Hindawi Publishing CorporationJournal of FluidsVolume 2013 Article ID 749271 8 pageshttpdxdoiorg1011552013749271

Research ArticleThermal Jump Effects on Boundary Layer Flow of a Jeffrey FluidNear the Stagnation Point on a StretchingShrinking Sheet withVariable Thermal Conductivity

M A A Hamad1 S M AbdEl-Gaied1 and W A Khan2

1 Mathematics Department Faculty of Science Assiut University Assiut 71516 Egypt2 Department of Engineering Sciences PN Engineering College National University of Science Pakistan

Correspondence should be addressed to S M AbdEl-Gaied sagaied123gmailcom

Received 27 June 2013 Accepted 29 October 2013

Academic Editor Boming Yu

Copyright copy 2013 M A A Hamad et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Amathematical model will be analyzed in order to study the effects of thermal jump and variable thermal conductivity on flow andheat transfer near the stagnation point on a stretchingshrinking sheet in a Jeffrey fluid The highly nonlinear partial differentialequation of Jeffrey fluid flow along with the energy equation are transformed to an ordinary system using nondimensionaltransformationsThe arising equations are solved for temperature velocity shear stress and heat flux using finite differencemethodThe effect of the influences parameters is discussed For nonradiation regular viscous fluid our results are as that by Nazar et al(2002)

1 Introduction

It is well known that the thermophysical properties of afluid play an important role in the engineering applicationsin aerodynamics geothermal systems crude oil extractionsground water pollution thermal insulation heat exchangerstorage of nuclear waste and so forth convective flows overbodies The change in the thermal conductivity with temper-ature is an important property [1ndash5] Prasad and Vajravelu[6] investigated the effect of variable thermal conductivity ina nonisothermal sheet stretching through power law fluidswhile Prasad et al [4] reported similar studies for viscoelasticfluids Abel et al [7] studied combined effects of thermalbuoyancy and variable thermal conductivity on amagnetohy-drodynamic flow and the associated heat transfer in a power-law fluid past a vertical stretching sheet in the presence of anonuniformheat sourceThe general findings of these studieswere that the effects of variable thermal conductivity increasethe shear stress The temperature at wall increase with anincrease in variable thermal conductivity by Seddeek et al [8]Prasad et al [9] found that the variable thermal conductivityhas an impact in enhancing the skin friction coefficient

hence fluids with less thermal conductivity may be opted foreffective cooling Abel et al [10] concluded that the variablethermal conductivity increases the temperature distributionin both prescribed surface temperature and prescribed heatflux cases Mahanti and Gaur [11] investigated the effectsof linearly varying viscosity and thermal conductivity onsteady free convective flow of a viscous incompressible fluidalong an isothermal vertical plate in the presence of heatsink Deissler [12] obtained that the effects of second-orderterms on the velocity and temperature jumps at a wall areby a physical derivation The analysis used the conceptsof effective mean free paths for momentum and energytransfer the effective mean free paths are obtained fromknown viscosities and thermal conductivities Rahman andEltayeb [13] studied numerically the convective slip flow ofslightly rarefied fluids over a wedge with thermal jump andtemperature dependent transport properties such as fluidviscosity and thermal conductivity Cipolla Jr [14] studiedthe temperature jump in polyatomic gas also Kao [15] andLatyshev and Yushkanov [16] studied the temperature jumpThe flow and heat transfer of Jeffry fluid near stagnationpoint on a stretchingshrinking sheet with parallel external

2 Journal of Fluids

flowwas investigated byTurkyilmazoglu andPop [17] Akramand Nadeem [18] discussed the peristaltic motion of a two-dimensional Jeffry fluid Authors in [19ndash22] studied moreproperties in Jeffrey fluid Different non-Newtonian fluidswere considered in studies by Pandey and Tripathi [23ndash25]and Tripathi [26]

Interest in boundary layer flow and heat transfer over astretching sheet has gained considerable attention because ofits application in industry andmanufacturing processes Suchapplications include polymer extrusion drawing of copperwires continuous stretching of plastic films and artificialfibers hot rolling wire drawing glass fiber metal extrusionand metal spinning For example Liu and Andersson [27]studied the heat transfer in a liquid film on an unsteadystretching sheet The effects of variable fluid properties andthermocapillarity on the flow of a thin film on an unsteadystretching sheet were studied by Dandapat et al [28] Hayatet al [29] investigated the peristaltic mechanism of a Jeffreyfluid in a circular tube Nadeem et al [30] analyzed theboundary layer flow of a Jeffrey fluid over an exponentiallystretching surfaceThe effects of thermal radiation are carriedout for two cases of heat transfer analysis known as (1) pre-scribed exponential order surface temperature (PEST) and(2) prescribed exponential order heat flux (PEHF) Hamad[31] studied the convective flow and heat transfer of anincompressible viscous nanofluid past a semi-infinite verticalstretching sheet in the presence of a magnetic field Hamadand Pop [32] studied theoretically the steady boundary layerflownear the stagnation-point flow on a permeable stretchingsheet in a porous medium saturated with a nanofluid and inthe presence of internal heat generationabsorption

The objective of the present study is to investigate thedynamics of the thermal boundary layer flow of a viscousincompressible Jeffrey fluid near the stagnation point ona stretching sheet taking into account the thermal jumpcondition at the surface Thus the main focus of the analysisis to investigate how the flow field temperature field shearstress and heat flux vary within the boundary layer withthermal jump at the wall when the thermal conductivity istemperature dependent The similarity equations are derivedand solved numerically with the widely used and robust com-puter algebra software Graphs and tables are presented toillustrate and discuss important hydrodynamic and thermalfeatures of the flow

2 Problem Formulation

Consider a steady two dimensional flow of an incompress-ible Jeffrey fluid near the stagnation point on a stretch-ingshrinking sheet The thermal conductivity is assumed tobe functions of temperature A thermal jump condition isassumed to occur at the wall We are considering Cartesiancoordinate system in such a way that 119909-axis is taken along thestretching sheet in the direction of the motion and 119910-axis isnormal to it The plate is stretched in the 119909-direction with avelocity 119906

119908= 119888119909 defined at 119910 = 0 The flow and heat transfer

characteristics under the boundary layer approximations aregoverned by the following equations

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910

= 119906119890

119889119906119890

119889119909+

]

1 + 1205741

[1205972119906

1205971199102

+ 1205742(119906

1205973119906

1205971199091205971199102+ V

1205973119906

1205971199103minus120597119906

120597119909

1205972119906

1205971199102

+120597119906

120597119910

1205972119906

120597 119909120597 119910)]

(2)

120588119862119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(120581 (119879)

120597119879

120597119910) (3)

with the boundary conditions (see Rahman and Eltayeb [13])

V = V119908(119909) 119906 = 119906

119908(119909) = 119888119909

119879jump = 119879119891minus 119879119908

= 1205821(

2

120590119879

minus 1)2120574

120574 + 1

120581 (119879)

120583119862119901

120597119879

120597119910at 119910 = 0

119906 = 119906119890(119909) = 119886119909

120597119906

120597119910= 0 119879 = 119879

infin

as 119910 997888rarr infin

(4)

Here119909 and119910 are theCartesian coordinates along the plate andnormal to it respectively 119906 and V are the velocity componentsalong 119909- and 119910-axes V

119908(119909) is the mass transfer velocity with

V119908(119909) lt 0 for suction and V

119908(119909) gt 0 for injection or

withdrawal119879 is the fluid temperature120572 is thermal diffusivity] is the kinematic viscosity 120574

1is the ratio of relaxation and

retardation times 1205742is the relaxation time 120574 is the ratio of

specific heats 120590119879is the thermal accommodation coefficient

1205821is the mean free path 120583 is the dynamic viscosity and 120581(119879)

is the thermal conductivity which can be following Chiam[1] written as

120581 = 120581infin(1 + 120576

119879 minus 119879infin

119879119908minus 119879infin

) (5)

where 120576 is the thermal conductivity parameterWe introduce now the following similarity variables

120595 = radic119886]119909119891 (120578) 120579 (120578) =(119879 minus 119879

infin)

(119879119908minus 119879infin)

120578 = radic119886

]119910

(6)

where 120595 is the stream function which is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Thus V

119908(119909) = minus radic119886]119904

where 119904 is the mass transfer parameter with 119904 gt 0 for suctionand 119904 lt 0 for injection respectively Substituting (5) and

Journal of Fluids 3

(6) into (2) and (3) the following set of ordinary differentialequations results in

119891101584010158401015840

+ (1 + 1205741) (11989111989110158401015840

minus 11989110158402

)

+ 120573 (119891101584010158402

minus 1198911198911015840101584010158401015840

) + (1 + 1205741) = 0

(1 + 120576 120579) 12057910158401015840

+ 12057612057910158402

+ Pr1198911205791015840 = 0

(7)

and the boundary conditions (4) become

119891 (0) = 119904 1198911015840

(0) = 120582

120579 (0) = 1 +2120574119879119904

120574 + 1Pr (1 + 120576120579 (0)) 120579

1015840

(0)

1198911015840

(infin) = 1 11989110158401015840

(infin) = 0 120579 (infin) = 0

(8)

where Pr = 120583119862119901120581infin

is the Prandtl number 120582 = 119888119886 is thestretching (120582 gt 0) or shrinking (120582 lt 0) parameter 120573 = 119888120574

2

is the Deborah number 119879119904= 1205821(2(120590119879minus 1))radic119886] is the slip

parameter and primes denote differentiation with respect to120578

21 Particular Case It is worth mentioning that for a regularviscous fluid (120573 = 120574

1= 0) (7) reduce to the steady state

equations from the paper by Nazar et al [33] when we neglectthe radiation effect

22 PhysicalQuantities Thephysical quantities of interest arethe skin friction coefficient 119862

119891and the local Nusselt number

Nu119909 which are defined as

119862119891=

120591119908

1205881199062119890(119909)

Nu119909=

119909119902119908

120581 (119879119908minus 119879infin) (9)

where 120591119908

is the skin friction or shear stress along thestretching surface and 119902

119908is the heat flux from the surface

which are given by

120591119908= 120583(

120597119906

120597119910)119910=0

119902119908= minus 120581(

120597119879jump

120597119910)119910=0

(10)

Using (6) we get

Re12119909

119862119891= 11989110158401015840

(0)

Reminus12119909

Nu119909

= minus2120574

120574 + 1119879119904Pr [(1 + 120576120579) 120579

10158401015840

(0) + 120576(1205791015840

(0))2

]

(11)

where Re119909= 119906119890(119909)119909] is the local Reynolds number

3 Results and Discussion

The transformed equations (7) with boundary conditions (8)are solved numerically by using a finite difference methodThe asymptotic boundary conditions at 120578 = infin are replacedby 120578 = 6 In Table 1 we have shown the variation of walltemperature and heat transfer rates with the Prandtl numbersfor three different values of Deborah numbers It is observed

0 1 2 3 4 5

Suction without stretching

0

1

02

04

06

08

s = 1 120582 = 0

f998400(120578

)

120578

1205741 = 1 2 3

120573 = 05

120573 = 30

Figure 1 Effect of 120573 and 1205741on the velocity profiles

that the wall temperature decreases whereas the heat transferrates increase with an increase in Prandtl number Howeverthe wall temperature increases slightly and heat transfervalues decrease slightly with an increase in Deborah numberwhereas both decrease with an increase in the ratio of specificheats This is shown in Tables 1(a) and 1(b) Tables 1(c)and 1(d) show the effects of slip temperature on the walltemperature and heat transfer rates for the constant thermalconductivity Table 1(e) shows the effects of thermal slip onthewall temperature and heat transfer rates when the thermalconductivity varies with temperature It can be seen that boththe wall temperature and heat transfer rates decrease withan increase in the Prandtl number due to decrease in thethermal conductivity The effects of the ratio of relaxationand retardation times Deborah number and suction andstretching parameters on the 119909-component of velocity areshown in Figures 1 and 2 Figure 1 shows the effects in theabsence of stretching It shows that the velocity increasesas the ratio of relaxation and retardation times increasesbut decreases with an increase in Deborah number Thevelocity boundary layer converges quickly for small Deborahnumbers In fact small Deborah numbers correspond tosituationswhere thematerial has time to relax (and behaves ina viscous manner) while high Deborah numbers correspondto situations where the material behaves rather elasticallyFigure 2 shows the effects of stretching parameter on thevelocity for different values ofDeborah number It is observedthat the velocity becomes constant when 120582 = 0 increaseswhen 120582 lt 1 and decreases when 120582 gt 0 Accordingly thevelocity decreases or increases with Deborah number when120582 lt 1 or 120582 gt 0

The effect of thermal conductivity parameter on temper-ature profiles is shown in Figure 3 for two different Prandtlnumbers It is observed that the thermal boundary layer

4 Journal of Fluids

Table 1 Wall temperature and heat transfer values when one has the following

(a) 119904 = 1 120582 = 05 1205741= 1 120576 = 0119879

119904= 01 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 09132 13014 09139 12922 09142 128652 07516 18627 07526 18553 07532 185093 05883 20586 05892 20540 05898 205124 04540 20474 04547 20447 04552 204315 03523 19430 03529 19415 03532 194056 02772 18069 02776 18060 02778 180557 02218 16676 02221 16670 02222 166678 01804 15368 01806 15364 01807 153629 01490 14184 01491 14181 01492 1418010 01248 13128 01249 13127 01250 13126

(b) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 01 120574 = 10

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 08753 12473 08761 12389 08766 123362 06686 16570 06698 16511 06705 164763 04878 17072 04888 17040 04894 170214 03567 16084 03573 16067 03577 160575 02661 14677 02666 14668 02669 146636 02036 13273 02039 13268 02041 132657 01597 12005 01599 12002 01600 120008 01279 10901 01281 10899 01282 108989 01045 09950 01046 09949 01047 0994810 00868 09132 00869 09131 00869 09131

(c) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06780 09661 06797 09611 06807 095792 03771 09344 03783 09326 03791 093143 02223 07778 02229 07771 02233 077674 01426 06431 01430 06428 01432 064265 00981 05411 00983 05410 00984 054096 00713 04644 00714 04643 00714 046437 00539 04055 00540 04054 00541 040548 00422 03592 00422 03592 00422 035929 00338 03221 00339 03220 00339 0322010 00277 02917 00278 02917 00278 02917

(d) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 1

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 05839 08321 05858 08284 05870 082602 02875 07125 02886 07114 02893 071083 01600 05600 01605 05596 01609 055944 00998 04501 01001 04500 01002 044995 00676 03730 00678 03729 00679 037296 00487 03171 00487 03171 00488 03171

Journal of Fluids 5

(d) Continued

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

7 00366 02753 00367 02752 00367 027528 00285 02429 00285 02429 00286 024299 00228 02172 00228 02172 00229 0217110 00187 01963 00187 01963 00187 01963

(e) 119904 = 1 120582 = 05 1205741= 1 120576 = minus05 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06884 14256 06899 14204 06908 068842 03812 11468 03824 11455 03831 038123 02236 08741 02242 08737 02246 022364 01431 06922 01434 06921 01436 014315 00983 05690 00985 05689 00986 009836 00713 04815 00715 04815 00715 007137 00540 04167 00541 04167 00541 005408 00422 03669 00422 03669 00423 004229 00338 03276 00339 03276 00339 0033810 00277 02958 00278 02958 00278 00277

0 1 2 3 4 5 6

120573 = 05 15 25

120573 = 05 15 25

1205741 = 1 s = 1

f998400(120578

)

120578

14

12

1

08

06

120582 = 05

120582 = 10

120582 = 15

Figure 2 Effect of 120573 and 120582 on the velocity profiles

thickness decreases with an increase in Prandtl number Asthe thermal conductivity parameter increases the tempera-ture in the thermal boundary layer increases The variaton ofskin frictionwith the ratio of relaxation and retardation timesfor different parameters is shown in Figures 4(a) and 4(b)When there is no stretching the skin friction increases withthe ratio of relaxation and retardation times and decreaseswith an increase in Deborah number As expected the skinfriction reduces with an increase in the suction parameterin both cases Comparing Figures 4(a) and 4(b) it can be

0 05 151 2 25 3

120576 = minus05 05 10

1205741 = 5

120573 = 05

120574 = 0

Ts = 1

120578

0

1

02

04

06

08

120579(120578

)

Pr = 1Pr = 3

Figure 3 Effect of Pr and 120576 on the temperature profiles

seen that the skin friction decreases with an increase in thestretching parameterThe variation in heat transfer rates withthe ratio of relaxation and retardation times is shown inFigures 5 and 6 for different values of suction and thermalconductivity parameters and Prandtl and Deborah numbersThe other parameters are kept constant As the ratio ofrelaxation and retardation times increases the heat transferrate increases For higher values of the suction parameter theheat transfer rates are found to be higher It is also observedthat the heat transfer rates decrease with an increase in

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

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2 Journal of Fluids

flowwas investigated byTurkyilmazoglu andPop [17] Akramand Nadeem [18] discussed the peristaltic motion of a two-dimensional Jeffry fluid Authors in [19ndash22] studied moreproperties in Jeffrey fluid Different non-Newtonian fluidswere considered in studies by Pandey and Tripathi [23ndash25]and Tripathi [26]

Interest in boundary layer flow and heat transfer over astretching sheet has gained considerable attention because ofits application in industry andmanufacturing processes Suchapplications include polymer extrusion drawing of copperwires continuous stretching of plastic films and artificialfibers hot rolling wire drawing glass fiber metal extrusionand metal spinning For example Liu and Andersson [27]studied the heat transfer in a liquid film on an unsteadystretching sheet The effects of variable fluid properties andthermocapillarity on the flow of a thin film on an unsteadystretching sheet were studied by Dandapat et al [28] Hayatet al [29] investigated the peristaltic mechanism of a Jeffreyfluid in a circular tube Nadeem et al [30] analyzed theboundary layer flow of a Jeffrey fluid over an exponentiallystretching surfaceThe effects of thermal radiation are carriedout for two cases of heat transfer analysis known as (1) pre-scribed exponential order surface temperature (PEST) and(2) prescribed exponential order heat flux (PEHF) Hamad[31] studied the convective flow and heat transfer of anincompressible viscous nanofluid past a semi-infinite verticalstretching sheet in the presence of a magnetic field Hamadand Pop [32] studied theoretically the steady boundary layerflownear the stagnation-point flow on a permeable stretchingsheet in a porous medium saturated with a nanofluid and inthe presence of internal heat generationabsorption

The objective of the present study is to investigate thedynamics of the thermal boundary layer flow of a viscousincompressible Jeffrey fluid near the stagnation point ona stretching sheet taking into account the thermal jumpcondition at the surface Thus the main focus of the analysisis to investigate how the flow field temperature field shearstress and heat flux vary within the boundary layer withthermal jump at the wall when the thermal conductivity istemperature dependent The similarity equations are derivedand solved numerically with the widely used and robust com-puter algebra software Graphs and tables are presented toillustrate and discuss important hydrodynamic and thermalfeatures of the flow

2 Problem Formulation

Consider a steady two dimensional flow of an incompress-ible Jeffrey fluid near the stagnation point on a stretch-ingshrinking sheet The thermal conductivity is assumed tobe functions of temperature A thermal jump condition isassumed to occur at the wall We are considering Cartesiancoordinate system in such a way that 119909-axis is taken along thestretching sheet in the direction of the motion and 119910-axis isnormal to it The plate is stretched in the 119909-direction with avelocity 119906

119908= 119888119909 defined at 119910 = 0 The flow and heat transfer

characteristics under the boundary layer approximations aregoverned by the following equations

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910

= 119906119890

119889119906119890

119889119909+

]

1 + 1205741

[1205972119906

1205971199102

+ 1205742(119906

1205973119906

1205971199091205971199102+ V

1205973119906

1205971199103minus120597119906

120597119909

1205972119906

1205971199102

+120597119906

120597119910

1205972119906

120597 119909120597 119910)]

(2)

120588119862119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(120581 (119879)

120597119879

120597119910) (3)

with the boundary conditions (see Rahman and Eltayeb [13])

V = V119908(119909) 119906 = 119906

119908(119909) = 119888119909

119879jump = 119879119891minus 119879119908

= 1205821(

2

120590119879

minus 1)2120574

120574 + 1

120581 (119879)

120583119862119901

120597119879

120597119910at 119910 = 0

119906 = 119906119890(119909) = 119886119909

120597119906

120597119910= 0 119879 = 119879

infin

as 119910 997888rarr infin

(4)

Here119909 and119910 are theCartesian coordinates along the plate andnormal to it respectively 119906 and V are the velocity componentsalong 119909- and 119910-axes V

119908(119909) is the mass transfer velocity with

V119908(119909) lt 0 for suction and V

119908(119909) gt 0 for injection or

withdrawal119879 is the fluid temperature120572 is thermal diffusivity] is the kinematic viscosity 120574

1is the ratio of relaxation and

retardation times 1205742is the relaxation time 120574 is the ratio of

specific heats 120590119879is the thermal accommodation coefficient

1205821is the mean free path 120583 is the dynamic viscosity and 120581(119879)

is the thermal conductivity which can be following Chiam[1] written as

120581 = 120581infin(1 + 120576

119879 minus 119879infin

119879119908minus 119879infin

) (5)

where 120576 is the thermal conductivity parameterWe introduce now the following similarity variables

120595 = radic119886]119909119891 (120578) 120579 (120578) =(119879 minus 119879

infin)

(119879119908minus 119879infin)

120578 = radic119886

]119910

(6)

where 120595 is the stream function which is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Thus V

119908(119909) = minus radic119886]119904

where 119904 is the mass transfer parameter with 119904 gt 0 for suctionand 119904 lt 0 for injection respectively Substituting (5) and

Journal of Fluids 3

(6) into (2) and (3) the following set of ordinary differentialequations results in

119891101584010158401015840

+ (1 + 1205741) (11989111989110158401015840

minus 11989110158402

)

+ 120573 (119891101584010158402

minus 1198911198911015840101584010158401015840

) + (1 + 1205741) = 0

(1 + 120576 120579) 12057910158401015840

+ 12057612057910158402

+ Pr1198911205791015840 = 0

(7)

and the boundary conditions (4) become

119891 (0) = 119904 1198911015840

(0) = 120582

120579 (0) = 1 +2120574119879119904

120574 + 1Pr (1 + 120576120579 (0)) 120579

1015840

(0)

1198911015840

(infin) = 1 11989110158401015840

(infin) = 0 120579 (infin) = 0

(8)

where Pr = 120583119862119901120581infin

is the Prandtl number 120582 = 119888119886 is thestretching (120582 gt 0) or shrinking (120582 lt 0) parameter 120573 = 119888120574

2

is the Deborah number 119879119904= 1205821(2(120590119879minus 1))radic119886] is the slip

parameter and primes denote differentiation with respect to120578

21 Particular Case It is worth mentioning that for a regularviscous fluid (120573 = 120574

1= 0) (7) reduce to the steady state

equations from the paper by Nazar et al [33] when we neglectthe radiation effect

22 PhysicalQuantities Thephysical quantities of interest arethe skin friction coefficient 119862

119891and the local Nusselt number

Nu119909 which are defined as

119862119891=

120591119908

1205881199062119890(119909)

Nu119909=

119909119902119908

120581 (119879119908minus 119879infin) (9)

where 120591119908

is the skin friction or shear stress along thestretching surface and 119902

119908is the heat flux from the surface

which are given by

120591119908= 120583(

120597119906

120597119910)119910=0

119902119908= minus 120581(

120597119879jump

120597119910)119910=0

(10)

Using (6) we get

Re12119909

119862119891= 11989110158401015840

(0)

Reminus12119909

Nu119909

= minus2120574

120574 + 1119879119904Pr [(1 + 120576120579) 120579

10158401015840

(0) + 120576(1205791015840

(0))2

]

(11)

where Re119909= 119906119890(119909)119909] is the local Reynolds number

3 Results and Discussion

The transformed equations (7) with boundary conditions (8)are solved numerically by using a finite difference methodThe asymptotic boundary conditions at 120578 = infin are replacedby 120578 = 6 In Table 1 we have shown the variation of walltemperature and heat transfer rates with the Prandtl numbersfor three different values of Deborah numbers It is observed

0 1 2 3 4 5

Suction without stretching

0

1

02

04

06

08

s = 1 120582 = 0

f998400(120578

)

120578

1205741 = 1 2 3

120573 = 05

120573 = 30

Figure 1 Effect of 120573 and 1205741on the velocity profiles

that the wall temperature decreases whereas the heat transferrates increase with an increase in Prandtl number Howeverthe wall temperature increases slightly and heat transfervalues decrease slightly with an increase in Deborah numberwhereas both decrease with an increase in the ratio of specificheats This is shown in Tables 1(a) and 1(b) Tables 1(c)and 1(d) show the effects of slip temperature on the walltemperature and heat transfer rates for the constant thermalconductivity Table 1(e) shows the effects of thermal slip onthewall temperature and heat transfer rates when the thermalconductivity varies with temperature It can be seen that boththe wall temperature and heat transfer rates decrease withan increase in the Prandtl number due to decrease in thethermal conductivity The effects of the ratio of relaxationand retardation times Deborah number and suction andstretching parameters on the 119909-component of velocity areshown in Figures 1 and 2 Figure 1 shows the effects in theabsence of stretching It shows that the velocity increasesas the ratio of relaxation and retardation times increasesbut decreases with an increase in Deborah number Thevelocity boundary layer converges quickly for small Deborahnumbers In fact small Deborah numbers correspond tosituationswhere thematerial has time to relax (and behaves ina viscous manner) while high Deborah numbers correspondto situations where the material behaves rather elasticallyFigure 2 shows the effects of stretching parameter on thevelocity for different values ofDeborah number It is observedthat the velocity becomes constant when 120582 = 0 increaseswhen 120582 lt 1 and decreases when 120582 gt 0 Accordingly thevelocity decreases or increases with Deborah number when120582 lt 1 or 120582 gt 0

The effect of thermal conductivity parameter on temper-ature profiles is shown in Figure 3 for two different Prandtlnumbers It is observed that the thermal boundary layer

4 Journal of Fluids

Table 1 Wall temperature and heat transfer values when one has the following

(a) 119904 = 1 120582 = 05 1205741= 1 120576 = 0119879

119904= 01 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 09132 13014 09139 12922 09142 128652 07516 18627 07526 18553 07532 185093 05883 20586 05892 20540 05898 205124 04540 20474 04547 20447 04552 204315 03523 19430 03529 19415 03532 194056 02772 18069 02776 18060 02778 180557 02218 16676 02221 16670 02222 166678 01804 15368 01806 15364 01807 153629 01490 14184 01491 14181 01492 1418010 01248 13128 01249 13127 01250 13126

(b) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 01 120574 = 10

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 08753 12473 08761 12389 08766 123362 06686 16570 06698 16511 06705 164763 04878 17072 04888 17040 04894 170214 03567 16084 03573 16067 03577 160575 02661 14677 02666 14668 02669 146636 02036 13273 02039 13268 02041 132657 01597 12005 01599 12002 01600 120008 01279 10901 01281 10899 01282 108989 01045 09950 01046 09949 01047 0994810 00868 09132 00869 09131 00869 09131

(c) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06780 09661 06797 09611 06807 095792 03771 09344 03783 09326 03791 093143 02223 07778 02229 07771 02233 077674 01426 06431 01430 06428 01432 064265 00981 05411 00983 05410 00984 054096 00713 04644 00714 04643 00714 046437 00539 04055 00540 04054 00541 040548 00422 03592 00422 03592 00422 035929 00338 03221 00339 03220 00339 0322010 00277 02917 00278 02917 00278 02917

(d) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 1

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 05839 08321 05858 08284 05870 082602 02875 07125 02886 07114 02893 071083 01600 05600 01605 05596 01609 055944 00998 04501 01001 04500 01002 044995 00676 03730 00678 03729 00679 037296 00487 03171 00487 03171 00488 03171

Journal of Fluids 5

(d) Continued

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

7 00366 02753 00367 02752 00367 027528 00285 02429 00285 02429 00286 024299 00228 02172 00228 02172 00229 0217110 00187 01963 00187 01963 00187 01963

(e) 119904 = 1 120582 = 05 1205741= 1 120576 = minus05 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06884 14256 06899 14204 06908 068842 03812 11468 03824 11455 03831 038123 02236 08741 02242 08737 02246 022364 01431 06922 01434 06921 01436 014315 00983 05690 00985 05689 00986 009836 00713 04815 00715 04815 00715 007137 00540 04167 00541 04167 00541 005408 00422 03669 00422 03669 00423 004229 00338 03276 00339 03276 00339 0033810 00277 02958 00278 02958 00278 00277

0 1 2 3 4 5 6

120573 = 05 15 25

120573 = 05 15 25

1205741 = 1 s = 1

f998400(120578

)

120578

14

12

1

08

06

120582 = 05

120582 = 10

120582 = 15

Figure 2 Effect of 120573 and 120582 on the velocity profiles

thickness decreases with an increase in Prandtl number Asthe thermal conductivity parameter increases the tempera-ture in the thermal boundary layer increases The variaton ofskin frictionwith the ratio of relaxation and retardation timesfor different parameters is shown in Figures 4(a) and 4(b)When there is no stretching the skin friction increases withthe ratio of relaxation and retardation times and decreaseswith an increase in Deborah number As expected the skinfriction reduces with an increase in the suction parameterin both cases Comparing Figures 4(a) and 4(b) it can be

0 05 151 2 25 3

120576 = minus05 05 10

1205741 = 5

120573 = 05

120574 = 0

Ts = 1

120578

0

1

02

04

06

08

120579(120578

)

Pr = 1Pr = 3

Figure 3 Effect of Pr and 120576 on the temperature profiles

seen that the skin friction decreases with an increase in thestretching parameterThe variation in heat transfer rates withthe ratio of relaxation and retardation times is shown inFigures 5 and 6 for different values of suction and thermalconductivity parameters and Prandtl and Deborah numbersThe other parameters are kept constant As the ratio ofrelaxation and retardation times increases the heat transferrate increases For higher values of the suction parameter theheat transfer rates are found to be higher It is also observedthat the heat transfer rates decrease with an increase in

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 3

(6) into (2) and (3) the following set of ordinary differentialequations results in

119891101584010158401015840

+ (1 + 1205741) (11989111989110158401015840

minus 11989110158402

)

+ 120573 (119891101584010158402

minus 1198911198911015840101584010158401015840

) + (1 + 1205741) = 0

(1 + 120576 120579) 12057910158401015840

+ 12057612057910158402

+ Pr1198911205791015840 = 0

(7)

and the boundary conditions (4) become

119891 (0) = 119904 1198911015840

(0) = 120582

120579 (0) = 1 +2120574119879119904

120574 + 1Pr (1 + 120576120579 (0)) 120579

1015840

(0)

1198911015840

(infin) = 1 11989110158401015840

(infin) = 0 120579 (infin) = 0

(8)

where Pr = 120583119862119901120581infin

is the Prandtl number 120582 = 119888119886 is thestretching (120582 gt 0) or shrinking (120582 lt 0) parameter 120573 = 119888120574

2

is the Deborah number 119879119904= 1205821(2(120590119879minus 1))radic119886] is the slip

parameter and primes denote differentiation with respect to120578

21 Particular Case It is worth mentioning that for a regularviscous fluid (120573 = 120574

1= 0) (7) reduce to the steady state

equations from the paper by Nazar et al [33] when we neglectthe radiation effect

22 PhysicalQuantities Thephysical quantities of interest arethe skin friction coefficient 119862

119891and the local Nusselt number

Nu119909 which are defined as

119862119891=

120591119908

1205881199062119890(119909)

Nu119909=

119909119902119908

120581 (119879119908minus 119879infin) (9)

where 120591119908

is the skin friction or shear stress along thestretching surface and 119902

119908is the heat flux from the surface

which are given by

120591119908= 120583(

120597119906

120597119910)119910=0

119902119908= minus 120581(

120597119879jump

120597119910)119910=0

(10)

Using (6) we get

Re12119909

119862119891= 11989110158401015840

(0)

Reminus12119909

Nu119909

= minus2120574

120574 + 1119879119904Pr [(1 + 120576120579) 120579

10158401015840

(0) + 120576(1205791015840

(0))2

]

(11)

where Re119909= 119906119890(119909)119909] is the local Reynolds number

3 Results and Discussion

The transformed equations (7) with boundary conditions (8)are solved numerically by using a finite difference methodThe asymptotic boundary conditions at 120578 = infin are replacedby 120578 = 6 In Table 1 we have shown the variation of walltemperature and heat transfer rates with the Prandtl numbersfor three different values of Deborah numbers It is observed

0 1 2 3 4 5

Suction without stretching

0

1

02

04

06

08

s = 1 120582 = 0

f998400(120578

)

120578

1205741 = 1 2 3

120573 = 05

120573 = 30

Figure 1 Effect of 120573 and 1205741on the velocity profiles

that the wall temperature decreases whereas the heat transferrates increase with an increase in Prandtl number Howeverthe wall temperature increases slightly and heat transfervalues decrease slightly with an increase in Deborah numberwhereas both decrease with an increase in the ratio of specificheats This is shown in Tables 1(a) and 1(b) Tables 1(c)and 1(d) show the effects of slip temperature on the walltemperature and heat transfer rates for the constant thermalconductivity Table 1(e) shows the effects of thermal slip onthewall temperature and heat transfer rates when the thermalconductivity varies with temperature It can be seen that boththe wall temperature and heat transfer rates decrease withan increase in the Prandtl number due to decrease in thethermal conductivity The effects of the ratio of relaxationand retardation times Deborah number and suction andstretching parameters on the 119909-component of velocity areshown in Figures 1 and 2 Figure 1 shows the effects in theabsence of stretching It shows that the velocity increasesas the ratio of relaxation and retardation times increasesbut decreases with an increase in Deborah number Thevelocity boundary layer converges quickly for small Deborahnumbers In fact small Deborah numbers correspond tosituationswhere thematerial has time to relax (and behaves ina viscous manner) while high Deborah numbers correspondto situations where the material behaves rather elasticallyFigure 2 shows the effects of stretching parameter on thevelocity for different values ofDeborah number It is observedthat the velocity becomes constant when 120582 = 0 increaseswhen 120582 lt 1 and decreases when 120582 gt 0 Accordingly thevelocity decreases or increases with Deborah number when120582 lt 1 or 120582 gt 0

The effect of thermal conductivity parameter on temper-ature profiles is shown in Figure 3 for two different Prandtlnumbers It is observed that the thermal boundary layer

4 Journal of Fluids

Table 1 Wall temperature and heat transfer values when one has the following

(a) 119904 = 1 120582 = 05 1205741= 1 120576 = 0119879

119904= 01 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 09132 13014 09139 12922 09142 128652 07516 18627 07526 18553 07532 185093 05883 20586 05892 20540 05898 205124 04540 20474 04547 20447 04552 204315 03523 19430 03529 19415 03532 194056 02772 18069 02776 18060 02778 180557 02218 16676 02221 16670 02222 166678 01804 15368 01806 15364 01807 153629 01490 14184 01491 14181 01492 1418010 01248 13128 01249 13127 01250 13126

(b) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 01 120574 = 10

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 08753 12473 08761 12389 08766 123362 06686 16570 06698 16511 06705 164763 04878 17072 04888 17040 04894 170214 03567 16084 03573 16067 03577 160575 02661 14677 02666 14668 02669 146636 02036 13273 02039 13268 02041 132657 01597 12005 01599 12002 01600 120008 01279 10901 01281 10899 01282 108989 01045 09950 01046 09949 01047 0994810 00868 09132 00869 09131 00869 09131

(c) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06780 09661 06797 09611 06807 095792 03771 09344 03783 09326 03791 093143 02223 07778 02229 07771 02233 077674 01426 06431 01430 06428 01432 064265 00981 05411 00983 05410 00984 054096 00713 04644 00714 04643 00714 046437 00539 04055 00540 04054 00541 040548 00422 03592 00422 03592 00422 035929 00338 03221 00339 03220 00339 0322010 00277 02917 00278 02917 00278 02917

(d) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 1

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 05839 08321 05858 08284 05870 082602 02875 07125 02886 07114 02893 071083 01600 05600 01605 05596 01609 055944 00998 04501 01001 04500 01002 044995 00676 03730 00678 03729 00679 037296 00487 03171 00487 03171 00488 03171

Journal of Fluids 5

(d) Continued

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

7 00366 02753 00367 02752 00367 027528 00285 02429 00285 02429 00286 024299 00228 02172 00228 02172 00229 0217110 00187 01963 00187 01963 00187 01963

(e) 119904 = 1 120582 = 05 1205741= 1 120576 = minus05 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06884 14256 06899 14204 06908 068842 03812 11468 03824 11455 03831 038123 02236 08741 02242 08737 02246 022364 01431 06922 01434 06921 01436 014315 00983 05690 00985 05689 00986 009836 00713 04815 00715 04815 00715 007137 00540 04167 00541 04167 00541 005408 00422 03669 00422 03669 00423 004229 00338 03276 00339 03276 00339 0033810 00277 02958 00278 02958 00278 00277

0 1 2 3 4 5 6

120573 = 05 15 25

120573 = 05 15 25

1205741 = 1 s = 1

f998400(120578

)

120578

14

12

1

08

06

120582 = 05

120582 = 10

120582 = 15

Figure 2 Effect of 120573 and 120582 on the velocity profiles

thickness decreases with an increase in Prandtl number Asthe thermal conductivity parameter increases the tempera-ture in the thermal boundary layer increases The variaton ofskin frictionwith the ratio of relaxation and retardation timesfor different parameters is shown in Figures 4(a) and 4(b)When there is no stretching the skin friction increases withthe ratio of relaxation and retardation times and decreaseswith an increase in Deborah number As expected the skinfriction reduces with an increase in the suction parameterin both cases Comparing Figures 4(a) and 4(b) it can be

0 05 151 2 25 3

120576 = minus05 05 10

1205741 = 5

120573 = 05

120574 = 0

Ts = 1

120578

0

1

02

04

06

08

120579(120578

)

Pr = 1Pr = 3

Figure 3 Effect of Pr and 120576 on the temperature profiles

seen that the skin friction decreases with an increase in thestretching parameterThe variation in heat transfer rates withthe ratio of relaxation and retardation times is shown inFigures 5 and 6 for different values of suction and thermalconductivity parameters and Prandtl and Deborah numbersThe other parameters are kept constant As the ratio ofrelaxation and retardation times increases the heat transferrate increases For higher values of the suction parameter theheat transfer rates are found to be higher It is also observedthat the heat transfer rates decrease with an increase in

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Journal of Fluids

Table 1 Wall temperature and heat transfer values when one has the following

(a) 119904 = 1 120582 = 05 1205741= 1 120576 = 0119879

119904= 01 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 09132 13014 09139 12922 09142 128652 07516 18627 07526 18553 07532 185093 05883 20586 05892 20540 05898 205124 04540 20474 04547 20447 04552 204315 03523 19430 03529 19415 03532 194056 02772 18069 02776 18060 02778 180557 02218 16676 02221 16670 02222 166678 01804 15368 01806 15364 01807 153629 01490 14184 01491 14181 01492 1418010 01248 13128 01249 13127 01250 13126

(b) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 01 120574 = 10

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 08753 12473 08761 12389 08766 123362 06686 16570 06698 16511 06705 164763 04878 17072 04888 17040 04894 170214 03567 16084 03573 16067 03577 160575 02661 14677 02666 14668 02669 146636 02036 13273 02039 13268 02041 132657 01597 12005 01599 12002 01600 120008 01279 10901 01281 10899 01282 108989 01045 09950 01046 09949 01047 0994810 00868 09132 00869 09131 00869 09131

(c) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06780 09661 06797 09611 06807 095792 03771 09344 03783 09326 03791 093143 02223 07778 02229 07771 02233 077674 01426 06431 01430 06428 01432 064265 00981 05411 00983 05410 00984 054096 00713 04644 00714 04643 00714 046437 00539 04055 00540 04054 00541 040548 00422 03592 00422 03592 00422 035929 00338 03221 00339 03220 00339 0322010 00277 02917 00278 02917 00278 02917

(d) 119904 = 1 120582 = 05 1205741= 1 120576 = 0 119879

119904= 05 120574 = 1

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 05839 08321 05858 08284 05870 082602 02875 07125 02886 07114 02893 071083 01600 05600 01605 05596 01609 055944 00998 04501 01001 04500 01002 044995 00676 03730 00678 03729 00679 037296 00487 03171 00487 03171 00488 03171

Journal of Fluids 5

(d) Continued

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

7 00366 02753 00367 02752 00367 027528 00285 02429 00285 02429 00286 024299 00228 02172 00228 02172 00229 0217110 00187 01963 00187 01963 00187 01963

(e) 119904 = 1 120582 = 05 1205741= 1 120576 = minus05 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06884 14256 06899 14204 06908 068842 03812 11468 03824 11455 03831 038123 02236 08741 02242 08737 02246 022364 01431 06922 01434 06921 01436 014315 00983 05690 00985 05689 00986 009836 00713 04815 00715 04815 00715 007137 00540 04167 00541 04167 00541 005408 00422 03669 00422 03669 00423 004229 00338 03276 00339 03276 00339 0033810 00277 02958 00278 02958 00278 00277

0 1 2 3 4 5 6

120573 = 05 15 25

120573 = 05 15 25

1205741 = 1 s = 1

f998400(120578

)

120578

14

12

1

08

06

120582 = 05

120582 = 10

120582 = 15

Figure 2 Effect of 120573 and 120582 on the velocity profiles

thickness decreases with an increase in Prandtl number Asthe thermal conductivity parameter increases the tempera-ture in the thermal boundary layer increases The variaton ofskin frictionwith the ratio of relaxation and retardation timesfor different parameters is shown in Figures 4(a) and 4(b)When there is no stretching the skin friction increases withthe ratio of relaxation and retardation times and decreaseswith an increase in Deborah number As expected the skinfriction reduces with an increase in the suction parameterin both cases Comparing Figures 4(a) and 4(b) it can be

0 05 151 2 25 3

120576 = minus05 05 10

1205741 = 5

120573 = 05

120574 = 0

Ts = 1

120578

0

1

02

04

06

08

120579(120578

)

Pr = 1Pr = 3

Figure 3 Effect of Pr and 120576 on the temperature profiles

seen that the skin friction decreases with an increase in thestretching parameterThe variation in heat transfer rates withthe ratio of relaxation and retardation times is shown inFigures 5 and 6 for different values of suction and thermalconductivity parameters and Prandtl and Deborah numbersThe other parameters are kept constant As the ratio ofrelaxation and retardation times increases the heat transferrate increases For higher values of the suction parameter theheat transfer rates are found to be higher It is also observedthat the heat transfer rates decrease with an increase in

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 5

(d) Continued

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

7 00366 02753 00367 02752 00367 027528 00285 02429 00285 02429 00286 024299 00228 02172 00228 02172 00229 0217110 00187 01963 00187 01963 00187 01963

(e) 119904 = 1 120582 = 05 1205741= 1 120576 = minus05 119879

119904= 05 120574 = 05

Pr 120573 = 05 120573 = 10 120573 = 15

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

1 06884 14256 06899 14204 06908 068842 03812 11468 03824 11455 03831 038123 02236 08741 02242 08737 02246 022364 01431 06922 01434 06921 01436 014315 00983 05690 00985 05689 00986 009836 00713 04815 00715 04815 00715 007137 00540 04167 00541 04167 00541 005408 00422 03669 00422 03669 00423 004229 00338 03276 00339 03276 00339 0033810 00277 02958 00278 02958 00278 00277

0 1 2 3 4 5 6

120573 = 05 15 25

120573 = 05 15 25

1205741 = 1 s = 1

f998400(120578

)

120578

14

12

1

08

06

120582 = 05

120582 = 10

120582 = 15

Figure 2 Effect of 120573 and 120582 on the velocity profiles

thickness decreases with an increase in Prandtl number Asthe thermal conductivity parameter increases the tempera-ture in the thermal boundary layer increases The variaton ofskin frictionwith the ratio of relaxation and retardation timesfor different parameters is shown in Figures 4(a) and 4(b)When there is no stretching the skin friction increases withthe ratio of relaxation and retardation times and decreaseswith an increase in Deborah number As expected the skinfriction reduces with an increase in the suction parameterin both cases Comparing Figures 4(a) and 4(b) it can be

0 05 151 2 25 3

120576 = minus05 05 10

1205741 = 5

120573 = 05

120574 = 0

Ts = 1

120578

0

1

02

04

06

08

120579(120578

)

Pr = 1Pr = 3

Figure 3 Effect of Pr and 120576 on the temperature profiles

seen that the skin friction decreases with an increase in thestretching parameterThe variation in heat transfer rates withthe ratio of relaxation and retardation times is shown inFigures 5 and 6 for different values of suction and thermalconductivity parameters and Prandtl and Deborah numbersThe other parameters are kept constant As the ratio ofrelaxation and retardation times increases the heat transferrate increases For higher values of the suction parameter theheat transfer rates are found to be higher It is also observedthat the heat transfer rates decrease with an increase in

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Fluids

1 1505 2

1

12

14

16

18

2

22

24

26120582 = 0

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(a)

1 1505 204

12

06

08

1

120582 = 05

120573 = 05 10 15

Re12x

Cf

1205741

s = 1

s = 3

(b)

Figure 4 Effect of 120573 and 120582 on the skin friction profiles

05 151 2076

08

084

088

092

096

1

s = 10

s = 15

120573 = 05

120573 = 10

120573 = 15

1205741

minus120579998400(0

)

120582 = 05

Pr = 1120576 = 1

120574 = 1

Ts = 0

Figure 5 Effect of 120573 and 119904 on the heat transfer rate

Deborah number This is shown in Figure 5 As evident fromFigure 6 the heat transfer rates increase with an increase inthe Prandtl number Figure 6 also shows that the heat transferrates decrease with an increase in the thermal conductivityparameter

4 Conclusions

Theeffects of thermal jump and variable thermal conductivityon flow and heat transfer near the stagnation point on a

05 151 21205741

Pr = 1

Pr = 2

Pr = 2

Pr = 2

minus120579998400(0

)

24

21

18

15

12

09

06

120576 = 01

120576 = 05

120576 = 1

120573 = 05 s = 1 120582 = 01 120574 = 1 Ts = 0

Figure 6 Effect of Pr and 120576 on the heat transfer rate

stretchingshrinking sheet are investigated numerically in aJeffrey fluid The effects of governing parameters includingratio of relaxation and retardation times 120574

1 Deborah number

120573 Prandtl number stretching parameter120582 suction parameter119904 and thermal conductivity parameter 120576 on the dimensionlessvelocity temperature skin friction and heat transfer ratesare investigated and are presented graphically and in tabularform We conclude the following

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 7

(a) The wall temperature increases slightly while heattransfer values decrease slightly with an increase inDeborah number

(b) The wall temperature and heat transfer decrease withan increase in the ratio of specific heats

(c) The decreases of Deborah number and the increasesof relaxation and retardation times leads to increasesin the velocity

(d) The skin friction decreases with an increase in thestretching parameter

(e) As the ratio of relaxation and retardation timesincreases the heat transfer rate increases

(f) For higher values of the suction parameter the heattransfer rates are found to be higher

(g) The heat transfer rates decrease with an increase inDeborah number

References

[1] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[2] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[3] P S Datti K V Prasad M S Abel and A Joshi ldquoMHD visco-elastic fluid flow over a non-isothermal stretching sheetrdquo Inter-national Journal of Engineering Science vol 42 no 8-9 pp 935ndash946 2004

[4] K V Prasad M S Abel and S K Khan ldquoMomentum and heattransfer in visco-elastic fluid flow in a porous medium over anon-isothermal stretching sheetrdquo International Journal of Num-erical Methods for Heat and Fluid Flow vol 10 no 8 pp 786ndash801 2000

[5] M S Abel KV Prasad andAMahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[6] KV Prasad andKVajravelu ldquoHeat transfer in theMHDflowofa power law fluid over a non-isothermal stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 52 no 21-22 pp 4956ndash4965 2009

[7] M S Abel P G Siddheshwar and NMahesha ldquoEffects of ther-mal buoyancy and variable thermal conductivity on the MHDflow and heat transfer in a power-law fluid past a verticalstretching sheet in the presence of a non-uniform heat sourcerdquoInternational Journal of Non-LinearMechanics vol 44 no 1 pp1ndash12 2009

[8] M A Seddeek S NOdda andM S Abdelmeguid ldquoNumericalstudy for the effects of thermophoresis and variable thermalconductivity on heat and mass transfer over an acceleratingsurface with heat sourcerdquo Computational Materials Science vol47 no 1 pp 93ndash98 2009

[9] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effect ofvariable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in Nonlinear

Science and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[10] M S Abel P G Siddheshwar and N Mahesha ldquoNumericalsolution of the momentum and heat transfer equations for ahydromagnetic flow due to a stretching sheet of a non-uniformproperty micropolar liquidrdquo Applied Mathematics and Compu-tation vol 217 no 12 pp 5895ndash5909 2011

[11] N C Mahanti and P Gaur ldquoEffects of varying viscosity andthermal conductivity on steady free convective flow and heattransfer along an isothermal vertical plate in the presence ofheat sinkrdquo Journal of Applied Fluid Mechanics vol 2 no 1 pp23ndash28 2009

[12] R G Deissler ldquoAn analysis of second-order slip flow and temp-erature-jump boundary conditions for rarefied gasesrdquo Interna-tional Journal of Heat and Mass Transfer vol 7 no 6 pp 681ndash694 1964

[13] M M Rahman and I A Eltayeb ldquoConvective slip flow of rare-fied fluids over a wedge with thermal jump and variable tran-sport propertiesrdquo International Journal of Thermal Sciences vol50 no 4 pp 468ndash479 2011

[14] J W Cipolla Jr ldquoHeat transfer and temperature jump in a poly-atomic gasrdquo International Journal ofHeat andMass Transfer vol14 no 10 pp 1599ndash1610 1971

[15] T-T Kao ldquoLaminar free convective heat transfer response alonga vertical flat plate with step jump in surface temperaturerdquoLetters inHeat andMass Transfer vol 2 no 5 pp 419ndash428 1975

[16] A V Latyshev and A A Yushkanov ldquoAn analytic solution ofthe problemof the temperature jumps and vapour density over asurface when there is a temperature gradientrdquo Journal of AppliedMathematics and Mechanics vol 58 no 2 pp 259ndash265 1994

[17] M Turkyilmazoglu and I Pop ldquoExact analytical solution for theflow and heat transfer near the stagnation point on a stretch-ingshrinking sheet in a Jeffrey fluidrdquo International Journal ofHeat and Mass Transfer vol 57 no 1 pp 82ndash88 2013

[18] S Akram and S Nadeem ldquoInfluence of induced magnetic fieldand heat transfer on the peristaltic motion of Jeffrey fluid inan asymmetric channel closed form solutionsrdquo Journal ofMagnetism and Magnetic Materials vol 328 pp 11ndash20 2013

[19] C E Siewert and D Valougeorgis ldquoThe temperature-jump pro-blem for a mixture of two gasesrdquo Journal of Quantitative Spect-roscopy and Radiative Transfer vol 70 no 3 pp 307ndash319 2001

[20] S Nadeem A Hussain and M Khan ldquoStagnation flow of a Jef-frey fluid over a shrinking sheetrdquo Zeitschrift fur NaturforschungA vol 65 no 6-7 pp 540ndash548 2010

[21] S K Pandey and D Tripathi ldquoUnsteady model of transporta-tion of Jeffrey-fluid by peristalsisrdquo International Journal of Bio-mathematics vol 3 no 4 pp 473ndash491 2010

[22] T Hayat M Awais S Asghar and A A Hendi ldquoAnalytic solu-tion for the magnetohydrodynamic rotating flow of Jeffrey fluidin a channelrdquo Journal of Fluids Engineering vol 133 no 6 Arti-cle ID 061201 2011

[23] S K Pandey andD Tripathi ldquoInfluence ofmagnetic field on theperistaltic flow of a viscous fluid through a finite-length cylin-drical tuberdquo Applied Bionics and Biomechanics vol 7 no 3 pp169ndash176 2010

[24] S K Pandey andD Tripathi ldquoEffects of non-integral number ofperistalticwaves transporting couple stress fluids in finite lengthchannelsrdquo Zeitschrift fur Naturforschung A vol 66 no 3-4 pp172ndash180 2011

[25] S K Pandey and D Tripathi ldquoUnsteady peristaltic flow ofmicro-polar fluid in a finite channelrdquo Zeitschrift fur Natur-forschung A vol 66 no 3-4 pp 181ndash192 2011

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Fluids

[26] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[27] I-C Liu and H I Andersson ldquoHeat transfer in a liquid film onan unsteady stretching sheetrdquo International Journal of ThermalSciences vol 47 no 6 pp 766ndash772 2008

[28] B S Dandapat B Santra and K Vajravelu ldquoThe effects of var-able fluid properties and thermocapillarity on the flow of a thinfilm on an unsteady stretching sheetrdquo International Journal ofHeat and Mass Transfer vol 50 no 5-6 pp 991ndash996 2007

[29] T Hayat N Ali and S Asghar ldquoAn analysis of peristaltic tran-sport for flow of a Jeffrey fluidrdquoActaMechanica vol 193 no 1-2pp 101ndash112 2007

[30] S Nadeem S Zaheer and T Fang ldquoEffects of thermal radiationon the boundary layer flow of a Jeffrey fluid over an exponen-tially stretching surfacerdquo Numerical Algorithms vol 57 no 2pp 187ndash205 2011

[31] M A AHamad ldquoAnalytical solution of natural convection flowof a nanofluid over a linearly stretching sheet in the presence ofmagnetic fieldrdquo International Communications inHeat andMassTransfer vol 38 no 4 pp 487ndash492 2011

[32] M A A Hamad and I Pop ldquoScaling Transformations for Bou-ndary Layer Flow near the Stagnation-Point on a HeatedPermeable Stretching Surface in a Porous Medium Saturatedwith a Nanofluid and Heat GenerationAbsorption EffectsrdquoTransport in Porous Media vol 87 no 1 pp 25ndash39 2011

[33] R Nazar N Amin D Filip and I Pop ldquoUnsteady boundarylayer flow in the region of the stagnation point on a stretchingsheetrdquo International Journal of Engineering Science vol 42 no11-12 pp 1241ndash1253 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of