WORDS A GRADE A

VOL. 18 NO. 4 (1995) 765-772

HEAT TRANSFER ANALYSIS OF A SECOND GRADE FLUID OVER A STRETCHING SHEET

C.E.MANESCHY

Department

of MechanicalEngineering UniversityofPittsburgh

Pittsburgh,

PA

15261 M.MASSOUDI

U.S.Department

of

Energy

Pittsburgh

Energy

Technology Center

Pittsburgh,

PA

15236

(Received November 15, 1993 and in revised form January 27, 1994)

ABSTRACT. The heat tranfer andflowof a non-Newtonian fluidpastastretchingsheet isanalyzed in this

paper.

Results in a non-dimensionalformarepresentedhere forthe velocityandtemperature profilesassumingdifferent kind of

boundary

conditions.

KEY

WORDS

AND PHRASES.

Non-Newtonian fluid,

Rheology,

Viscoelasticliquids.

1980 AMS

SUBJECT CLASSIFICATION CODE.

76A05

1.

INTRODUCTION.

Theboundary layerflow ofnon-Newtonianfluids overastretching sheet has been studied extensively in the recent years. This problem is of interest when a polymer sheet is extruded continuouslyfrom a die.

In

a recentpaper,

Maneschy,

etal investigatedthe flow of a secondgrade fluid over aporouselastic sheet due tostretchingandgavea brief review of thepreviousworks.

In

the present analysis, we extend the result of

Maneschy,

etal

[1]

^tostudythe heat transfer over a stretching sheet with suctionatthe surface.

We

will first giveabrief review ofthe heat transfer studies forthisproblem.

Erickson,etal. [2]discussedthe heat andmasstransfer overastretchingsheetwithsuction or injection.The fluid was assumedtobe an incompressibleviscous fluid and the surfacespeed was assumedtobe constant.

Later, Gupta

and

Gupta

[3]extended this workby assumingthat the surface speed^varieswiththe coordinate

along

theflow.

In

both oftheseinvestigations,results forvelocity, temperature and concentrationprofilesare presented.

Dutta,

etal. [4] analyzed thecase where a uniform heat flux isprescribedatthe surface and the surfaceisnonporous. Duttaand

Gupta

[5]also considered the case where thetemperatureof the sheetisvariable.

While n all these studiesthe fluid was assumedtobe Newtonian, Siddappaand Abel [6]

discussed this problem assuming a viscoelastic fluid with suction at the surface. They obtained analytic solution for the velocity and temperature fields. The effects of^vscou,, dissipation ^were gnored Inthspaper. Bujurke,etal. [7]assumedcubic profiles forvelocityand temperature fields and usedasecond orderfluidmodel.They, however,dd notconsiderthermodynamicalrestrictions imposedbythe Clausius-Duheminequality (cf. DunnandFosdick[8]).Char and Chert[9]considered the fluidtobe ofWaiters’liquid

B

model and theplate^wassubjectedtovariable heat flux.

Dandapat

and

Gupta

10]presented^acritical review ofpreviousworks where the fluid had been assumedtobe non-Newtonian.

In

particular, they showedthe errors and inconsistencies in the works ofSiddappa and Abel [6] andBujurke, etal. [7]. However, the analysis ofDandapat ^and

Gupta [10]

is also incompletein the sense that (i) nothermodynamicalconsiderations are taken intoaccount,and (ii) the effect ofviscousdissipationisneglected. Whilethe laterisjustifiableas anassumption,theformer could be considered as a serious flaw in the analysis. Two otherrecent works which need to be mentioned arethose ofSurmaDeviand Nath 11 andLawrenceandRao 12].

In

thefirstcase, the fluid was assumed to be asecond grade electrically conducting fluid. Lawrence and Rao [12]

basically consideredthe same problemas Dandapat and

Gupta

[10],except that they give ^more detailed solutionof the velocity andtemperaturefields, fordifferent values of the non-Newtonian parameter.

In

this paper, ^{we assume} the fluid is thermodynamically compatible in the sense that it satisfies the Clausis-Duhem inequality. The momentum equation is solved using the numerical method discussed in Maneschy, etal. [1] and the results are then used to find the temperature distribution in theboundary layer.

2. GOVERNING

EQUATIONS.

Weconsider the flow of asecond-gradefluidpast^aporous stretchingsheetcoincidingwith theplane y 0 andsubjectedtosuctionatthisplane. Two equal^andoppositeforces areapplied along the x-axis in such a

way

that the origin remains fixed and the velocity ofstretching is proportional^{tothedistancefrom the}origin.The sheet is assumedtohave constanttemperature

T

at its surface and

T..

^{is the}temperature of the ambient fluid. Under these assumptions,theboundary layerequations governingthe flow can be written as

cgu Ov

--+

=0 (2.1)

x

F ^{a (} a ul aua v a ul au av

v- -o ^{L [U j+ v- j (2.2)}

0T 0T o32T

u--- ⁺ v--- ^{K Oy--, (2.3)}

with

v #/p

k, a,/p,

where is the coefficient ofviscosity,/9the density of the fluid,

K

the thermalconductivityand

t

thematerialmodulus usuallyreferredtoasthe firstnormalstresscoefficient.The effects ofviscous dssipationareneglectedinequation(2.3).

Thesignof the parameter

o,

^andtherefore

k

^{inequation(2.2),}has been thesubjectof much controversy. Thermodynamic and compatibility in a sense that all motions of the fluid meetthe Clausius-Duheminequalityand theassumptionthat thespecificHelmholtz freeenergyof thefluidbe aminimum when thefluid is locallyat restrequire,amongother conditions, that

oq

> 0. Although experiments carried out in manydilute polymeric liquids, under the assumption that the fluid is second-grade, seem to imply that

tx

^{< 0, studies by Dunn} and Fosdick [8] and Fosdick and Rajagopal [13] show that when this is thecase therest state of the fluidis unstableor the fluid exhibitspropertiesthatareclearly physically unacceptable.Withoutgettinginto a detailed discussion ofthis matter, we will assume that the normal stresscoefficient

or

^is positive throughout this analysis. Readers should refertoa recentexhaustivecritical reviewby DunnandRajagopal 14]on the thermomechanics of fluid of differential type.

Theboundary^conditionsforthisproblemare

u=Cx"

v=-Vo" T=T

^{at y=0, C>0 (2.4)}

u-0"

T- T,

^as

^(2.5)

where V is the suctionvelocityattheplatesurface.

Letus introduce thefollowing dimensionless variables

x y

R1/2

^{u V}

By

^CL

-=--"

^y=

= ^{"= R="}

⁰⁼

^(2.6)

where R istheReynoldsnumber for the flow and

L

isthe characteristiclength. Ifwe further define the non-dimensionalvelocity componentsas

fi

2f’(y); =-f(y),

^(2.7)

then it can be shown thatequations

(2.1)-(2.3)

reduceto

(2.8)

0" + Pr fO"

^0,

^(2.9)

where

k _R"

^o

k=5- ^Pr -,

and theprimedenotes differentiation^withrespectto

y. In

deriving (2.9)thetemperature profile^was assumed in the form

(y).

Theboundaryconditionsare

f=-VRv2"

_CL

^f’=l"

^{0=0 at}

^.=0

^(2.10)

f’--+O"

^{0-+1 as}

^y-+oo.

^(2.11)

Itisimportantto notethat equation(2.8), (2.10)and

(2.11)

represent atwo-pointnon-linear fourth-order differential equation having only threeboundaryconditionsand, therefore, an additional conditionfor thefunction

f

^{isrequired.Sinceingeneralfar}awayfrom theplatethe flowisumform, Rajagopaland

Gupta

15] suggested usingthe condition that

->0 or

f"--->0

^as

NUMERICAL SOLUTION.

(2.12)

Equations (2.8)and(2.9)constitutethe final system of differentialequationstobe solved.

First, the non-linearequation

(2.8)

^issolved for thevelocityfunction

f(y)

^withtheresult, inturn, substituted in(2.9)todeterminethetemperaturedistribution

0(.7). Following

thequasi-linearization method

developed

byBellman and Kalaba 16], equation (2.9)canbe written in the form below

(3.1)

where

f

^and

f

^{representthecurrentand}previous approximationtothesolution, respectively.

Equation

(3.1)

is a linear

non-homogeneous

equationintheunknownfunction

f.

^Itssolution

can be written as a linear combination oftwo linearindependentsolutions to the homogeneous problems,

fm

^and

^fn2,

^{and aparticularsolution,}

fp,

^{such that}

f Cfm ⁺ C2f.2 ⁺ f.

^(3.2)

In

order to satisfy boundary conditions

(2.10)

and (2.11) and the linearindependence requirement,the three solutions above are assumed to have thefollowingvalues at y 0

’’]

(O,O,l,O)

[fm, f/, f/n,J

m

[fn2,.2,.,2," ’" ^{f"] (0,0,0,1)}

[f,, f/,, f/,’T f/"] (-c- ^R’/,l,

^O,

OI. ^(3.3)

For

each of thehomogeneousandparticularsolutions, equation (3.1)is numerically integratedto y y. usingthe

Runge-Kutta

integrationmethod. The values foundatthe endpointareusedtofind thepairof constantsC andC sothattheboundaryconditionsat y y..aremet.Therefore,

Cf,(y.)+ C2f2(y,,)+ f(y,,)=

⁰

Czf[_i"(y..)+ C2fffl(y.)+ f(yo.)=

^O.

(3.4)

Oncethe values for the constants are found the function

f

^canbe obtained fromequation

(3.2)

for all discretization points 0=

y,

< <_yj<...<

Yu

Y... ^This processis

repeated

until the

magnitude of the difference betweentwoconsecutivesiterationsfall belowaprescribedtolerance,at all points.

In

this work, a tolerance limit of 10^-7 was used. As the zeroth approximatnon ^to the solution,thefunction

f,(y)

y, (3.5)

sansfyingallboundaryconditions, was assumed.

The solution for the function

f

^issubstitutedinequation (2.9),which is then solved using the finite differencetechniquefor the temperatureprofile.

4. RESULTS

AND DISCUSSIONS

Theresults obtainedby solvingthe differentialequationsderived in theprevioussectionswill be now outlined for differentvalues of the folowingparameters

k

otR.

^V

_RV

^l)

PL

^a

CL ^Pr --K

Foragivenflow, these parameters accountfor the effects of the normalstresscoefficient,

oq,

the suction velocity, V and the Prandtl number, Pr,onthe solution.

The value of

f

^{asafunctionof y isplottedinfigure} for different values of k, assumingno- suction attheplatewall(a 0).Thisfunction represents the normal component of thevelocityfor theflow.Asitcanbe seen fromfigure1, the magnitude of the normalvelocitycomponent^increases significantlyforhighvalues of the normalstresscoefficient,

ot.

0 0

Fig. 1. Function

f

for Different Values of k.

Figure2 indicates thetemperaturevariationwhen

Pr

1.0 and a 0, for different values of k.

It

canbe seen thatincreasingthe normalstresscoefficienthasthe effect ofdecreasingthe thermal boundary layer thickness. Figure 3 displays the temperature profiles for different values of

Pr,

assuming k 1.0 andno suction at the wall (a 0.0). As is expected, for higher values of the Prandtl number,thereis a decrease in thethermal boundary layer. The variationof thetemperature withthesuctionvelocitysshowninfigure4for k 1.0 and Pr=1.0. Similartotheresults observed in theothertwocases, as thesuctionvelocityincreasesthethermal

boundary

layerdecreases.

F,- ,7

!/ ,_, _:o.

k =lO.O

Fig.2.

Temperature

_Profileas Function of

Non-Newtonian

Coefficient.

k-l.O ^qO.O

Fig.3.

Temperature

Profile as Function ofPrandtlNumber.

LEGEND

0.50 a 0.25 a O.OO

Fig. 4.

Temperature

^ProfileasFunction as SuctionVelocity.

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porous

elastic sheet due to stretching," To appear^{in the} Journal ofMathematical and Physical Science

(1993).

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plate

with suctionorinjection," I&ECFund., Vol.5

(1966),

p.19.

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p.53.

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R.L.

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