Research Article Mathematical Analysis on Heat Transfer ...

Research ArticleMathematical Analysis on Heat Transfer duringPeristaltic Pumping of Fractional Second-Grade Fluid througha Nonuniform Permeable Tube

Siddharth Shankar Bhatt1 Amit Medhavi2 and R S Gupta1

1Department of Applied Sciences and Humanities Kamla Nehru Institute of Technology Sultanpur Uttar Pradesh 228118 India2Department of Mechanical Engineering Kamla Nehru Institute of Technology Sultanpur India

Correspondence should be addressed to Siddharth Shankar Bhatt shankarbhatt56gmailcom

Received 8 March 2016 Accepted 19 May 2016

Academic Editor Jose M Montanero

Copyright copy 2016 Siddharth Shankar Bhatt et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This mathematical study is related to heat transfer under peristaltic flow of fractional second-grade fluid through nonuniformcylindrical tube with permeable wallsThe analysis is performed under low Reynolds number and long wavelength approximationThe analytical solution for pressure gradient friction force and temperature field is obtainedThe effects of appropriate parameterssuch as Grashof number nonuniformity of tube permeability of tube wall heat sourcesink parameter material constant fractionaltime derivative parameter and amplitude ratio on pressure rise friction force and temperature distribution are discussed It is foundthat an increase in amplitude ratio andmaterial constant causes increase in pressure but increase in nonuniformity of the tube causesdecrease in pressure It is also observed that variation of friction force against flow rate shows opposite behavior to that of pressureIncrease in temperature is also observed due to increase in heat sourcesink parameter at inlet as well as downstream

1 Introduction

Theword ldquoperistalsisrdquo originated from the Greek word ldquoperi-staltikosrdquo which means clasping and compressing Peristalsisis a mechanism of fluid transport through deformable vesselswith the aid of a progressive contractionexpansion wavealong the vessel It is an important mechanism of fluidtransport in different parts of the entire physiological systemPeristaltic flow appears in urine transport from kidney tobladder the movement of spermatozoa in the ductus effer-entes of the male reproductive tract and in the vasomotion ofsmall blood vessels such as arterioles venules and capillariesPeristaltic motion finds application in industry such as heartlung machine and roller pump Peristaltic motion was firststudied clinically in an article given by Bayliss and Starling[1] and much later Latham [2] theoretically investigatedperistalsis using fluid mechanics principles The work ofJaffrin and Shapiro [3] throws light on various parametersinvolved in the analysis of peristaltic pumping Kumar et al[4] studied unsteady peristaltic pumping in finite length tubewith permeable wall

Interaction of peristalsis with heat transfer plays animportant role in biomedical science Models of microvascu-lar heat transfer are being increasingly used for optimizingthermal therapies such as hyperthermia treatment Thermo-dynamic aspects of blood become significant in processeslike oxygenation and hemodialysis Victor and Shah [5]studied heat transfer to blood flowing in a tube Srinivasand Kothandapani [6] investigated peristaltic transport inan asymmetric channel with heat transfer Muthuraj andSrinivas [7] studied mixed convective heat and mass transferin a vertical wavy channel with traveling thermal wavesand porous medium Lots of investigations have been donefor uniform channel or tube however most physiologi-cal vessels for example ureters esophagus intestine andductus efferentes of the reproductive tract possess nonuni-form geometries Some researcher have used nonuniformgeometry for analysis such as Radhakrishnamacharya andRadhakrishna [8] who discussed heat transfer to peristaltictransport in a nonuniform channel Ellahi et al [9] discussedeffect of heat and mass transfer on peristaltic flow in anonuniform rectangular ductHowever the study of heat flow

Hindawi Publishing CorporationJournal of FluidsVolume 2016 Article ID 7506953 8 pageshttpdxdoiorg10115520167506953

2 Journal of Fluids

during peristalsis has not been given much consideration byinvestigators

Most physiological fluids exhibit both viscous and elasticproperties Viscoelastic models are derived by using classicalmechanics laws that is Newtonrsquos law for viscous liquids andHookersquos law for elastic solids In modern days fractional cal-culus is a rapidly growing field of research in physics biologyand medical engineering By using fractional calculus theviscoelastic behavior of fluid can be successfully explainedFractional second-grade calculus operator is actually a gener-alization to deal with integrals and derivatives of nonintegerorder Fractional second-grade model can be obtained byreplacing ordinary time derivative to fractional time deriva-tiveThismodel is applied to the study ofmovement of chymethrough small intestine esophagus and so forth Using theconcept of fractional calculus some visoelastic models havebeen developed such as fractional Maxwell fractional Zenerfractional anti-Zener or fractional Jeffrey and fractionalBurgersrsquo models Number of researchers has studied unsteadyflows of viscoelastic fluids using different models like frac-tional Maxwell model fractional second-grade fluid modelfractional Burgersrsquo model and fractional generalized Burg-ersrsquo model and fractional Oldroyed-B model in channelsannulus or tubes Qi and Xu [10] studied unsteady flow ofviscoelastic fluid with fractionalMaxwell model in a channelTripathi [11] explored the transportation of a viscoelastic fluidwith fractional second-grade model by peristalsis throughcylindrical tube under the assumptions of long wavelengthand low Reynolds number Tripathi and Beg [12] studiedperistaltic propulsion of generalized Burgersrsquo fluid through anonuniform porous medium with chyme dynamics throughdiseased intestine Hameed et al [13] analysed the heattransfer on peristaltic flow of the fractional second-gradefluid confined in a uniform cylindrical tube in presenceof magnetic field Rathod and Tuljappa [14] have studiedperistaltic flow of fractional second-grade fluid through acylindrical tube with heat transfer under the assumption oflong wavelength and low Reynolds number assumption

The objective of this study is to investigate the effectof various concerned parameters on fractional second-grade fluid with heat transfer and peristaltic flow throughnonuniform tube with permeable walls The effects of theseparameters have been studied under longwave length and thelow Reynolds number approximation The problem is solvedanalytically by the use of fractional calculus The obtainedexpressions are utilized to discuss the influences of variousphysical parameters

2 Basic Definitions

Definition 1 TheRiemann-Liouville fractional integral oper-ator of order 120572 gt 0 of a function 119891(119909) (0infin) rarr 119877 is givenby [15]

119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585 120572 gt 0 119909 gt 0 (1)

x

120578a

b

H(X t)

R

Figure 1 Geometry of the problem

Definition 2 The fractional derivative of order 120572 gt 0 of acontinuous function 119891(119909) (0infin) rarr 119877 is given by [15]

119863120572

119891 (119909) =

1

Γ (119898 minus 120572)

(

119889

119889119909

)

119898

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585

for 119898 minus 1 lt 120572 le 119898 119898 isin 119873 119909 gt 0 119891 isin 119862119898

minus1

(2)

where119898 = [120572]+1 provided that right-hand side is point-wisedefined on (0infin)

Remark 3 For example 119891(119909) = 119909120573 we quote for 120573 gt minus1 in

(2) one can get

119863120572

119909120573

=

Γ (120573 + 1)

Γ (120573 minus 120572 + 1)

119909120573minus120572

(3)

giving in particular 119863120572119909120573minus119899 = 0 119899 = 1 2 3 119873 where 119873is the smallest integer greater than or equal to 120572

3 Mathematical Modelling

Consider the flow of an incompressible fractional second-grade fluid (as shown in Figure 1) due to peristaltic transportinduced by sinusoidal wave trains propagating with constantspeed 119888 The temperature of walls of tube is 119879

0 The consti-

tutive equation for viscoelastic fluid with fractional second-grade model is given by

= 120583 (1 +1205821

120572 120597120572

120597

120572) (4)

where and 1205821is time shear stress rate of shear strain

and material constant respectively 120583 is viscosity and 120572 isfractional time derivative parameters such that 0 lt 120572 le 1This model reduces to second-grade models when 120572 = 1

and classical Navier-Stokes model is obtained by substituting1205821= 0The geometry of wall surface is given by

119867 = 119886 + 1198611015840

119883 + 119887 sin 2120587

120578

(119883 minus 119888119905) (5)

Journal of Fluids 3

where 119886 119887 120578 119888 and 119905 are radius of the tube at inlet waveamplitude wave length wave propagation speed and timerespectively 1198611015840 is a constant whose magnitude depends onthe length of tube

The governing equations of the motion of viscoelasticfluid with fractional second-grade model through inclinedtube for axisymmetric flow are given by

continuity equation

1

119877

120597 (119877119881)

120597119877

+

120597119880

120597119883

= 0 (6)

momentum equation

120588(

120597119880

120597119905

+ 119880

120597119880

120597119883

+ 119881

120597119880

120597119877

)

= minus

120597119901

120597119883

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

+

1205972

119880

1205971198832]

+ 1205881198921205721(119879 minus 119879

0)

120588 (

120597119881

120597119905

+ 119880

120597119881

120597119883

+ 119881

120597119881

120597119877

)

= minus

120597119901

120597119877

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877119881)

120597119877

+

1205972

119881

1205971198832]

(7)

energy equation

120588119888119901(

120597119879

120597119905

+ 119880

120597119879

120597119883

+ 119881

120597119879

120597119877

)

= 119870(

1205972

119879

1205971198772+

1

119877

120597119879

120597119877

+

1205972

119879

1205971198832) + 119876

0

(8)

We introduce nondimensional parameters

1198811015840

=

119881

119888120575

1198801015840

=

119880

119888

1198831015840

=

119883

120578

1198771015840

=

119877

119886

1205821015840

=

1205821119888

120578

120575 =

119886

120578

1199011015840

=

1199011198862

120583119888120578

Re =120588119886119888120575

120583

120579 =

119879 minus 1198790

1198790

Pr =120583119888119901

119870

Gr =12058811989212057211198862

1198790

120583119888

1199051015840

=

119888119905

120578

120573 =

1198862

1198760

1198701198790

ℎ =

119867

119886

(9)

where 120588 is density of fluid 1198760is the constant heat and

119901 119880 119881 119877 120583 119870 1205721 Pr 120573 and Gr stand for pressure

axial velocity radial velocity radial coordinate coefficientof viscosity thermal conductivity coefficient of expansionPrandtl number sourcesink parameter Grashof numberrespectively

Using nondimensional parameters given in (9) and apply-ing long wavelength and low Reynolds number approxima-tion (7) and (8) reduce to (after dropping primes)

120597119901

120597119883

= (1 + 120582120572120597120572

120597119905120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

]

+ 120579Gr

120597119901

120597119877

= 0

1205972

120579

1205971198772+

1

119877

120597120579

120597119877

+ 120573 = 0

(10)

The boundary conditions in dimensionless form are given asfollows

120597120579

120597119877

= 0 at 119877 = 0 120579 = 0 at 119877 = ℎ

120597119880

120597119877

= 0 at 119877 = 0 119880 = minus119896

120597119880

120597119877

at 119877 = ℎ

(11)

where 119896 is the slip parameter including slip [16]The nondimensional wall surface geometry ℎ is given as

ℎ = 1 + 119861119883 + 0 sin 2120587 (119883 minus 119905) (12)

where 119861 = 1198611015840

120578119886 0 = 119887119886 is amplitude ratio

4 Journal of Fluids

On solving (10) with boundary condition (11) the follow-ing is obtained

(1 + 120582120572120597120572

120597119905120572)119880

= (

1198772

4

minus

ℎ2

4

minus

119896ℎ

2

)

120597119901

120597119883

minus

Gr12057364

(41198772

ℎ2

minus 1198774

minus 3ℎ4

minus 4119896ℎ3

)

(13)

120579 =

120573

4

(ℎ2

minus 1198772

) (14)

The volumetric flow rate is calculated as

119876 = int

ℎ

0

2120587119877119880119889119877 (15)

Using (13) in (15) the following is obtained

(1 + 120582120572120597120572

120597119905120572)

119876

120587

= (minus

ℎ4

8

minus

119896ℎ3

2

)

120597119901

120597119883

+

4Gr120573ℎ6

192

+

Gr120573119896ℎ5

16

(16)

Equation (16) gives

120597119901

120597119883

=

(1 + 120582120572

(120597120572

120597119905120572

)) (119876120587) minus 2Gr120573ℎ696 minus Gr120573119896ℎ516(minusℎ48 minus 119896ℎ

32)

(17)

It is observed that as 120573 rarr 0 and 119896 rarr 0 (12) and (15) reduceto the corresponding result of Tripathi [11]

Pressure rise and friction force at wall are given by

Δ119901 = int

1

0

119889119901

119889119883

119889119883

119865 = int

1

0

ℎ2

(minus

119889119901

119889119883

)119889119883

(18)

4 Result and Discussion

MATHEMATICA package is used to see quantitative effectsof various parameters involved in the result on the pumpingcharacteristic and heat transfer Extensive computation hasbeen performed to reveal the influence of geometry hydro-dynamical parameters on pressure distributation frictionalforce and heat transfer Parameters analyzed are frictionalparameters nonuniformity of the geometry 119861 amplituderatio 0 material constant 120582 Grashof number Gr heatsourcesink parameter 120573 and slip parameter 119896 To discussthe results for the above obtained quantities the form of the

05 10 15 20 25 30 35minus5

5

10

15

20

25

30

120572 = 02

120572 = 04

120572 = 06

Δp

k = 0

k = 01

Q

Figure 2 Pressure versus averaged flow rate for various values of 120572at 120582 = 1 119861 = 01 119905 = 05 120601 = 04 Gr = 3 and 120573 = 5

05 10 15 20 25

10

20

30

40

120573 = 0

120573 = 2

120573 = 4

Δp

k = 0

k = 01

Q


instantaneous volume flow rate119876(119883 119905) periodic in (119883minus119905) isassumed as follows [17]

119876 (119883 119905)

120587

=

119876

120587

minus

1206012

2

+

21198611015840

120578119883

119886

120601 sin 2120587 (119883 minus 119905)

+ 2120601 sin 2120587 (119883 minus 119905)

+ 1206012

(sin 2120587 (119883 minus 119905))2

(19)

where 119876120587 is time average of flow over one period of thewave The above form of 119876(119883 119905) has been assumed in viewof the fact that a constant value of 119876(119883 119905) gives Δ119901 negativeand consequently there would no pumping action in the tubewall

Figures 2ndash7 are plotted to see the variation of pressuredistribution for different physical parameters Figure 2 showsthat pressure increases with increase in fractional parameter120572 for given flow rate Maximum pressure is obtained at zeroflow rate It is also observed that pressure decreases withincreases in slip parameter Figure 3 reveals the fact that

Journal of Fluids 5

Gr = 4Gr = 6Gr = 8

05 10 15 20 25 30

10

20

30

40

Δp

k = 0

k = 01

Q

Figure 4 Pressure versus averaged flow rate for various values of Grat 120582 = 1 119861 = 01 119905 = 05 120601 = 04 120573 = 5 and 120572 = 02

B = 0B = 01B = 02

05 10 15 20 25 30minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

Figure 5 Pressure versus averaged flow rate for various values of 119861at 120582 = 1 Gr = 3 119905 = 05 120601 = 04 120573 = 5 and 120572 = 02

pressure increases with increase in sourcesink parameter 120573for fixed flow rate For given pressure flow rate increaseswith increase in 120573 in pumping as well as copumping regionAgain maximum pressure is obtained for zero flow ratewhich is a strong trend of peristaltic fluid dynamics It canbe also seen that pressure decreases with increases in slipparameter Figure 4 shows the effects of Grashof Number GrIt is clear from Figure 4 that for given pressure flow rateincreases with increase in Grashof Number Gr in pumpingas well as co-pumping region It can be also seen thatpressure increases with increase in Gr for given flow rateAgain it is observed that pressure decreases with increasesin slip parameter The nonuniformity parameter will exertan important influence on pressure distribution in peristalticregime Figure 5 shows the effect of change in nonuniformityparameter 119861 on pressure It can be seen from Figure 5 thaton increasing the value of 119861 pressure decreases for a givenvalue of flow rate Pressure decreases with increases in slipparameter at zero flow rate The variation of pressure againstflow rate for various values of amplitude ratio 0 is presented

05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120601 = 02120601 = 03120601 = 04


05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120582 = 1120582 = 2120582 = 3

Figure 7 Pressure versus averaged flow rate for various values of120601 120582 at 120601 = 04 Gr = 3 119905 = 05 119861 = 01 120573 = 5 and 120572 = 02

in Figure 6 It is clear from the Figure 6 that for given flowrate pressure increases with increases in 0 Figure 7 showsthe variation of pressure with change in material constant 120582It is evident from the Figure 7 that on increasing materialconstant pressure also increases Pressure decreases withincreases in slip parameter at zero flow rate

Figures 8ndash13 are plotted to examine the variation offriction force against flow rate for different physical param-eters It is evident from Figures 8ndash13 that there exist directproportionality between frictional force (119865) and flow rate(119876) irrespective of the parameter varied Figure 8 shows thevariation of frictional force against flow rate with changein fractional parameter 120572 It is clear from the Figure 8 thatfrictional force decreases with increase in 120572 and it is observedthat frictional force is higherwith increase in slip parameter atzero flow rate Figure 9 depicts the variation of frictional forceagainst flow rate for different values of sourcesink parameter120573 It can be examined from Figure 9 that frictional forcedecreases with increase in 120573 for given flow rate The greatestvalue of friction force is associated with lowest value of 120573Frictional force is higher with increase in slip parameter at

6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q

Figure 8 Friction force versus averaged flow rate for various valuesof 120572 at 120582 = 1 119905 = 05 120601 = 04 119861 = 01 Gr = 3 and 120573 = 5

05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q

Figure 9 Friction force versus averaged flow rate for various valuesof 120572 120573 at 120582 = 1 119905 = 05 120601 = 04 119861 = 01 Gr = 3 and 120572 = 02

k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q

Figure 10 Friction force versus averaged flow rate for various valuesof Gr at 120582 = 1 119905 = 05 120601 = 04 119861 = 01 120572 = 02 and 120573 = 5

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q

Figure 11 Friction force versus averaged flow rate for various valuesof 119861 at 120582 = 1 119905 = 05 120601 = 04 Gr = 3 120572 = 02 and 120573 = 5

05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579

Figure 14 Effect of 120573 on temperature at 120601 = 04 119861 = 01 and 119905 = 05

at inlet as well as downstream

zero flow rate Figure 10 portrays the variation of frictionalforce against flow rate for different values of Grashof NumberGr It is observed fromFigure 10 that frictional force decreaseswith increase in Gr for given flow rate Frictional force ishigher with increase in slip parameter at zero flow rate It isclear from the Figure 11 that frictional force decreases withincrease in 119861 Frictional force is higher with increase in slipparameter at zero flow rate The effect of amplitude ratio 0

on pressure rise is illustrated in Figure 12 It is noted thatfrictional force decreaseswith increase in 0 for given flow rateFrictional force is higher with increase in slip parameter atzero flow rate The effect of material constant 120582 on pressurerise is illustrated in Figure 13 It is evident from Figure 13that frictional force decreases with increase in 120582 For lowflow rate frictional force is negative but becomes positive forhigher flow rate Frictional force is higher with increase in slipparameter at zero flow rate

Figures 14ndash16 show the effects of various parameterson temperature Temperature increases with increase insourcesink parameter120573 at inlet aswell as downstreamas por-trayed in Figure 14 Temperature is higher at inlet but as wemove downstream temperature gradually decreases Figure 15illustrates the variation of temperaturewith amplitude ratio atinlet and downstream It is clear that temperature decreaseswith increase in amplitude ratio at inlet and remains constantat downstream Figure 16 depicts the variation of temperaturewith nonuniformity parameter 119861 It is clear that temperatureremains unaffected with increase in 119861 at inlet but increasesdownstream It is also observed that temperature is higher atinlet

5 Conclusion

In this paper viscoelastic fluid flow with fractional second-grade model for heat effect with peristaltic transport throughnonuniform inclined tube with permeable wall Exact expres-sion for pressure gradient and temperature are obtainedunder long wavelength and low Reynolds number Effects

02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579



of fractional parameter nonuniformity of tube permeabilityof wall material constant sourcesink parameter amplituderatio and Grashof number are studied Fractional calculustheory is used to find pressure gradient expression Thefollowing conclusion can be summarized

(1) The linear relation is found between pressure andflow

(2) The linear relation is found between friction force andflow

(3) The pressure function increases with increasingvalue of fractional parameter sourcesink parameterGrashof Number and amplitude ratio

(4) The variation of friction force against flow rate showsopposite behavior to that of pressure

(5) Temperature increases with increasing value ofsourcesink parameter at inlet as well as downstreamof the tube

8 Journal of Fluids

(6) Temperature remains constant with increasing valueof nonuniformity parameter and amplitude ratio atinlet butmay increase with increase in nonuniformityand decreases with increases in amplitude ratio atdownstream

Competing Interests

The authors declare that they have no competing interests

References

[1] W M Bayliss and E H Starling ldquoThe movement and innerva-tion of the small intestinerdquoThe Journal of Physiology vol 24 no2 pp 99ndash143 1899

[2] T W Latham Fluid motion in a peristaltic pump [MS thesis]Massachusetts Institute of Technology Cambridge Mass USA1966

[3] M Y Jaffrin and A H Shapiro ldquoPeristaltic pumpingrdquo AnnualReview of Fluid Mechanics vol 3 pp 13ndash36 1971

[4] Y V K R Kumar P S V H N K Kumari M V RMurthy andS Sreenadh ldquoUnsteady peristaltic pumping in a finite lengthtube with permeable wallrdquo Journal of Fluids Engineering vol132 no 10 Article ID 101201 4 pages 2010

[5] S A Victor and V L Shah ldquoHeat transfer to blood flowing in atuberdquo Biorheology vol 12 no 6 pp 361ndash368 1975

[6] S Srinivas and M Kothandapani ldquoPeristaltic transport in anasymmetric channel with heat transfermdasha noterdquo InternationalCommunications in Heat and Mass Transfer vol 35 no 4 pp514ndash522 2008

[7] R Muthuraj and S Srinivas ldquoMixed convective heat and masstransfer in a vertical wavy channel with traveling thermalwaves and porous mediumrdquo Computers amp Mathematics withApplications vol 59 no 11 pp 3516ndash3528 2010

[8] G Radhakrishnamacharya andV Radhakrishna ldquoHeat transferto peristaltic transport in a non-uniform channelrdquo DefenceScience Journal vol 43 no 3 pp 275ndash280 1993

[9] R Ellahi M M Bhatti and K Vafai ldquoEffects of heat and masstransfer on peristaltic flow in a non-uniform rectangular ductrdquoInternational Journal of Heat andMass Transfer vol 71 pp 706ndash719 2014

[10] H Qi and M Xu ldquoUnsteady flow of viscoelastic fluid withfractional Maxwell model in a channelrdquo Mechanics ResearchCommunications vol 34 no 2 pp 210ndash212 2007

[11] D Tripathi ldquoPeristaltic flow of a fractional second grade fluidthrough a cylindrical tuberdquoThermal Science vol 15 supplement2 pp 167ndash173 2011

[12] D Tripathi and O A Beg ldquoPeristaltic propulsion of generalizedBurgersrsquo fluids through a non-uniform porous medium astudy of chyme dynamics through the diseased intestinerdquoMathematical Biosciences vol 248 pp 67ndash77 2014

[13] M Hameed A A Khan R Ellahi and M Raza ldquoStudy ofmagnetic and heat transfer on the peristaltic transport of afractional second grade fluid in a vertical tuberdquo EngineeringScience and Technology an International Journal vol 18 no 3pp 496ndash502 2015

[14] V P Rathod and A Tuljappa ldquoPeristaltic flow of fractionalsecond grade fluid through a cylindricakl tube with heattransferrdquo Journal of Chemical Biological and Physical Sciencesvol 5 pp 1841ndash1855 2015

[15] V K Narla K M Prasad and J V Ramanamurthy ldquoPeristalticmotion of viscoelastic fluid with fractional second grade modelin curved channelsrdquo Chinese Journal of Engineering vol 2013Article ID 582390 7 pages 2013

[16] P G Saffman ldquoOn the boundary condition at the surface of aporous mediumrdquo Studies in Applied Mathematics vol 50 no 2pp 93ndash101 1971

[17] L M Srivastava and V P Srivastava ldquoPeristaltic transport ofa power-law fluid application to the ductus efferentes of thereproductive tractrdquo Rheologica Acta vol 27 no 4 pp 428ndash4331988

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2 Journal of Fluids

during peristalsis has not been given much consideration byinvestigators

Most physiological fluids exhibit both viscous and elasticproperties Viscoelastic models are derived by using classicalmechanics laws that is Newtonrsquos law for viscous liquids andHookersquos law for elastic solids In modern days fractional cal-culus is a rapidly growing field of research in physics biologyand medical engineering By using fractional calculus theviscoelastic behavior of fluid can be successfully explainedFractional second-grade calculus operator is actually a gener-alization to deal with integrals and derivatives of nonintegerorder Fractional second-grade model can be obtained byreplacing ordinary time derivative to fractional time deriva-tiveThismodel is applied to the study ofmovement of chymethrough small intestine esophagus and so forth Using theconcept of fractional calculus some visoelastic models havebeen developed such as fractional Maxwell fractional Zenerfractional anti-Zener or fractional Jeffrey and fractionalBurgersrsquo models Number of researchers has studied unsteadyflows of viscoelastic fluids using different models like frac-tional Maxwell model fractional second-grade fluid modelfractional Burgersrsquo model and fractional generalized Burg-ersrsquo model and fractional Oldroyed-B model in channelsannulus or tubes Qi and Xu [10] studied unsteady flow ofviscoelastic fluid with fractionalMaxwell model in a channelTripathi [11] explored the transportation of a viscoelastic fluidwith fractional second-grade model by peristalsis throughcylindrical tube under the assumptions of long wavelengthand low Reynolds number Tripathi and Beg [12] studiedperistaltic propulsion of generalized Burgersrsquo fluid through anonuniform porous medium with chyme dynamics throughdiseased intestine Hameed et al [13] analysed the heattransfer on peristaltic flow of the fractional second-gradefluid confined in a uniform cylindrical tube in presenceof magnetic field Rathod and Tuljappa [14] have studiedperistaltic flow of fractional second-grade fluid through acylindrical tube with heat transfer under the assumption oflong wavelength and low Reynolds number assumption

The objective of this study is to investigate the effectof various concerned parameters on fractional second-grade fluid with heat transfer and peristaltic flow throughnonuniform tube with permeable walls The effects of theseparameters have been studied under longwave length and thelow Reynolds number approximation The problem is solvedanalytically by the use of fractional calculus The obtainedexpressions are utilized to discuss the influences of variousphysical parameters

2 Basic Definitions

Definition 1 TheRiemann-Liouville fractional integral oper-ator of order 120572 gt 0 of a function 119891(119909) (0infin) rarr 119877 is givenby [15]

119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585 120572 gt 0 119909 gt 0 (1)

x

120578a

b

H(X t)

R

Figure 1 Geometry of the problem

Definition 2 The fractional derivative of order 120572 gt 0 of acontinuous function 119891(119909) (0infin) rarr 119877 is given by [15]

119863120572

119891 (119909) =

1

Γ (119898 minus 120572)

(

119889

119889119909

)

119898

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585

for 119898 minus 1 lt 120572 le 119898 119898 isin 119873 119909 gt 0 119891 isin 119862119898

minus1

(2)

where119898 = [120572]+1 provided that right-hand side is point-wisedefined on (0infin)

Remark 3 For example 119891(119909) = 119909120573 we quote for 120573 gt minus1 in

(2) one can get

119863120572

119909120573

=

Γ (120573 + 1)

Γ (120573 minus 120572 + 1)

119909120573minus120572

(3)

giving in particular 119863120572119909120573minus119899 = 0 119899 = 1 2 3 119873 where 119873is the smallest integer greater than or equal to 120572

3 Mathematical Modelling

Consider the flow of an incompressible fractional second-grade fluid (as shown in Figure 1) due to peristaltic transportinduced by sinusoidal wave trains propagating with constantspeed 119888 The temperature of walls of tube is 119879

0 The consti-

tutive equation for viscoelastic fluid with fractional second-grade model is given by

= 120583 (1 +1205821

120572 120597120572

120597

120572) (4)

where and 1205821is time shear stress rate of shear strain

and material constant respectively 120583 is viscosity and 120572 isfractional time derivative parameters such that 0 lt 120572 le 1This model reduces to second-grade models when 120572 = 1

and classical Navier-Stokes model is obtained by substituting1205821= 0The geometry of wall surface is given by

119867 = 119886 + 1198611015840

119883 + 119887 sin 2120587

120578

(119883 minus 119888119905) (5)

Journal of Fluids 3



continuity equation

1

119877

120597 (119877119881)

120597119877

+

120597119880

120597119883

= 0 (6)

momentum equation

120588(

120597119880

120597119905

+ 119880

120597119880

120597119883

+ 119881

120597119880

120597119877

)

= minus

120597119901

120597119883

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

+

1205972

119880

1205971198832]

+ 1205881198921205721(119879 minus 119879

0)

120588 (

120597119881

120597119905

+ 119880

120597119881

120597119883

+ 119881

120597119881

120597119877

)

= minus

120597119901

120597119877

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877119881)

120597119877

+

1205972

119881

1205971198832]

(7)

energy equation

120588119888119901(

120597119879

120597119905

+ 119880

120597119879

120597119883

+ 119881

120597119879

120597119877

)

= 119870(

1205972

119879

1205971198772+

1

119877

120597119879

120597119877

+

1205972

119879

1205971198832) + 119876

0

(8)


1198811015840

=

119881

119888120575

1198801015840

=

119880

119888

1198831015840

=

119883

120578

1198771015840

=

119877

119886

1205821015840

=

1205821119888

120578

120575 =

119886

120578

1199011015840

=

1199011198862

120583119888120578

Re =120588119886119888120575

120583

120579 =

119879 minus 1198790

1198790

Pr =120583119888119901

119870

Gr =12058811989212057211198862

1198790

120583119888

1199051015840

=

119888119905

120578

120573 =

1198862

1198760

1198701198790

ℎ =

119867

119886

(9)





120597119901

120597119883

= (1 + 120582120572120597120572

120597119905120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

]

+ 120579Gr

120597119901

120597119877

= 0

1205972

120579

1205971198772+

1

119877

120597120579

120597119877

+ 120573 = 0

(10)


120597120579

120597119877

= 0 at 119877 = 0 120579 = 0 at 119877 = ℎ

120597119880

120597119877

= 0 at 119877 = 0 119880 = minus119896

120597119880

120597119877

at 119877 = ℎ

(11)


ℎ = 1 + 119861119883 + 0 sin 2120587 (119883 minus 119905) (12)

where 119861 = 1198611015840


4 Journal of Fluids


(1 + 120582120572120597120572

120597119905120572)119880

= (

1198772

4

minus

ℎ2

4

minus

119896ℎ

2

)

120597119901

120597119883

minus

Gr12057364

(41198772

ℎ2

minus 1198774

minus 3ℎ4

minus 4119896ℎ3

)

(13)

120579 =

120573

4

(ℎ2

minus 1198772

) (14)


119876 = int

ℎ

0

2120587119877119880119889119877 (15)


(1 + 120582120572120597120572

120597119905120572)

119876

120587

= (minus

ℎ4

8

minus

119896ℎ3

2

)

120597119901

120597119883

+

4Gr120573ℎ6

192

+

Gr120573119896ℎ5

16

(16)

Equation (16) gives

120597119901

120597119883

=

(1 + 120582120572

(120597120572

120597119905120572


32)

(17)



Δ119901 = int

1

0

119889119901

119889119883

119889119883

119865 = int

1

0

ℎ2

(minus

119889119901

119889119883

)119889119883

(18)



05 10 15 20 25 30 35minus5

5

10

15

20

25

30

120572 = 02

120572 = 04

120572 = 06

Δp

k = 0

k = 01

Q


05 10 15 20 25

10

20

30

40

120573 = 0

120573 = 2

120573 = 4

Δp

k = 0

k = 01

Q



119876 (119883 119905)

120587

=

119876

120587

minus

1206012

2

+

21198611015840

120578119883

119886

120601 sin 2120587 (119883 minus 119905)

+ 2120601 sin 2120587 (119883 minus 119905)

+ 1206012

(sin 2120587 (119883 minus 119905))2

(19)



Journal of Fluids 5

Gr = 4Gr = 6Gr = 8

05 10 15 20 25 30

10

20

30

40

Δp

k = 0

k = 01

Q


B = 0B = 01B = 02

05 10 15 20 25 30minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q



05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120601 = 02120601 = 03120601 = 04


05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120582 = 1120582 = 2120582 = 3




6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q


05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q


k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q


05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 3



continuity equation

1

119877

120597 (119877119881)

120597119877

+

120597119880

120597119883

= 0 (6)

momentum equation

120588(

120597119880

120597119905

+ 119880

120597119880

120597119883

+ 119881

120597119880

120597119877

)

= minus

120597119901

120597119883

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

+

1205972

119880

1205971198832]

+ 1205881198921205721(119879 minus 119879

0)

120588 (

120597119881

120597119905

+ 119880

120597119881

120597119883

+ 119881

120597119881

120597119877

)

= minus

120597119901

120597119877

+ 120583(1 +1205821

120572 120597120572

120597

120572)[

1

119877

120597 (119877119881)

120597119877

+

1205972

119881

1205971198832]

(7)

energy equation

120588119888119901(

120597119879

120597119905

+ 119880

120597119879

120597119883

+ 119881

120597119879

120597119877

)

= 119870(

1205972

119879

1205971198772+

1

119877

120597119879

120597119877

+

1205972

119879

1205971198832) + 119876

0

(8)


1198811015840

=

119881

119888120575

1198801015840

=

119880

119888

1198831015840

=

119883

120578

1198771015840

=

119877

119886

1205821015840

=

1205821119888

120578

120575 =

119886

120578

1199011015840

=

1199011198862

120583119888120578

Re =120588119886119888120575

120583

120579 =

119879 minus 1198790

1198790

Pr =120583119888119901

119870

Gr =12058811989212057211198862

1198790

120583119888

1199051015840

=

119888119905

120578

120573 =

1198862

1198760

1198701198790

ℎ =

119867

119886

(9)





120597119901

120597119883

= (1 + 120582120572120597120572

120597119905120572)[

1

119877

120597 (119877 (120597119880120597119877))

120597119877

]

+ 120579Gr

120597119901

120597119877

= 0

1205972

120579

1205971198772+

1

119877

120597120579

120597119877

+ 120573 = 0

(10)


120597120579

120597119877

= 0 at 119877 = 0 120579 = 0 at 119877 = ℎ

120597119880

120597119877

= 0 at 119877 = 0 119880 = minus119896

120597119880

120597119877

at 119877 = ℎ

(11)


ℎ = 1 + 119861119883 + 0 sin 2120587 (119883 minus 119905) (12)

where 119861 = 1198611015840


4 Journal of Fluids


(1 + 120582120572120597120572

120597119905120572)119880

= (

1198772

4

minus

ℎ2

4

minus

119896ℎ

2

)

120597119901

120597119883

minus

Gr12057364

(41198772

ℎ2

minus 1198774

minus 3ℎ4

minus 4119896ℎ3

)

(13)

120579 =

120573

4

(ℎ2

minus 1198772

) (14)


119876 = int

ℎ

0

2120587119877119880119889119877 (15)


(1 + 120582120572120597120572

120597119905120572)

119876

120587

= (minus

ℎ4

8

minus

119896ℎ3

2

)

120597119901

120597119883

+

4Gr120573ℎ6

192

+

Gr120573119896ℎ5

16

(16)

Equation (16) gives

120597119901

120597119883

=

(1 + 120582120572

(120597120572

120597119905120572


32)

(17)



Δ119901 = int

1

0

119889119901

119889119883

119889119883

119865 = int

1

0

ℎ2

(minus

119889119901

119889119883

)119889119883

(18)



05 10 15 20 25 30 35minus5

5

10

15

20

25

30

120572 = 02

120572 = 04

120572 = 06

Δp

k = 0

k = 01

Q


05 10 15 20 25

10

20

30

40

120573 = 0

120573 = 2

120573 = 4

Δp

k = 0

k = 01

Q



119876 (119883 119905)

120587

=

119876

120587

minus

1206012

2

+

21198611015840

120578119883

119886

120601 sin 2120587 (119883 minus 119905)

+ 2120601 sin 2120587 (119883 minus 119905)

+ 1206012

(sin 2120587 (119883 minus 119905))2

(19)



Journal of Fluids 5

Gr = 4Gr = 6Gr = 8

05 10 15 20 25 30

10

20

30

40

Δp

k = 0

k = 01

Q


B = 0B = 01B = 02

05 10 15 20 25 30minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q



05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120601 = 02120601 = 03120601 = 04


05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120582 = 1120582 = 2120582 = 3




6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q


05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q


k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q


05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



4 Journal of Fluids


(1 + 120582120572120597120572

120597119905120572)119880

= (

1198772

4

minus

ℎ2

4

minus

119896ℎ

2

)

120597119901

120597119883

minus

Gr12057364

(41198772

ℎ2

minus 1198774

minus 3ℎ4

minus 4119896ℎ3

)

(13)

120579 =

120573

4

(ℎ2

minus 1198772

) (14)


119876 = int

ℎ

0

2120587119877119880119889119877 (15)


(1 + 120582120572120597120572

120597119905120572)

119876

120587

= (minus

ℎ4

8

minus

119896ℎ3

2

)

120597119901

120597119883

+

4Gr120573ℎ6

192

+

Gr120573119896ℎ5

16

(16)

Equation (16) gives

120597119901

120597119883

=

(1 + 120582120572

(120597120572

120597119905120572


32)

(17)



Δ119901 = int

1

0

119889119901

119889119883

119889119883

119865 = int

1

0

ℎ2

(minus

119889119901

119889119883

)119889119883

(18)



05 10 15 20 25 30 35minus5

5

10

15

20

25

30

120572 = 02

120572 = 04

120572 = 06

Δp

k = 0

k = 01

Q


05 10 15 20 25

10

20

30

40

120573 = 0

120573 = 2

120573 = 4

Δp

k = 0

k = 01

Q



119876 (119883 119905)

120587

=

119876

120587

minus

1206012

2

+

21198611015840

120578119883

119886

120601 sin 2120587 (119883 minus 119905)

+ 2120601 sin 2120587 (119883 minus 119905)

+ 1206012

(sin 2120587 (119883 minus 119905))2

(19)



Journal of Fluids 5

Gr = 4Gr = 6Gr = 8

05 10 15 20 25 30

10

20

30

40

Δp

k = 0

k = 01

Q


B = 0B = 01B = 02

05 10 15 20 25 30minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q



05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120601 = 02120601 = 03120601 = 04


05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120582 = 1120582 = 2120582 = 3




6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q


05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q


k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q


05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 5

Gr = 4Gr = 6Gr = 8

05 10 15 20 25 30

10

20

30

40

Δp

k = 0

k = 01

Q


B = 0B = 01B = 02

05 10 15 20 25 30minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q



05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120601 = 02120601 = 03120601 = 04


05 10 15 20 25minus5

5

10

15

20

25

30

Δp

k = 0

k = 01

Q

120582 = 1120582 = 2120582 = 3




6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q


05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q


k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q


05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



6 Journal of Fluids

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0k = 01

120572 = 02120572 = 04120572 = 06

Q


05 10 15 20 25 30

minus15

minus10

minus5

5

10

F

k = 0k = 01

120573 = 0120573 = 2120573 = 4

Q


k = 01

05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

F

k = 0

Gr = 4Gr = 6Gr = 8

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

B = 0B = 01B = 02

Q


05 10 15 20 25 30

minus30

minus20

minus10

10

k = 01

F

k = 0

120601 = 02120601 = 04120601 = 06

Q


05 10 15 20 25 30

minus20

minus15

minus10

minus5

5

10

k = 01

F

k = 0

120582 = 1120582 = 2120582 = 3

Q


Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 7

120573 = 4120573 = 6120573 = 8

x = 02

x = 0

02 04 06 08 10 12 14R

05

10

15

20

120579






5 Conclusion


02 04 06 08 10 12 14

05

10

15

120601 = 02120601 = 04120601 = 06

x = 02

x = 0

R

120579



02 04 06 08 10 12 14

05

10

15

B = 0B = 01B = 02

x = 02x = 0

R

120579









8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



8 Journal of Fluids


Competing Interests


References























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Mathematical Analysis on Heat Transfer ...

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