Viscoelastic Fluid Natural Comvection

8/3/2019 Viscoelastic Fluid Natural Comvection

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LAMINAR NATURAL CONVECTION HEAT TRANSFER TOA VISCOELASTIC FLUID

A V SH ENO Y and R A MASH ELKARtDepar tment of Chemical Engneermg, Umverslty of Salford, Salford M5 4WT, England(Recerued 20 June 1977, accepted 25 September 197)

Abstract-A theoret lcal analysis of Lammar natural convectlon heat transfer to a viscoelastlc fluid has been done bythe approximate Integral method It has been observed that a simllanty solution exmt s only for the case of a secondorder fluid m the stagnatron reson of a constant temperat ure heated hor izontal cylinder Lack of properexpenmental data pr&vented a quantltatlve comparison with the theore tlcal analysts, however, a quahtatlvecomparison between the theory and the avrulable experimental data has shown good agreement

lNTRODlJCTtONTher e has been a continued Int erest m the mvestlgatlonof natur al convection heat transfer to non-Newtomanflulds The studies for melastlc non-Newtoman fluldsappear to have been very comprehensively done andrehable theore tical analyses[l, 4, 5, 7, 10-13, 18, 28, 32,35) and exper tmental mvestlgattons[5, 7-13, 17. 23-27)are avadable which provide rehab le design mformatlonUnfortunately, the same sltuatlon does not exist m thecase of non-Newtoman fltuds, w&h exhibit viscoelastl-city Although a number of theor etical mvestlgatr ons[2,21,22,30,31] have been made in this area , we shall showm the followmg that none of these analyses are physlc-ally and mathematically sound Only one expenmentalstudy I I91 exists to-date where the problem ofnatur al convection heat transfer to a horizonta l cylinderhas been studled m moderate ly vlscoelastlc drag reduc-mg polymer solutions

In the followmg, we shall bnefly r eview the exlstmgtheor etical analyses which have been perfor med withspecral refer ence to natur al convection heat tr ansfer mvlscoelastlc fluids The boundar y value problems m non-Newtoman fluid mechamcs are notonously difficultbecause of the non-hnmty m the constltutlve equattonsand not too surpnsmgly, all the theor etical solutionswhich have been pubhshed to-date corr espond to thesimplest asymptotlcally valid forms of the constltutlveequation

Amato and Tlen[Z] consldered the problem of natur alconvection heat transfer from a vertical plate to anOldr oyd fluid It can be shown that their govemmgboundary layer equation ts Incorrect in that rt does notcontam the denvatlve of the pnmary normal str essdtfference ter m but only the denvatlve of a smgle normalstress term Further more, m equatmg the buoyancyand vtscous ter ms m the momentum boundar y layerand m equating the convectlon and conduction ter msm their ener gy equation. they have equated theexponents over the respective non-dtmenstonal ter msThis IS mathematically unsound Theu final result

WEP D, N ational Chemical Laboratones. Poona 41100s. In&a

mdlcates that theu Nusselt number depends only upon avlscoelastlclty number, which IS simply a rat io of thematenat parameters of the vlscoelastic flutd underconslderatlon Int ultlvely one would expect the natur alconvection process m a vtscoelastlc flmd to depend upona dlmenslonless parameter which IS rat to of the charac-terlstlc time of the fluid (having a matenal parametercombination) and a charactenstlc tune of the process

Mlshra[22] has consldered the problem of natur alconvection heat transfer to a vertical plate for a secondorder fluid He has conducted a search for self-similarsolut# ons, wherein tt is found that the slmrlarrty solutionexists only for the pragmatically umnterestmg case of theexcess wall temperat ure varymg linear ly with the dls-tance along the wall h&h ra s solution leads to aconstant momentum and ther mal boundary layer thick-ness, which IS physIcally unsound m the case of avertical plate Mlshra[21] has consrdered the same prob-lem for a Walters B fluld It IS readtly seen that thegovermng boundar y layer equations are the same as m[22] and consequently the same comments hold here too

A number of pragmatically unmterestmg cases such asunsteady natur al convection for a vlscoelastic fluid pastan mfinite plate with constant suction with[31] orwlthout[30] viscous dt sslpatlon have been worked outbut the results are of only margmal Interest

It IS clear from the dIscussIon m the foregoing thatthe problem of natur al convection beat transfer In vu~ o-elastic fluids remams to be solved corr ectly In thepresent work we analyse the problem of natur al con-vection heat tr ansfer to a vlscoelastlc fluid The govern-ing equations are carefully derived and the physicallyrealistic cases where slmdarlty solutron s may exist aresearched Soltit lons to such cases are then obtamed andthe mfluence of vtscoelasticlty IS clearly estabhshedExperimental data m the literature appears to be msupport of our findmgs

GOVERNI NG ONsER VATroN QuATIoiQJFor two dlmenalonal flow over an object mdlcated m

Fe 1 the governmg equations of conservation of mass,769


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770 A V SHENOV and R A MASHELKARmfferenha tmg eqn (8) &r t x gives

FI N 1 SchematI c diagram of flow past a curved surface

momentum and ener gy could be wrltten as

(2)p C (3)

(4)

(The meanmg of the symbols used 1s given m the Nota-tlon ) The boundary conditions on velocity and tempera-ture are

u(x, 0) = 0(x, 0) = 0u(x, 8) = U(X, 6) = 0

(5)T(x. 0) = Tw

T(x, 8~) = T.s

The exact solution of the set of eqns (lH 5) IS , of course,quite dticult to obtam and consequently we shall solvethese by mahng the usual boundary layer ap-proxlmatlons The vahdlty of such appr oxunatlons forvlscoelastlc flulds has been described by White andMetzner [34] and Whlte[33] and consequently, we shallnot discuss these m detail By usmg these ap-proxlmat lons, eqns (1) and (2) remam unchanged Equa-tlon (3) can be slmphfied to

o=-?!?+?52ay ayand eqn (4) can b&slmphfied to

(7)

Integrat mg eqn (6) from y =0 to y = y for any given x,we obtam

P(~~Y)-P(x,o)=Ty,(x,y)-Tyy(x,o) (8)

aa Y) ap(x. 0)--= adx, Y) _ a7,,(x, 0)ax ax C3X aX (9)Substltutmg for (dplax) from eqn (9) mto eqn (2)

[ u 2%+ u * I [1ax apk 0) : a4x. Y) _ a d x , 0)a~ p a~ ax ax I(10)As v + m, eqn (10) becomes

0 = _ r dP(z 0)_P- C dx drw (x, 0) + drY,.(x, ao)dx dx I fx (111Notmg that the fluld IS at rest at mfimty, we obtamT,.~(x.CQ) 0 and consequently

(12)Combmmg eqns (lo), (11) and (12) the resultmg equationcan be re-arr anged asu~+~au~123+la(,_ay P ay P ax 7,,) +fx 1-F( > (13)

Now the body force term can be taken asfi = -g(x) (14)

and by usmg Boussmesq appr oxlmatlon, the density maybe related to the temperature by

%=1+/3(T-T,) (15)Substltutlon of eqns (14) and (15) in eqn (13) gives

u~+udUJ~+la(,_,y)ay P ay P ax+&MT - T-J (16)

Note that the 1 h s represents the mertlal term, the firstterm on the r h s IS the vtscous stress, the second IS theelastic stress and the last IS the buoyancy term For aNewtonian or a purely wscous (melastlc fluld), we have7.X.X TYY 0 and the classlcal equations of natur al con-vectlon for two-dImensIona flow are recovered For avlscoelastlc flmd. ru - ryy# 0 and consequently theseelastic (or normal stresses) moddy the velocity field andhence the tempera ture field The simultaneous solutionof eqns (l), (16) and (7) with boundar y condltlons (5) wdlbe the objective of the present study

CON!THTUTWE EQUATIONA speck consfitutlve equation will have to be chosento solve eqn (16) The flulds chosen ~111 be described by


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Lammu natural convectIon heat transfer to a vlscoelashc Ruld 771the constltutlve equation

7,, = p(ii)B:~,+w@)B:*r5%

-~((ii)B;rZ>where

B;;, = g= v,,, + gJmD ,,,

(17)

(18)Bi~+,,=E$ (19)

and the time derlvatlve S/i% IS defined as.SB&, _ aBfln,---+ VB&,k- v',B;~,st at

-v,B;:> (20)fi repr esents the second mvanant of B f{, and p, w and Aare functions of l? only The use of such constltutlveequation for solution of boundar y layer flows of elasticflulds has been well described by Denn[6] and Kale etal [ 161 The J ustdicatlon IS essentially due to the fact thateqn (17) repr esents the behavlour of elastic flutds exactlym vlscometrlc flows and that for the two dlmenslonalboundar y layer flow under conslderatlon, the dommantter ms m the rat e of stram tensor are those which appearm vlscometrlc flows The functions J L and A may oftenbe expr essed as power functions

&i)=K $=i[ I n--III2A(fi)=m ifi[ 1s-2V2

(21)

(22)The form of w(J7I ) 1s ummportant m the present case,since the terms m which it appear s vamshes m twodunenslonal flows Note that with A = o = 0 and withp(n) gven by eqn (21), the so called O stwald-de-Waelepower-law behavlour IS represented

Applymg the usual boundar y-layer appr oximations, thestress components may be expr essed as

,,=K($)'-m (3-[u~+v$+2~3(23)

and

TX% Tyy =zm g( > (24)APPRoxlM.4TEmTEGRALTEczpRQuE EaHlfmINTIm

Substltutaon of eqns (23) and (24) m eqn (16) and thesimultaneous solution of eqns (I ), (16) and (7) withboundary condrtlons (5) IS the task at hand and m spite ofthe slmphficatlons made b y us, It 1s a formidable task to

solve these non-linear pa rt ial dlfferentlai equations Weshall hence use the appr oximate integra l technr que forthe solutton of these eq uations

It has been a common practice m the integr al solutionof natur al convection problems to asssume that thether mal and the momentum boundar y layer thickn essesare equal We will disregar d this practice and assumethem to be unequal

Let S be the momentum boundar y layer thukness and& be the ther mal boundary layer thickness Then eqn (16)can be Integrated across t he momentum boundar y layerwath the help of equation of contmulty to obtain

(25)which on substitut ion from eqns (23) and (24) becomes

(26)Equation (7) on the other hand can be Integrated acrossthe thermal boundary layer to obtam

&J(u~dy=-(&)(g),=o27)0A further order of magnitude anaI ysls of eqns (26) and(27) can be made by assuming that u - O(U,), x - O(L)and y - O (S) or y - O (&) depending upon the momentumor energy equation being constdered It can be readilyshown that for large values of a character lstlc P randt lnumber Pr , defined as

.x FgfJ(Tw _ T_)] [&,(Tw _ T,)](-2)/(+ 2) (30)


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772 A V SHEUOKandA charactenstlc velocity for the flow under conslderat lon1s defined as

UC = t/(L&(Tw - Tel) = { ($) IBg(Tw - T,r}"(m+z'(31)

The non-dImensI onal varlab1es can now be defined as

T-T,*=T w -T , (32)

Assummg further that the gravity field g(x) 1s given asg(x) = gxlp (33)

we obtam eqns (26) and (27) (on neglectmg Iner tia) as.

(34)and

u,e dy, = _ Gr.x--~"+'""+2" (a$)x 1 YI=o

(3%where 8, 1s the non-dlmenslonal momentum boundar ylayer thtckness

&, 1s the nondrmenslonal thermalthickness

(36)

boundary layer

Gr, IS a local distance based Grashof numberGrX _ p2x + %MTw - Tm)12-K2

Pr, IS a local distance based P randt l numberp,., _ $V (~)z -+:,.-,,,N+ I,,[gS(Tw

P- T-11CKa1n/c2(n+))

and W t IS a Welssenberg number defined as

2m & --Cl~r+tl/~n+a=-P K [/jg(T,,.l-_)](~~-))~(+2)

(37)

(38)

(39)

(40)

R A MASHELKARNote that the combmatlon (Grx (n -Y(2 (n +m n ~2 /Prx ) ISindependent of x and can be regar ded as constant dunngfurther analysis Equat ions (34) and (35) will be solvedsubJect t o the boundary condlttons

11I(XL, ) = ttl(XI, 0) = 0,eh, 0) = 1,

Ul(Xl, so =0ml, &A = 0 (41)

SOLUTION OF THE MOMENTUM -INTEGRAL JlQIJATlONSIn the usual tr adltlon of the Integral solution. the

specdicatlon of the velocity profile (u,) and the tempera-ture profile (e), should yield two ordmary dlfferentlalequations which ~111 have to be solved simultaneouslyApart from the boundar y condltlons specified m eqn (41)we also need to tmpose certam compatlblhty condltlonsm order to be able to choose the proper form of thevelocity profile This matter has been largely ignored mthe prior lrterature at least as far as the natural con-vectlon problem 1s concerned We shall, however,consider these condttlons rather carefully smce non-tnvlal dtfficulttes ~111 arr se other wise In the solution ofthe problem

For smoothness at the edges of the momentum andther mal boundar y layer, we must have

au,( >=0 at y,=S,aYIand

a e( )y1 =0 at ye=&,(42)

(43)

The different ial form of eqn (34) must be satisfied at thewall (y, = 0) glvmg

x,ne+- -a u , m , a au1 ( 1 --a y , a Yl ( 1x , a h =0 at y1=0Fur ther more, the dlfferentlal form of eqn (35) must alsobe satMed at the wall, gvmng

2e=O at y,=OayrA further r esmctlon arlses when one apphes theddferentt lal form of the eqn (34) at the edge of themomentum boundar y layer It follows stralghtforwar dlythat smce (auMay&,-a, = 0, we have 8 = 0 at the ed geof the momentum boundar y layer However, 8 = 0 at theedge of the ther mal boundar y layer by the very defimtlonof the thermal boundary layer We thus conclude that thethermal boundary layer cannot extend beyond themomentum boundar y layer or else drop e(8,) # 0 Hencewe shall consider only the sltuatlon where

s,-cl8, w


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L.aminarnatural convectIon heat transfer to a vlscoclasuc fluid n3By assuming polynomial forms for the velocity andtemperature profiles of the form

andNBT) = 7 6,q; (47)

(where v = (YJ&) and VT = (y,/&,)) It can be readilyshown t hat for satisfying all the boundar y and compatl-bdlty condltlons except for eqn (44), we need at leastI = 6 for U,(T) and I = 4 for @VT)Application of all the boundary and compatlbditycondltrons makes It possible to determine a, and b, andthe results can be shown to reduce to

and8(m) = (1 f TJ T)(l -m-J 3 (49)

The coefficient C IS as yet unspecdied and wdl bedetermined by usmg compatlbdlty condition (44)SEARCH OF A S- sOINTION

Equations (34) and (35) are now solved with thesubstitution of eqns (48) and (49) whereupon with somere-arrangement we obtain

and(n--IY(2(n+lMn+2))& [&Cf(a)l =& Grx pr, (51)

wherea=&18 (52)

f(a)= [~*-~..+~*3-3.4+~=5-~=6](53)

and(54)

For solvmg eqns (SO)and (51). we assumeST, = B,x,81 = Bzx,

andC = Bs,~ (55)

and on substttutlon of eqn (55) m eqns (SO )and (Sl), weget

0 = & B,x,+~ - $ xl(-) + IW,

and+ [sq - (s - l)t, ~x,--(--1 (56)

The condrtions for a smulanty solution to exist arededuced from eqns (56) and (57) I t readdy follows thatr = t and furthermorep+r=n(q-r)=sq-(s-l)r-1

andr+q-1=--r

solving for r , q and 3 we ob tamr=t=z

(58)

(59)

3n+I+2pq= 3n+ls- @+1)(3n+l)(3P + 1)

WJ)

We now examme the reahstlc values of s, n, p and q forwhich a solutmn wtll extst Apparently the power-lawmdlces and the normal stress dtierence function mduzesstrongly Influence the development of the boundary layerthicknesses and the velocrty field, a sltuatron not un-common in the forced convectton flows of vtscoelastlcflulds [33]Case 1

Exammmon of the shear stress shear rate and thenormal stress difference shear rate functions mdtcatesthat m the low shear regon (O( l) set-), we have s =n + 1 Large amount of expenmental data exists m thisregon (see, e g 3 and 20) In this case then we obtainfrom eqn (60) p = n and consequently r = t = 0 A spe-clalcaseofs =n+lextstswheos=2aodn=l,whlchIS the so called *second or der flurd In this case t henP = 1 IO other words, the gravrty field should be of thetype

g(x) = gx1 (61)This correspoucJs to the stagnatmn remon of a honzontalcyhnder with g(x) = g sm x1 - gxl (0 < x, < r /6)


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774 A V SHEN~Vnd R A MASHELKARWe thus conclude that for a second or der fluld. thestagnation r epon of a horizontal cylinder pr ovides aphysically reahstlc solution It IS Impor tant to note herethat smce the shear rates m the natural convection flowsare likely to be generally quite low, the second order flowbehavlour LS hkely to be closely appr oached andconsequently the solution I S even more meanmgful Weshall hence discuss this solution In some detad

Case 2In the intermediate shear range (0(50-X10) see-), wehave s - 2n (see 3, 14 and 29) Substltutmg s = 2n m eqn(60) we obtain p = (n + 1)/(3n - 1) and r=t=(1 - n)/(l - 3n) For the physical situaUon examined byus p 2 0 and furthermore since the boundar y layerthickness can either remam constant or mcrease with x,,we have r = f 2 0 only It can be readdy shown that thisIS mpossible to satisfy simultaneously unless R = 1, s = 2and p = 1, which IS the case already considered Henceno physically realistic solution ex ists in this region

Case 3We now examine whether or not a similarity solutionexists for the case of a vertical plate maintained atconstant wall temperature In this case p = 0 Consldera-tlon of eqn (60) gves s = 3n + 1 No thuds a re known toexist for which the r elation I S known to hold andconsequenily It must be concluded that no reallstlcsolution ex ists m this case As a tr ivial consequence ofthis observation, we can also conclude that for the caseof a second order fluid, no similarity solution exists forthe isothermal vertical hot plateCase 4As a special case, we consider a purely VI SCOUSlu id ,for which WI = 0 and s = 0 We then have for the wellstudied case of vertical isothermal plat e @ = 0 ), r = t =n/(3n + 1) This variation of boundar y layer thickness LSquite m order with the theoretical predictions and theexpert mental observations, and repr esents a reahstlcphysical situationThus, from the foregoing four cases I t can beconcluded tha t, for a vlscoelastic fluid, slmllar lty solutionexists only for the special case of a second order fluid Inthe stagnation region of a constant temperature heatedhomontal cylmder

For this particular case of s = 2, n = 1, p = 1, q = 3/2and r = r = 0, eqns (56) and (57) can then be slmphfied togive3 B3 10 2O=i$X-B+- WI,%2 99 B2Bn2Bf(a) = &

(62)

The thir d equation for the solution of B,, B2 and Bs canbe obtamed from eqn (44) which a lone was not satisfiedm makmg the choice for the velocity and temperatureprofiles Thus, we have and 3

wherewl, = & &S(Tw - Tm ) 2 n

P K2 ISolving (62). (63). (64). we have .

980a2 II4Bt = (297 Q - 501fk)

pr- 114

(64)

(65)

(66)

(67)

(68)The relationshIp between (I and W d can be obtained fromthe following expression which emerges out by properarr angement of the above equations

245f(a) 495a(3a - 1)(297a - 50) (297a - 50) I2 _ Wls2

Pr (69)Now the local Nusselt number 1s defined as

(7Oa)

(70b)

= f ,3rx14X,i4 (7Oc)IFor the stagnat ion region of a constant temperatureheated honzontal cylinder we have p = 1 and L, = R(radius of the cylinder), thus mvtng

Iv&=2 E(297a - 5O)f(a) 4Gr 1,4pr1,4 X 4980a2 1 x 0 5 (71)The average Nusselt number can now be easily obtainedas

Nu,,~ = 2 (297a - S O)f (a ) 4 GrR,,4Prl,4980a I (72)where

Gm = p2R+ 2M3(Tw - T-)1*-KZ (73)

(74)The results of the above analysis are borne out by Figs 2


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Lammar natural convectIon heat transfer to a vlscoelasuc fluid 775

03 1 I I 1 I 1 I I , OD 002 0.04 006 006 01 0 12 0 14 0 16 018 02 02 2(Wr, )VPr

FIN 2 Var iation of the ratlo of the boundary layer thrck nesses with vlscoelastrclty

0495

0485

,rf, I-

1 -

04=oo I I I I I I I I I 1002 I004 006 008 01 0 12 014 016 Ol6 02 022( Wls)2/Pr

Fig 3 Vanatlon of the average Nusselt number with vlscoelastlcrty

DISCUsslONIt IS mterestrng to note that the influence of VISCO -elastlclty on Nusselt numbers depends upon the mag-mtude of the W elssenberg number At small Welssen-

berg numbers, the Nusselt number appear s to go througha margmd enhancement, however, at larger Welssenbergnumbers, there 1s a marked reduction Based on thematerml par ameter data for viscoelastic fluids and thenatura l convection process par ameters, it would appearthat the range of 10e3 < WI < 10 IS of Inter est It wouldthus seem that the net influence wdl largely depend uponthe combmatlon of process and material parameters asqven m eqn (65)Expenmental data on natur al convectlon heat transferfrom a horizontal cyhnder provided by Lyons et al 1191to moderately elastic drag reducmg polyethylene oxidesolutions (10&1000ppm) indeed show that with m-creased polymer concentrat ion (Incr eased elastlclty and

Welssenberg number), there I S a decrease m Nusseltnumbers m comparison to the Newtoruan value Ourtheoretical pred ictlons, thus appear to be borne out bythe exper Imental data Unfortunately, no quantitativecomparison can be made due to the fact that no materialparameter data (such as relaxa tion trmes) have beenobtained by Lyons et al [19] and furthermore, ouranalysis IS pertment only to the stagnatlon ree;lon of thehorizontal cylinderIt IS mterestmg to note also, that Amato and TEn[2]also observed tha t Nusselt numbers for vlscoelastlc fluidseither increased or decreased depending upon the valueof a vlscoelastlclty number Although the observation ISparallel to the one m this work, due to the hmltatlonsmdlcated m the I ntroduction this may be consldered asa mere comcidenceAcknowlrdgemenr-The financial support of the Bntlsh GasCorporation dunng the tenure of ttus work I S great ly appr eciated


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A V SHHNOY and R A MASHELKAR76

gGr,G~ R

MYrATIoNnth &vhn-Enckson acceleratton tensorspectic heat per untt massbody force term parallel to x-dwctronbody force term parallel to ydtrectronacceleratron due to gravrtycomponent of the acceleration due to gravrty m thex-duectlonconuugate metrr c tensorcharacterrstlc generahsed Grashof number defined by

eqn (29)generahsed Grashof number based on the radtus of thecyhnder defined by eqn (73)generahsed local Grashof number defined by eqn (38)thermal conductrvrtymatenal constantcharactenstrc lengthmatertal constantexponent m shear stress power-lawlocal Nusselt number defined by eqn (7Oa)average Nusselt numberpressurePrandtl number defined by eqn (74)charactenstrc generahsed P randtl number dehrt ed by eqn(28)generahsed local Prandtl number defined by eqn (39)exponent m normal stress power-lawtemperaturetemperature of the sohd surfacetemperature of the bulk of the Ruedveloctty component along x-co-ordmatedlmenslonless veloctty component deCned by eqn (32)charactenstlc velocrtyvelocity component along y-co-ordmatedrmenslonless velocity component defined by eqn (32)velocny vectordrstance along the curved surfacedunenstonless dtstance d efined by eqn (32)distance normal to the curved surfacedunensionless drstance defined by cqn (32)generahsed Werssenberg number defined by eqn (40)Wetssenberg number for a second order fltnd defined byeqn (65)

Greek symbolsrat10 of the thermal boundarv laver thuzkness to themomentum boundary layer tinckitessexpanston coefficient of the Butd defined by eqn (I S)momentum boundary layer thtck nessdtmenstonless momentum boundary layer thtcknessdefined m eqn (36)thermal boundary layer thwzknessdunenstonless thermal boundary layer thickness definedm eqn (37)stmdanty vart able defined by eqn (47)sundart ty vart abk defined by eqn (47)dtmensronloss temperahue ddbrence de6ned m eqn (32)matenal functton of flvtscostty of tho second order Btndmaterml function of Rdenstiy of the fluid at temperatur e Tdensuy of the flmd at temperatur e T,,normal stress m the x-due&onshearmgstressmthex-ydtrecttonnormal stress m the ydnectron

111M131141PIWI171181[91

AcnvosA,AIChE J 19606584Amato W S and Tlen C Chem Engng Prog Svmp SerNo 102 197066 92Brodnvan I G . Gaskms F H and Phthoooff W TransSac iheol 195; 1 109 *-Chen T Y and Wollersherm D E Trans ASME J HeatTrans 1973 95 123Dale 1 D and Emery A F, Trans ASME, J Heat Trans19729464Denn M hi, Chem Engng S cr 1967 22 395Emery A F, Chl H W and Dale J D, Trans ASME, JHeat Trans 1971 93 164

DOI[IIIWI11311141

::i;t1711181

1191

Emery A F , Dreger W W. W yche D L and Yang A,Tmns ASME. J Heat Tmns 1975 97 366Emery A F, Yang A and Wilson J R . ASME paper No76-HT-46 presented at the 16th National Heat TransferConf , MLSSOU~.ug 8-11, 1976FUJII , Mlyatake 0, Fu~ u M and Tanaka H , nt ChemEngng 1972 12 729FUJI]T , Mlyatake 0, FUJI ~M , Tanaka H and MurakamrK , Int J Heat Mass Tmns 1973 16 2177FUJII , Mryatake 0, FUJIIM , Tanaka H and MurakamrK , Int .I Heat Mass Tr uns 1974 17 149Gentry C C and Wollershelm D E , Trans ASME, J HeatTmns 1974 % 3Gmn R F and Metzner A B Prac 4th Int RheofCongr l%S 2 383HellumsJ D andChurchfflS W,AJ ChEJ 196410110Kale D D , Mashelkar R A and Ulbrecht J , Rheo Acta1975 14 631Kim C B and Woller shetm D E , Truns ASME, J HeatTrans 1976 98 144Kleppe J and Marner W J Trans ASME I Heat Tmns1972 94 371Lyons D W, White J W and Hatcher J D, Ind EngChem Fundls 1972 11 586[20] Markovltz H ,Pmc 4th Int Rhea! Congr 1%5 1 I89[2l] Mlshr a S P , hdtan Chem Engr I%6 g 28

7 devtatonc str ess tensor0 matenal function of Iifi second utvarlant of Si\,

REFERENCES

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Viscoelastic Fluid Natural Comvection

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