 The same observations were recorded by

Misra and Shit [17] for the Herschel-Bulkley fluidThe variations of skin friction 120591

1at the midpoint of the

stenosis with stenosis height 1205751198770for different values of the

yield stress 120591119888with 119896 = 7 and 119871

0119871 = 03 are sketched in

Figure 5 It is noted that the skin friction decreasesmarginallyas the yield stress increases

The estimates of skin friction 1205911for different values of

axial variable 119911 and viscosity coefficient 119896 with 120591119888= 005 and


1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911

Figure 4 Variations of skin friction 1205911with axial distance for

different percentages of stenosis 1198710119871 and yield stress 120591

119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911

Figure 5 Variations of skin friction 1205911with stenosis height 119886 for

different values of yield stress 120591119888

30 stenosis are computed in Table 1 It is observed that theskin friction increases very slightly with the increase of theviscosity coefficient

In order to compare our results with those of Misra andShit [17] the variations of skin friction 120591

1with axial distance

for different fluids are shown in Figure 6 for 30 of stenosiswith 119896 = 4 and 120591

119888= 005 It is observed that the plot of the

skin friction of Casson fluid model lies between those of theHerschel-Bulkley fluid model with 119899 = 1 and 119899 = 105 Herethe skin friction is normalized with respect to the normalartery with the same fluid as was done byMisra and Shit [17]

Themathematical form of cosine curve-shaped geometryfor stenosis given by Chaturani and Ponnalagar Samy [16] isreproduced as follows

119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)

For computation values of the parameters are taken as 119871 =10 119889 = 35 and 119871

0= 3

The variations of skin friction 1205911with axial distance for

Casson fluid model with cosine- and bell-shaped stenosisgeometrieswith 30 stenosis andwith 119896 = 4 and 120591

119888= 005 are

shown in Figure 7 (the data for plotting the graph is computedfrom (27)) It is found that the skin friction in cosine curve-shaped stenosed arteries is considerably higher than that inbell-shaped stenosed arteries

412 Effects of Non-Newtonian Behavior on Skin-FrictionWe can study the effects of different non-Newtonian fluidsif we normalize it with respect to Newtonian fluid as done

Table 1 Variations of the skin friction 1205911with axial distance for

different values of viscosity with 120591119888= 005 and 119871

0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7

minus15 101069 1010799 1010864minus1 1104858 1105946 1106599minus05 1490229 1495693 1498970 1915269 1926048 193251605 1490229 1495693 1498971 1104858 1105946 110659915 101069 1010799 1010864

112141618

2

1205911


Bingham fluid Casson fluid

Herschel-Bulkley fluid with n = 095


Herschel-Bulkley fluid withn=105

z


different fluids

Cosine curve-shaped geometry


z

112141618

2

1205911Bell-shaped geometry

Figure 7 Variations of skin friction 1205911for Casson fluid with axial

distance for different arterial geometries

by [15 16 18ndash20] In this case the following results are inagreement with the results given by these authors but areopposite in nature to those given in Section 411 except forthe variations of skin friction with axial distance for differentvalues of stenosis length as given in Figure 8


different values of 1198710119871 and for a given 119896 = 4 and 120591

119888= 005

are illustrated in Figure 9 It is observed that the skin frictionincreases considerably with the increase in the stenosislength

Figure 10 sketches the variations of skin friction 1205912with

axial distance for different values of yield stress with 119896 = 4

and 1198710119871 = 03 It is observed that the skin friction increases

marginally with the increase in the yield stress 120591119888of the fluid

The variations of skin friction 1205912at the midpoint of the



0119871 = 03 are shown in


08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0

Figure 8 Comparison of cosine- and bell-shaped stenosis shapes


z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912


different values of stenosis size 1198710119871

Figure 11 It is clear that the skin friction increases marginallywhen the yield stress increases

Figure 12 depicts the variations of skin friction 1205912with

axial distance for different values of viscosity coefficient 119896 for120591119888= 005 and 30 of stenosis One can notice that the skin

friction decreases slightly as viscosity increasesFigure 13 shows the variations of skin friction 120591

2with

axial distance for different fluid models using cosine-shapedstenosis with 30 of stenosis yield stress 120591

119888= 005 and 119896 = 4

It is observed that the skin friction of Casson fluid modellies between those of the Herschel-Bulkley fluid model with119899 = 095 and 119899 = 1


different fluid models using bell-shaped stenosis with 30of stenosis yield stress 120591

119888= 005 and 119896 = 4 are shown in

Figure 14 It is observed that the skin friction of Casson fluidmodel lies between those of theHerschel-Bulkley fluidmodelwith 119899 = 095 and 119899 = 1

Figure 15 shows the variations of skin friction 1205912with

axial distance for Casson fluid model with cosine- and bell-shaped geometries with 30 stenosis 119896 = 4 and 120591

119888= 005

(data computed using (28)) It is seen that the cosine-shapedstenosis has a greater width than that of bell-shaped stenosisas depicted in Figure 8

Misra and Shit [17] have analyzed the variations of skinfriction with axial distance for different values of power lawindex and yield stress and reported that the skin frictiondecreases with increase in yield stress If normalized withNewtonian fluid the skin friction increases considerablywhen the yield stress increases with axial distance for 30stenosis 119896 = 4 as displayed in Figure 16


z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0

Figure 11 Variations of skin friction 1205912with stenosis height 120575119877

0for


42 Resistance to Flow The variations of resistance to flow 1205821

with the height of the stenosis 1205751198770for different values of

stenosis length 1198710119871with viscosity coefficient 119896 = 4 and yield

stress 120591119888= 005 are shown in Figure 17 It is found that the flow

resistance increases nonlinearly with the increase in the ste-nosis size

Using the result (23) the variations of resistance to flow1205822with stenosis height for different values of stenosis length

for the given 119896 = 4 and 120591119862= 005 are shown in Figure 18

It is observed that the flow resistance decreases considerablywhen the length of the stenosis increases Also the variationsof resistance to flow 120582

2with stenosis height for different

values of yield stress and for 119896 = 4 and 1198710119871 = 03 is shown in

Figure 19 It is clear that the resistance to flow increases signif-icantly with the increase in the yield stress

43 Flow Rate Figure 20 depicts the variation of flow ratewith axial distance for different values of viscosity coefficient119896 and yield stress 120591

119888with 30 of stenosis One can notice that

the flow rate decreases significantly with the increase of eitherthe viscosity coefficient or the yield stress It is also observedthat the flow rate decreases very significantly (nonlinearly) atthe throat of the stenosis for lower values of either the yieldstress or viscosity coefficient and slowly (almost linearly) forhigher values of either the yield stress or viscosity coefficient

44 Physiological Application To highlight some possibleclinical applications of the present study the data used bySankar [21] (the radii of different arteries and flow rate) areused to compute the physiologically important flow quanti-ties such as resistance to flow and skin friction The valuesof the parameters are taken as 119896 = 35 and 120591

119888= 004 The

dimensionless resistance to flow 1205822and skin friction 120591

2are


Table 2 Estimates of resistance to flow 1205822and skin friction (dimensionless) 120591

2in arteries with different radii with 119871

0119871 = 1

Blood vessels1205911015840

119888

119886 = 010 119886 = 015

Radius 1198770(cm)

120582 120591 120582 120591Flow rate 119876 (cm3 sminus1)Aorta

1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912


different values of viscosity 119896

0

05

1

15

2

25

Bingham fluid

Casson fluid Herschel-Bulkley fluid with n = 095



z

1205912


different fluid models with cosine-shaped stenosis

computed for arteries with different radii and with119898 = 2 and1198710119871 = 1 from (23) and (28) (normalized with Newtonian

fluid in stenosed artery) and are presented in Table 2 It isnoted that the resistance to flow increases considerably whenthe stenosis height increases and skin friction significantlyincreases with the increase of stenosis height

The percentages of increase in resistance to flow and skinfriction over that for uniform diameter tube (no stenosis)and for arteries with different radii are computed in Table 3One can observe that the percentages of increase in resistance

Bingham fluid




z

005

1

15

2

25

1205912


different fluid models with bell-shaped stenosis


z

1

12141618

2

1205912


Bell-shaped geometry

Figure 15 Variations of skin friction 1205912of Casson fluid model with

axial distance for different geometries

to flow and skin friction increase significantly when stenosisheight increases Comparisons of resistance to flow (120582

2) in

bell-shaped stenosed arteries and cosine curve-shapedstenosed arteries with different values of stenosis sizeparameters 119886 119871

0 and 119898 are given in Table 4 It is recorded

that the resistance to flow in bell-shaped stenosed arteryincreases considerably when the length of the stenosisincreases and also it significantly increases when stenosisheight increases But it marginally decreases with the increaseof the stenosis length parameter 119898 It is also found that for


Table 3 Estimates of the percentage of increase in resistance to flow 1205822and skin friction 120591

2for arteries with different radii with 119871

0119871 = 1

and 120591119888= 004 dyn cmminus2

Blood vessels 1205911015840

119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10

minus2 2031 3225 3400 5423Femoral 5718029 times 10

minus3 2113 3400 3544 5728Carotid 4571948 times 10

minus3 2127 3431 3569 5782Coronary 8733753 times 10

minus4 2200 3586 3697 6053Arteriole 5629388 times 10

minus3 2114 3403 3545 5732

Table 4 Estimates of percentage of increase in resistance to flow 1205822for arteries with different radii and with (i) bell-shaped stenosis (ii) cosine

curve-shaped stenosis

Blood vessels

Bell-shaped stenosis Cosine curve-shaped stenosis1205822() 120582

2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15

Aorta 406 609 433 1042 1531 428 1579Femoral 423 634 450 1087 1599 445 1678Carotid 425 638 453 1095 1611 450 1696Coronary 440 660 468 1135 1672 471 1786Arteriole 423 634 450 1088 1600 447 1679

0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591

Figure 16 Variations of skin frictionwith axial distance for differentvalues of power index and yield stress 120591

119867

a given set of values of the parameters the percentage ofincrease in resistance to flow in the case of bell-shaped ste-nosed artery is slightly lower than that of cosine curve-shapedstenosed artery This is physically verified with the depth ofthe stenosis for cosine-shaped stenosis (shown in Figure 8)being more the resistance to flow is higher in this typeof stenosis geometry compared to the bell-shaped stenosisgeometry

5 Conclusion

The present study analyzed the steady flow of blood in a nar-row artery with bell-shaped mild stenosis treating blood asCasson fluid and the results are compared with the results ofMisra and Shit for Herschel-Bulkley fluidmodel [17] and also

08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0

Figure 17 Variations of flow resistance 1205821with stenosis height 120575119877

0

for different values of stenosis length 1198710119871

08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0


with the results of Chaturani and Ponnalagar Samy [16] (forblood flow in cosine curve-shaped stenosed arteries treatingblood as Herschel-Bulkley fluidmodel)Themain findings ofthe present mathematical analysis are as follows


1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0

for different values of yield stress 120591119888

00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z

Figure 20 Variations of the rate of flow with axial distance fordifferent values of viscosity 119896 and yield stress 120591

119888

(i) Whether normalized with respect to non-NewtonianorNewtonian fluid in normal artery (ie in both casesof the effect of stenosis and the effect of non-Newto-nian fluid) we notice the following

(1) Skin friction increaseswith the increase of depthand length of the stenosis

(2) Skin friction of Casson fluid model is signifi-cantly lower than that of the Herschel-Bulkleyfluid model

(3) Skin friction in bell-shaped stenosed arteryis considerably lower than that in the cosinecurve-shaped stenosed artery

(ii) The effect of stenosis on blood flow is that the resis-tance to flow increases when either the stenosis lengthor depth increases

(iii) The effect of non-Newtonian fluid on blood flow isthat the resistance to flow increases significantly withthe increase of yield stress but it decreases wheneither the stenosis length or depth increases

(iv) The percentage of increase in resistance to flow in thecase of bell-shaped stenosed artery is slightly lowerthan that of the cosine curve-shaped stenosed arteryin the case of normalization with respect to Newto-nian fluid

(v) Flow rate decreases with the increase of the yieldstress and viscosity coefficient

Hence in view of the results obtained we conclude thatthe present study may be considered as an improvement in

the studies of the mathematical modeling of blood flow innarrow arteries with mild stenosis

Nomenclature

120591 Shear stress120591119888 Yield stress for Casson fluid

120591119877 Skin friction in stenosed artery

1205911 Nondimensional skin friction normalized with

non-Newtonian fluid1205912 Nondimensional skin friction normalized with

Newtonian fluid120591119873 Skin friction in normal artery normalized with

non-Newtonian fluid120591119873119890

Skin friction in normal artery normalized withNewtonian fluid

120575 Stenosis height120582 Flow resistance1205821 Nondimensional flow resistance normalized with

non-Newtonian fluid1205822 Nondimensional flow resistance normalized with

Newtonian fluid120582119873 Flow resistance in normal artery normalized with


Flow resistance in normal artery normalized withNewtonian fluid

119876 Volumetric flow rate119903 Radial coordinate119911 Axial coordinate119906 Radial velocity1198770 Radius of the normal artery

119877(119911) Radius of the artery in the stenosed portion119871 Half-length of segment of the narrow artery1198710 Half-length of the stenosis

119901 Pressure119896 Viscosity coefficient

Acknowledgments

The authors thank S V Subhashini Assistant Professor (SG)Department of Mathematics Anna University Chennai forher useful suggestions and timely help for improving thequality of the paper The research work is supported by theResearch University grant of the Universiti Sains MalaysiaMalaysia RU Grant no 1001PMATHS811177

References

[1] M E Clark JM Robertson and L C Cheng ldquoStenosis severityeffects for unbalanced simple-pulsatile bifurcation flowrdquo Jour-nal of Biomechanics vol 16 no 11 pp 895ndash906 1983

[2] D Liepsch M Singh andM Lee ldquoExperimental analysis of theinfluence of stenotic geometry on steady flowrdquo Biorheology vol29 no 4 pp 419ndash431 1992

[3] S Cavalcanti ldquoHemodynamics of an artery with mild stenosisrdquoJournal of Biomechanics vol 28 no 4 pp 387ndash399 1995

[4] V P Rathod and S Tanveer ldquoPulsatile flow of couple stress fluidthrough a porous medium with periodic body acceleration and


magnetic fieldrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 32 no 2 pp 245ndash259 2009

[5] G R Cokelet ldquoThe rheology of human bloodrdquo in BiomechanicsY C Fung et al Ed p 63 Prentice-hall Englewood Cliffs NJUSA 1972

[6] W Boyd Text Book of Pathology Structure and Functions inDiseases Lea and Febiger Philadelphia Pa USA 1963

[7] B C Krishnan S E Rittgers and A P Yoganathan BiouidMechanics The Human Circulation Taylor and Francis NewYork NY USA 2012

[8] G W S Blair ldquoAn equation for the flow of blood plasma andserum through glass capillariesrdquo Nature vol 183 no 4661 pp613ndash614 1959

[9] A L Copley ldquoApparent viscosity and wall adherence of bloodsystemsrdquo in Flow Properties of Blood and Other Biological Sys-tems A L Copley and G Stainsly Eds Pergamon PressOxford UK 1960

[10] N Casson ldquoRheology of disperse systemsrdquo in Flow Equation forPigment Oil Suspensions of the Printing Ink Type Rheology ofDisperse Systems C C Mill Ed pp 84ndash102 Pergamon PressLondon UK 1959

[11] EWMerrill AM Benis E RGilliland TK Sherwood andEW Salzman ldquoPressure-flow relations of human blood in hollowfibers at low flow ratesrdquo Journal of Applied Physiology vol 20no 5 pp 954ndash967 1965

[12] S Charm and G Kurland ldquoViscometry of human blood forshear rates of 0-100000 secminus1rdquo Nature vol 206 no 4984 pp617ndash618 1965

[13] GW S Blair andD C SpannerAn Introduction to BiorheologyElsevier Scientific Oxford UK 1974

[14] J Aroesty and J F Gross ldquoPulsatile flow in small blood vesselsI Casson theoryrdquo Biorheology vol 9 no 1 pp 33ndash43 1972

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

[16] P Chaturani and V R Ponnalagar Samy ldquoA study of non-Newtonian aspects of blood flow through stenosed arteries andits applications in arterial diseasesrdquo Biorheology vol 22 no 6pp 521ndash531 1985

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

[18] D S Sankar ldquoTwo-fluid nonlinear mathematical model forpulsatile blood flow through stenosed arteriesrdquo Bulletin of theMalaysian Mathematical Sciences Society vol 35 no 2A pp487ndash498 2012

[19] D S Sankar and K Hemalatha ldquoPulsatile flow of Herschel-Bulkley fluid through stenosed arteriesmdasha mathematicalmodelrdquo International Journal of Non-Linear Mechanics vol 41no 8 pp 979ndash990 2006

[20] S U Siddiqui N K Verma and R S Gupta ldquoA mathematicalmodel for pulsatile flow of Herschel-Bulkley fluid throughstenosed arteriesrdquo Journal of Electronic Science and Technologyvol 4 pp 49ndash466 2010

[21] D S Sankar ldquoPerturbation analysis for blood flow in stenosedarteries under body accelerationrdquo International Journal of Non-linear Sciences and Numerical Simulation vol 11 no 8 pp 631ndash653 2010

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Stochastic AnalysisInternational Journal of


Charm and Kurland [12] pointed out in their experimen-tal findings that the Casson fluidmodel could be the best rep-resentative of blood when it flows through narrow arteries atlow shear rates and that it could be applied to human blood ata wide range of hematocrit and shear rates Blair and Spanner[13] reported that blood behaves like aCasson fluid in the caseof moderate shear rate flows and it is appropriate to assumeblood as a Casson fluid Aroesty and Gross [14] have devel-oped a Casson fluid theory for pulsatile blood flow throughnarrow uniform arteries Chaturani and Samy [15] analyzedthe pulsatile flow of Casson fluid through stenosed arteriesusing the perturbation method

Herschel-Bulkley fluid is also a non-Newtonian fluidwithyield stress which is more general in the sense that it containstwo parameters such as the yield stress and power law indexwhereas the Casson fluid has only one parameter which is theyield stress Herschel-Bulkley fluidrsquos constitutive equation canbe reduced to the constitutive equations of Newtonian Powerlaw and Bingham fluid models by taking appropriate valuesto the parameters Chaturani and Ponnalagar Samy [16] ana-lyzed the steady flow of Herschel-Bulkley fluid for blood flowthrough cosine-shaped stenosed arteries

Misra and Shit [17] analyzed the steady flow of Herschel-Bulkley fluid for blood flow in narrow arteries with bell-shaped mild stenosis The mathematical modeling of Cassonfluid model for steady flow of blood in narrow arteries withbell-shaped mild stenosis was not studied by anyone so faraccording to the knowledge of the authors Hence in thepresent study a mathematical model is developed to analyzethe blood flow at low shear rates in narrow arteries with mildbell-shaped stenosis treating blood as Casson fluid modelThe results of the present study are compared with the resultsof Misra and Shit [17] and Chaturani and Ponnalagar Samy[16] and some possible clinical applications to the presentstudy are also given

2 Formulation

Let us consider an axially symmetric laminar steady andfully developed flow of a non-Newtonian incompressible vis-cous fluid (blood) in the axial direction (119911) through a circularartery with bell-shaped mild stenosis The non-Newtonianbehavior of the flowing blood is characterized by Casson fluidmodel The artery wall is assumed to be rigid (due to thepresence of the stenosis) and the artery is assumed to be longenough so that the entrance and end effects can be neglectedin the arterial segment under study A cylindrical polar coor-dinate system (119903 120595 119911) is used to analyze the blood flow where119903 and 119911 are the variables taken in the radial and axial direc-tions respectively and120595 is the azimuthal angleThegeometryof the arterial segment with mild constriction is shown inFigure 1

Since the blood flow in narrow arteries is slow themagnitude of the inertial forces is negligibly small and thusthe inertial terms in the momentum equations are neglectedSince the considered flow is unidirectional and is in the axialdirection the radial component of the momentum equation

r

L0L0

0

R(z)R0

LL

120575

Z

Figure 1 Geometry of the arterial segment with stenosis

is ignored The axial component of the momentum equationis simplified to the following

minus119889119901

119889119911=1

119903

119889 (119903120591)

119889119903 (1)

where 120591 is the shear stress and 119901 is the pressure The con-stitutive equation (relationship between the shear stress andstrain rate) of Casson fluid model is defined as follows

minus119889119906

119889119903= 119891 (120591) =

1


2

120591 ge 120591119888

0 120591 le 120591119888

(2)

where 119906 is the velocity of blood in the axial direction 120591119888is the

yield stress and 119896 is the viscosity coefficient of Casson fluidThe geometry of the segment of the narrow artery with

mild bell-shaped stenosis ismathematicaly defined as follows

119877 (119911) = 1198770[1 minus

120575

1198770

119890minus1198982120598211991121198772

0] (3)

where 1198770is the radius of the normal artery 119877(119911) is the radius

of the artery in the stenosis region 120575 is the depth of thestenosis at its throat 119898 is a parametric constant and 120598 char-acterizes the relative length of the constriction defined as 120598 =11987701198710

Equation (3) can be rewritten as

119877 (119911)

1198770

= 1 minus 119886119890minus1198871199112

(4)

where 119886 = 1205751198770and 119887 = 119898212059821198772

0 Note that 119886 and 119887 are the

parameters in the nondimensional form corresponding to themaximum projection of the stenosis at its throat and variablelength of the stenosis in the segment of the narrow arteryunder study respectively Equation (3) spells out that the bell-shaped geometry has the advantage of having two parameterssuch as 119886 and 119887 compared to cosine-curve shaped geometrywhich has only one parameter namely the maximum depthof the stenosis [15] In the bell-shaped stenosis geometry bykeeping 119887 as variable and 119886 as constant one can generatearteries with different stenosis lengths with the same maxi-mumdepth of the stenosis 119886 and also by keeping 119886 as variableand 119887 as constant one can generate arteries with differentmaximum depths with the same length of the stenosis Thedifferent stenosis shapes obtained by varying the stenosisheight 119886 with 30 stenosis that is 119871

0= 15 are shown in


04

05

06

07

08

09

1


RR

0

a = 01

a = 02

a = 03

a = 04

a = 05

z







0= 15) It



119906 = 0 at 119903 = 119877 (119911) (5)


120591 is finite at 119903 = 0 (6)



120591 = minus119903

2

119889119901

119889119911 (7)


120591119877= minus

119877

2

119889119901

119889119911 (8)


119876 =1205871198773

1205913

119877

int

120591119877

0

1205912119891 (120591) 119889120591 (9)



119876 =1205871198773

1205913

119877

int

120591119877

120591119862

1205912 1


2

119889120591 (10)


119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) minus1

21(120591119888

120591119877

)

4

(11)


119888120591119877)4 in


119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) (12)


minus119889119901

119889119911=128120591119888

49119877+8

119877(119876119896

1205871198773minus120591119888

147) +

64

7119877

radic119896119876120591119888

1205871198773minus1205912

119888

147

(13)


minus119889119901

119889119911=128120591119888

49119877+8119876119896

1205871198774+64

7119877

radic119896119876120591119888

1205871198773 (14)


1at 119911 = minus119871 and 119901 = 119901

2at 119911 = 119871 we

obtain

1199011minus 1199012=128120591119888

491198770

int

119871

minus119871

119889119911

(1198771198770)+8119876119896

12058711987740

int

119871

minus119871

119889119911

(1198771198770)4

+64

7radic119876119896120591119888

1205871198775

0

int

119871

minus119871

119889119911

(1198771198770)52

(15)


1199011minus 1199012=256

49

120591119888

1198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119876119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic119876119896120591119888

1205871198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(16)


120582 =1199011minus 1199012

119876 (17)



06

07

08

09

1R

R0

z

L0 = 05

L0 = 15

L0 = 1

L0 = 2

(a)


z

06

07

08

09

1

RR

0

m = 3m = 4

m = 6m = 5

m = 2m = 1

(b)




120582 =256120591119888

491198761198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic

119896120591119888

1205871198761198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(18)



following

120582119873=16119871

1198770

119896

1205871198773

0

+16120591119888

49119876+8

7radic

119896120591119888

1205871198761198773

0

(19)


1205821=120582

120582119873

= 1 minus1198710

119871+1

119871(

(1612059111988849119876) 119868

1+ (119896120587119877

3

0) 1198682+ (87)radic(119896120591

119888120587119876119877

3

0)1198683

(1612059111988849119876) + (119896120587119877

3

0) + (87)radic119896120591

119888120587119876119877

3

0

) (20)

where 1198681= int1198710

0(119889119911(119877119877

0)) 1198682= int1198710

0(119889119911(119877119877

0)4) and 119868

3=

int1198710

0119889119911(119877119877

0)(52)




1 1198682 and 119868

3are


1198681= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)

1198682= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)4

1198683= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)52

(21)


1198681=1198710

2[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

+1

(1 minus 119886119890minus(1198871198712


]

1198682=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

4

+1

(1 minus 119886119890minus(1198871198712


4

]

]


1198683=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

52

+1

(1 minus 119886119890minus(1198871198712


52

]

]

(22)


119873119890


1205822=

120582

120582119873119890

= (1 +16

49(1205871205911198621198773

0

119876119896) +

8

7

radic1205871205911198621198773

0

119876119896)


119871) +

1

119871(16

49(1205871205911198621198773

0

119876119896) 1198681

+1198682+8

7

radic1205871205911198621198773

0

1198761198961198683)

(23)

where

120582119873119890

=16119896119871

12058711987740

(24)


119888= 0


120591119877= minus

119877

2sdot119889119901

119889119911=64

49120591119888+4119876119896

1205871198773+32

7

radic119896119876120591119888

1205871198773 (25)



120591119873=64

49120591119888+4119876119896

1205871198773

0

+32

7radic119896119876120591119888

1205871198773

0

(26)


1205911=120591119877

120591119873

=4119876119896 + (327) (119876119896120591

1198881205871198773

0)12

(1198771198770)32

+ (6449) 1205911198881205871198773

0(1198771198770)3

(1198771198770)3

4119876119896 + (327) (1198761198961205911198881205871198773

0)12

+ (6449) 1205911198881205871198773

0

(27)


1205912=120591119877

120591119873119890

=1

(1198771198770)3+16

49(1205871198773

0120591119862

119876119896)

+8

7

1

(1198771198770)32

radic1205871198773

0120591119862

119876119896

(28)

where

120591119873119890

=4119896119876

1205871198773

0

(29)




119888 00ndash05 dynecm2 119898 = 2 119871 =



41 Skin-Friction



0119871 and



119888 but it












1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911



119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911










119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)


0= 3



119888= 005 are





0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7


112141618

2

1205911






z


different fluids



z

112141618

2







119888= 005











08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










04

05

06

07

08

09

1


RR

0

a = 01

a = 02

a = 03

a = 04

a = 05

z







0= 15) It



119906 = 0 at 119903 = 119877 (119911) (5)


120591 is finite at 119903 = 0 (6)



120591 = minus119903

2

119889119901

119889119911 (7)


120591119877= minus

119877

2

119889119901

119889119911 (8)


119876 =1205871198773

1205913

119877

int

120591119877

0

1205912119891 (120591) 119889120591 (9)



119876 =1205871198773

1205913

119877

int

120591119877

120591119862

1205912 1


2

119889120591 (10)


119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) minus1

21(120591119888

120591119877

)

4

(11)


119888120591119877)4 in


119876 =1205871198773

41198961205911198771 minus

16

7(120591119888

120591119877

)

12

+4

3(120591119888

120591119877

) (12)


minus119889119901

119889119911=128120591119888

49119877+8

119877(119876119896

1205871198773minus120591119888

147) +

64

7119877

radic119896119876120591119888

1205871198773minus1205912

119888

147

(13)


minus119889119901

119889119911=128120591119888

49119877+8119876119896

1205871198774+64

7119877

radic119896119876120591119888

1205871198773 (14)


1at 119911 = minus119871 and 119901 = 119901

2at 119911 = 119871 we

obtain

1199011minus 1199012=128120591119888

491198770

int

119871

minus119871

119889119911

(1198771198770)+8119876119896

12058711987740

int

119871

minus119871

119889119911

(1198771198770)4

+64

7radic119876119896120591119888

1205871198775

0

int

119871

minus119871

119889119911

(1198771198770)52

(15)


1199011minus 1199012=256

49

120591119888

1198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119876119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic119876119896120591119888

1205871198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(16)


120582 =1199011minus 1199012

119876 (17)



06

07

08

09

1R

R0

z

L0 = 05

L0 = 15

L0 = 1

L0 = 2

(a)


z

06

07

08

09

1

RR

0

m = 3m = 4

m = 6m = 5

m = 2m = 1

(b)




120582 =256120591119888

491198761198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic

119896120591119888

1205871198761198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(18)



following

120582119873=16119871

1198770

119896

1205871198773

0

+16120591119888

49119876+8

7radic

119896120591119888

1205871198761198773

0

(19)


1205821=120582

120582119873

= 1 minus1198710

119871+1

119871(

(1612059111988849119876) 119868

1+ (119896120587119877

3

0) 1198682+ (87)radic(119896120591

119888120587119876119877

3

0)1198683

(1612059111988849119876) + (119896120587119877

3

0) + (87)radic119896120591

119888120587119876119877

3

0

) (20)

where 1198681= int1198710

0(119889119911(119877119877

0)) 1198682= int1198710

0(119889119911(119877119877

0)4) and 119868

3=

int1198710

0119889119911(119877119877

0)(52)




1 1198682 and 119868

3are


1198681= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)

1198682= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)4

1198683= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)52

(21)


1198681=1198710

2[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

+1

(1 minus 119886119890minus(1198871198712


]

1198682=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

4

+1

(1 minus 119886119890minus(1198871198712


4

]

]


1198683=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

52

+1

(1 minus 119886119890minus(1198871198712


52

]

]

(22)


119873119890


1205822=

120582

120582119873119890

= (1 +16

49(1205871205911198621198773

0

119876119896) +

8

7

radic1205871205911198621198773

0

119876119896)


119871) +

1

119871(16

49(1205871205911198621198773

0

119876119896) 1198681

+1198682+8

7

radic1205871205911198621198773

0

1198761198961198683)

(23)

where

120582119873119890

=16119896119871

12058711987740

(24)


119888= 0


120591119877= minus

119877

2sdot119889119901

119889119911=64

49120591119888+4119876119896

1205871198773+32

7

radic119896119876120591119888

1205871198773 (25)



120591119873=64

49120591119888+4119876119896

1205871198773

0

+32

7radic119896119876120591119888

1205871198773

0

(26)


1205911=120591119877

120591119873

=4119876119896 + (327) (119876119896120591

1198881205871198773

0)12

(1198771198770)32

+ (6449) 1205911198881205871198773

0(1198771198770)3

(1198771198770)3

4119876119896 + (327) (1198761198961205911198881205871198773

0)12

+ (6449) 1205911198881205871198773

0

(27)


1205912=120591119877

120591119873119890

=1

(1198771198770)3+16

49(1205871198773

0120591119862

119876119896)

+8

7

1

(1198771198770)32

radic1205871198773

0120591119862

119876119896

(28)

where

120591119873119890

=4119896119876

1205871198773

0

(29)




119888 00ndash05 dynecm2 119898 = 2 119871 =



41 Skin-Friction



0119871 and



119888 but it












1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911



119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911










119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)


0= 3



119888= 005 are





0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7


112141618

2

1205911






z


different fluids



z

112141618

2







119888= 005











08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











06

07

08

09

1R

R0

z

L0 = 05

L0 = 15

L0 = 1

L0 = 2

(a)


z

06

07

08

09

1

RR

0

m = 3m = 4

m = 6m = 5

m = 2m = 1

(b)




120582 =256120591119888

491198761198770

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)

+16119896

12058711987740

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)4

+128

7radic

119896120591119888

1205871198761198775

0

(119871 minus 1198710) + int

1198710

0

119889119911

(1198771198770)52

(18)



following

120582119873=16119871

1198770

119896

1205871198773

0

+16120591119888

49119876+8

7radic

119896120591119888

1205871198761198773

0

(19)


1205821=120582

120582119873

= 1 minus1198710

119871+1

119871(

(1612059111988849119876) 119868

1+ (119896120587119877

3

0) 1198682+ (87)radic(119896120591

119888120587119876119877

3

0)1198683

(1612059111988849119876) + (119896120587119877

3

0) + (87)radic119896120591

119888120587119876119877

3

0

) (20)

where 1198681= int1198710

0(119889119911(119877119877

0)) 1198682= int1198710

0(119889119911(119877119877

0)4) and 119868

3=

int1198710

0119889119911(119877119877

0)(52)




1 1198682 and 119868

3are


1198681= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)

1198682= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)4

1198683= int

1198710

0

119889119911

(1 minus 119886119890minus1198871199112

)52

(21)


1198681=1198710

2[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

+1

(1 minus 119886119890minus(1198871198712


]

1198682=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

4

+1

(1 minus 119886119890minus(1198871198712


4

]

]


1198683=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

52

+1

(1 minus 119886119890minus(1198871198712


52

]

]

(22)


119873119890


1205822=

120582

120582119873119890

= (1 +16

49(1205871205911198621198773

0

119876119896) +

8

7

radic1205871205911198621198773

0

119876119896)


119871) +

1

119871(16

49(1205871205911198621198773

0

119876119896) 1198681

+1198682+8

7

radic1205871205911198621198773

0

1198761198961198683)

(23)

where

120582119873119890

=16119896119871

12058711987740

(24)


119888= 0


120591119877= minus

119877

2sdot119889119901

119889119911=64

49120591119888+4119876119896

1205871198773+32

7

radic119896119876120591119888

1205871198773 (25)



120591119873=64

49120591119888+4119876119896

1205871198773

0

+32

7radic119896119876120591119888

1205871198773

0

(26)


1205911=120591119877

120591119873

=4119876119896 + (327) (119876119896120591

1198881205871198773

0)12

(1198771198770)32

+ (6449) 1205911198881205871198773

0(1198771198770)3

(1198771198770)3

4119876119896 + (327) (1198761198961205911198881205871198773

0)12

+ (6449) 1205911198881205871198773

0

(27)


1205912=120591119877

120591119873119890

=1

(1198771198770)3+16

49(1205871198773

0120591119862

119876119896)

+8

7

1

(1198771198770)32

radic1205871198773

0120591119862

119876119896

(28)

where

120591119873119890

=4119896119876

1205871198773

0

(29)




119888 00ndash05 dynecm2 119898 = 2 119871 =



41 Skin-Friction



0119871 and



119888 but it












1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911



119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911










119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)


0= 3



119888= 005 are





0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7


112141618

2

1205911






z


different fluids



z

112141618

2







119888= 005











08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198683=1198710

2

[

[

1

(1 minus 119886119890minus(1198871198712

012)(4+2radic3))

52

+1

(1 minus 119886119890minus(1198871198712


52

]

]

(22)


119873119890


1205822=

120582

120582119873119890

= (1 +16

49(1205871205911198621198773

0

119876119896) +

8

7

radic1205871205911198621198773

0

119876119896)


119871) +

1

119871(16

49(1205871205911198621198773

0

119876119896) 1198681

+1198682+8

7

radic1205871205911198621198773

0

1198761198961198683)

(23)

where

120582119873119890

=16119896119871

12058711987740

(24)


119888= 0


120591119877= minus

119877

2sdot119889119901

119889119911=64

49120591119888+4119876119896

1205871198773+32

7

radic119896119876120591119888

1205871198773 (25)



120591119873=64

49120591119888+4119876119896

1205871198773

0

+32

7radic119896119876120591119888

1205871198773

0

(26)


1205911=120591119877

120591119873

=4119876119896 + (327) (119876119896120591

1198881205871198773

0)12

(1198771198770)32

+ (6449) 1205911198881205871198773

0(1198771198770)3

(1198771198770)3

4119876119896 + (327) (1198761198961205911198881205871198773

0)12

+ (6449) 1205911198881205871198773

0

(27)


1205912=120591119877

120591119873119890

=1

(1198771198770)3+16

49(1205871198773

0120591119862

119876119896)

+8

7

1

(1198771198770)32

radic1205871198773

0120591119862

119876119896

(28)

where

120591119873119890

=4119896119876

1205871198773

0

(29)




119888 00ndash05 dynecm2 119898 = 2 119871 =



41 Skin-Friction



0119871 and



119888 but it












1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911



119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911










119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)


0= 3



119888= 005 are





0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7


112141618

2

1205911






z


different fluids



z

112141618

2







119888= 005











08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

12

14

16

18

2


120591c = 00 L0L = 03

120591c = 01 L0L = 03

120591c = 05 L0L = 03

L0L = 03 120591c = 005

L0L = 02 120591c = 005

L0L = 01 120591c = 005

z

1205911



119888

112141618

22224

0 005 01 015 02 025

120591c = 005

120591c = 01

120591c = 05

120591c = 00

a = 120575R0

1205911










119877 (119911)

1198770

= 1 minus120575

21198770

[1 + cos 21205871198710

(119911 minus 119889 minus1198710

2)]

119889 le 119911 le 119889 + 1198710

(30)


0= 3



119888= 005 are





0119871 = 03

119911120591

119896 = 2 119896 = 4 119896 = 7


112141618

2

1205911






z


different fluids



z

112141618

2







119888= 005











08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

085

09

095

1

Cosine curve

Bell-shaped curve


z

RR

0



z

L0L = 01

L0L = 02

L0L = 03

1

12

14

16

18

2

1205912







2with


119888= 005 and 119896 = 4




119888= 005 and 119896 = 4 are shown in




119888= 005




z

112141618

2422

21205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005



1

15

2

25

3

0 005 01 015 02 025

1205912 120591c = 01

120591c = 00

120591c = 05

120591c = 005

a = 120575R0


0for

















119888= 004 The


2are




0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0119871 = 1


119888

119886 = 010 119886 = 015

Radius 1198770(cm)


1529080 times 10minus2 1567142 1722748 1745439 2008973

(1 7167)Femoral

5718029 times 10minus3 1429659 1581644 1598551 1856356

(05 1963)Carotid

4571948 times 10minus3 1407374 1558726 1574716 1831521

(04 1257)Coronary

8733753 times 10minus4 1303768 1451998 1463802 1715671

(015 347)Arteriole

5629388 times 10minus3 1428014 1579953 1596792 1854524

(0008 000008)


z

k = 2

k = 4

k = 7

112141618

2

1205912



0

05

1

15

2

25

Bingham fluid




z

1205912






Bingham fluid




z

005

1

15

2

25

1205912




z

1

12141618

2

1205912






2) in







0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0119871 = 1

and 120591119888= 004 dyn cmminus2


119888

119886 = 010 119886 = 015

120582 () 120591 () 120582 () 120591 ()Aorta 1529080 times 10





minus3 2114 3403 3545 5732



Blood vessels


2()

119886 = 01 119886 = 01 119886 = 01 119886 = 02 119886 = 02 119886 = 01 119886 = 02

119898 = 2 119898 = 2 119898 = 3 119898 = 3 119898 = 2

1198710= 10 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 15 119871

0= 10 119871

0= 15


0811214161822224


z

120591H = 00

120591H = 03

120591H = 05

n = 095

n = 1

n = 105120591


119867


5 Conclusion


08

1

12

14

0 005 01 015 02 025

L0L = 01L0L = 02

L0L = 03

1205821

a = 120575R0


0


08

1

12

14

16

18

0 005 01 015 02 025a = 120575R0

L0L = 01 L0L = 02

L0L = 03

1205822


0




1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

12

14

16

18

1205822

0 005 01 015 02 025a = 120575R0

120591c = 005120591c = 01

120591c = 05

120591c = 00


0


00002000400060008

0010012001400160018

k = 2 120591c = 005

k = 4 120591c = 005

k = 7 120591c = 005

120591c = 00 k = 4

120591c = 01 k = 4

120591c = 05 k = 4

Q(c

m3s

)


z


119888











Nomenclature
















Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Mathematical Analysis of Casson Fluid...

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