Laminar natural convection heat transfer to a viscoelastic fluid

LAMINAR

NATURAL CONVECTION A VISCOELASTIC A

Department

V

SHENOY

and R

of Chemical

Engneermg,

Umverslty

20 June

1977, accepted

(Recerued

A

HEAT FLUID

TRANSFER

TO

MASHELKARt of Salford, Salford 25 September

M5 4WT, England

197)

Abstract-A theoretlcal analysis of Lammar natural convectlon heat transfer to a viscoelastlc fluid has been done by the approximate Integral method It has been observed that a simllanty solution exmts only for the case of a second order fluid m the stagnatron reson of a constant temperature heated horizontal cylinder Lack of proper expenmental data pr&vented a quantltatlve comparison with the theoretlcal analysts, however, a quahtatlve comparison between the theory and the avrulable experimental data has shown good agreement

lNTRODlJCTtON

mdlcates that theu Nusselt number depends only upon a vlscoelastlclty number, which IS simply a ratio of the matenat parameters of the vlscoelastic flutd under conslderatlon Intultlvely one would expect the natural convection process m a vtscoelastlc flmd to depend upon a dlmenslonless parameter which IS ratto of the characterlstlc time of the fluid (having a matenal parameter combination) and a charactenstlc tune of the process Mlshra[22] has consldered the problem of natural convection heat transfer to a vertical plate for a second order fluid He has conducted a search for self-similar solut#ons, wherein tt is found that the slmrlarrty solution exists only for the pragmatically umnterestmg case of the excess wall temperature varymg linearly with the dlstance along the wall h&hra’s solution leads to a constant momentum and thermal boundary layer thickness, which IS physIcally unsound m the case of a vertical plate Mlshra[21] has consrdered the same problem for a Walter’s B fluld It IS readtly seen that the govermng boundary layer equations are the same as m [22] and consequently the same comments hold here too A number of pragmatically unmterestmg cases such as unsteady natural convection for a vlscoelastic fluid past an mfinite plate with constant suction with[31] or wlthout[30] viscous dtsslpatlon have been worked out but the results are of only margmal Interest It IS clear from the dIscussIon m the foregoing that the problem of natural convection beat transfer In vu~oelastic fluids remams to be solved correctly In the present work we analyse the problem of natural convection heat transfer to a vlscoelastlc fluid The governing equations are carefully derived and the physically realistic cases where slmdarlty solutrons may exist are searched Soltitlons to such cases are then obtamed and the mfluence of vtscoelasticlty IS clearly estabhshed Experimental data m the literature appears to be m support of our findmgs

There has been a continued Interest m the mvestlgatlon of natural convection heat transfer to non-Newtoman flulds The studies for melastlc non-Newtoman flulds appear to have been very comprehensively done and rehable theoretical analyses[l, 4, 5, 7, 10-13, 18, 28, 32, 35) and expertmental mvestlgattons[5, 7-13, 17. 23-27) are avadable which provide rehable design mformatlon Unfortunately, the same sltuatlon does not exist m the case of non-Newtoman fltuds, w&h exhibit viscoelastlcity Although a number of theoretical mvestlgatrons[2, 21,22,30,31] have been made in this area, we shall show m the followmg that none of these analyses are physlcally and mathematically sound Only one expenmental the problem of where to-date study I I91 exists natural convection heat transfer to a horizontal cylinder has been studled m moderately vlscoelastlc drag reducmg polymer solutions In the followmg, we shall bnefly review the exlstmg theoretical analyses which have been performed with specral reference to natural convection heat transfer m vlscoelastlc fluids The boundary value problems m nonNewtoman fluid mechamcs are notonously difficult because of the non-hnmty m the constltutlve equattons and not too surpnsmgly, all the theoretical solutions which have been pubhshed to-date correspond to the simplest asymptotlcally valid forms of the constltutlve equation Amato and Tlen[Z] consldered the problem of natural convection heat transfer from a vertical plate to an Oldroyd fluid It can be shown that their govemmg boundary layer equation ts Incorrect in that rt does not contam the denvatlve of the pnmary normal stress dtfference term but only the denvatlve of a smgle normal stress term Furthermore, m equatmg the buoyancy and vtscous terms m the momentum boundary layer and m equating the convectlon and conduction terms m their energy equation. they have equated the exponents over the respective non-dtmenstonal terms This IS mathematically unsound Theu final result WEPD,

National

Chemical

Laboratones.

GOVERNING CONsERVATroNEQuATIoiQJ For two dlmenalonal flow over an object mdlcated m Fe 1 the governmg equations of conservation of mass,

Poona 41100s. In&a 769

A V

770

SHENOV

and R A

MASHELKAR

mfferenhatmg

eqn (8) &r t x gives

aa

---=

Substltutmg

u

[

2%+

ax

ap(x. 0)

Y)

ax

Y) _ a7,,(x,

adx,

ax

0)

(9)

aX

C3X

for (dplax) from eqn (9) mto eqn (2)

u

I p1 [apk =

*

a~

Y) _ adx, 0)

0) : a4x.

ax

a~

ax

I

(10) As v + m, eqn (10) becomes FIN 1 SchematIc diagram of flow past a curved surface 0 =

momentum

r

dP(z 0) -_ P- C dx

_

I x

0) + drY,.(x, ao) + f

drw(x,

dx

dx

and energy could be wrltten as Notmg that the fluld IS at rest at mfimty, T,.~(x.CQ)= 0 and consequently

we obtam

(12)

(2) p

C

(3)

Combmmg eqns (lo), (11) and (12) the resultmg can be re-arranged as

(4)

u~+~au~123+la(,_

ay

(The meanmg of the symbols used 1s given m the Notatlon ) The boundary conditions on velocity and temperature are u(x, 0) = 0(x, 0) = 0 u(x, 8) = U(X, 6) = 0 (5)

(111

P

ay

7,,)

P ax

+fx

(

1-F

equation

>

(13)

Now the body force term can be taken as (14)

fi = -g(x) and by usmg Boussmesq approxlmatlon, be related to the temperature by

the density

may

T(x. 0) = Tw

%=1+/3(T-T,)

T(x, 8~) = T.s

The exact solution of the set of eqns (lH5) IS, of course, quite dticult to obtam and consequently we shall solve these by mahng the usual boundary layer approxlmatlons The vahdlty of such approxunatlons for vlscoelastlc flulds has been described by White and Metzner[34] and Whlte[33] and consequently, we shall not discuss these m detail By usmg these approxlmatlons, eqns (1) and (2) remam unchanged Equatlon (3) can be slmphfied to

o=-?!?+?52

ay

and eqn (4) can b&slmphfied

ay

to (7)

Integratmg we obtam

eqn (6) from

y =0 to y = y for any given x,

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

(8)

Substltutlon

(15)

of eqns (14) and (15) in eqn (13) gives

u~+udUJ~+la(,_,y)

ay

P

ay

+&MT

P

ax - T-J

(16)

Note that the 1 h s represents the mertlal term, the first term on the r h s IS the vtscous stress, the second IS the elastic stress and the last IS the buoyancy term For a Newtonian or a purely wscous (melastlc fluld), we have 7.X.X - TYY= 0 and the classlcal equations of natural convectlon for two-dImensIona flow are recovered For a vlscoelastlc flmd. ru - ryy# 0 and consequently these elastic (or normal stresses) moddy the velocity field and hence the temperature field The simultaneous solution of eqns (l), (16) and (7) with boundary condltlons (5) wdl be the objective of the present study CON!THTUTWE

EQUATION

A speck consfitutlve equation will have to be chosen to solve eqn (16) The flulds chosen ~111 be described by

Lammu the

constltutlve

natural convectIon

heat transfer to a vlscoelashc Ruld

equation

7,, =

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

-~((ii)B;rZ>

(17)

B;;, = g”=v’,,, + gJmD‘,,,

(18)

where

Bi~+,,=E$

(19)

and the time derlvatlve

771

solve these non-linear partial dlfferentlai equations We shall hence use the approximate integral technrque for the solutton of these equations It has been a common practice m the integral solution of natural convection problems to asssume that the thermal and the momentum boundary layer thicknesses are equal We will disregard this practice and assume them to be unequal Let S be the momentum boundary layer thukness and & be the thermal boundary layer thickness Then eqn (16) can be Integrated across the momentum boundary layer wath the help of equation of contmulty to obtain

S/i% IS defined as

.

SB&, _ aBfln, ---+ at st

V’B&,k-

v',B;~,

(25) (20)

-v’,B;:>

fi represents the second mvanant of B f{, and p, w and A are functions of l? only The use of such constltutlve equation for solution of boundary layer flows of elastic flulds has been well described by Denn[6] and Kale et al [ 161 The Justdicatlon IS essentially due to the fact that eqn (17) represents the behavlour of elastic flutds exactly m vlscometrlc flows and that for the two dlmenslonal boundary layer flow under conslderatlon, the dommant terms m the rate of stram tensor are those which appear m vlscometrlc flows The functions JL and A may often be expressed as power functions

which on substitution

from eqns (23) and (24) becomes

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

across

&J”(u~dy=-(&)(g),=o (27)

0


&i)=K

[

$=i

I

(21)

ifi

1

(22)

Cs-2V2

A(fi)=m

[

The form of w(J7I) 1s ummportant m the present case, since the terms m which it appears vamshes m two dunenslonal flows Note that with A = o = 0 and with p(n) gven by eqn (21), the so called Ostwald-de-Waele power-law behavlour IS represented Applymg the usual boundary-layer approximations, the stress components may be expressed as

,,=K($)'-m

(3’-‘[u~+v$+2~3 (23)

and

TX% -

Tyy

=zm

(g’ >

APPRoxlM.4TEmTEGRALTEczpRQuE

A further order of magnitude anaIysls 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 momentum or energy equation being constdered It can be readily shown that for large values of a characterlstlc Prandtl number Pr, defined as

. x

Substltutaon of eqns (23) and (24) m eqn (16) and the simultaneous solution of eqns (I), (16) and (7) with boundary condrtlons (5) IS the task at hand and m spite of the slmphficatlons made by us, It 1s a formidable task to

_

T_)]<3(--1)ti(2(n+l))

(28)

the mertlal terms m the equation of motion are neghglble m relation to the terms of the r h s , this bemg a very reasonable assumption for non-Newtonian fluids which generally have very high consistencies (see, e g a slmllar assumption used by Acnvos [ l] m the analysis of natural convection phenomenon m power-law flu&) A Grashof number for the case under conslderatlon can be defined by considering the ratio of the buoyancy force to the VISCOUSforce as

(24)

EaHlfmINTIm

FgfJ(Tw

Grc

=

P’L,““Wgkf

- T#-”

(29)

Since there IS no characterlstlc length for the external flow bemg consldered, we use the method of Hellurns and Churchill [ 151 and choose L, to make Gr, = I Thus 2hn+2>

L,=

0

$

[&,(Tw

_ T,)](“-2)/(“+2)

(30)

772

A

A charactenstlc 1s defined as

velocity

UC = t/(L&(Tw

SHEUOK and R

V

for the flow under conslderatlon

- Tel) = {

($) IBg(Tw

- T,r}"(m+z'

eh,

further that the gravity

field g(x) 1s given as

g(x) = gxlp

(33)

we obtam eqns (26) and (27) (on neglectmg

Inertia) as.

(34) and dy,

=

Gr.x--‘~"+'""+2"

_

0) = 1,

1

(- >

au, =0 aYI

YI=o

(3% momentum

boundary

nondrmenslonal

ae

thermal

boundary

layer

(ay1)

Gr, IS a local distance

based Grashof

GrX _ p2x “+%MTw

K2

Pr, IS a local distance p,., _ $V

and Wt IS a Welssenberg

y,=S,

(42)

at

ye=&,

(43)

’ ax,(ah 1 ( 1 --

(38)

a

w,

au1

=0

at

at the

y1=0

2

at

e=O ayr

(~)z”-+‘:,.-,,,N”+I,,[gS(Tw P

- 1n/c2(n+ 1)) T-11CKa

=0

a au, m+ ay, aYl

based Prandtl number

-

at

Furthermore, the dlfferentlal form of eqn (35) must also be satMed at the wall, gvmng

number

- Tm)12-”

JlQIJATlONS

The differential form of eqn (34) must be satisfied wall (y, = 0) glvmg

x,ne+(37)

(41)

&A = 0

and

(36) &, 1s the thickness

ml,

In the usual tradltlon of the Integral solution. the specdicatlon of the velocity profile (u,) and the temperature profile (e), should yield two ordmary dlfferentlal equations which ~111 have to be solved simultaneously Apart from the boundary condltlons specified m eqn (41) we also need to tmpose certam compatlblhty condltlons m order to be able to choose the proper form of the velocity profile This matter has been largely ignored m the prior lrterature at least as far as the natural convectlon problem 1s concerned We shall, however, consider these condttlons rather carefully smce nontnvlal dtfficulttes ~111 arrse otherwise In the solution of the problem For smoothness at the edges of the momentum and thermal boundary layer, we must have

(a$)

x

where 8, 1s the non-dlmenslonal layer thtckness

so = 0

Ul(Xl,

SOLUTION OF THE MOMENTUM -INTEGRAL

(32)

*=Tw-T,

u,e

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

varlab1es can now be defined as

T-T,

Assummg

MASHELKAR

Note that the combmatlon (Grx(n-‘Y(2(n+‘mn~2”/Prx) IS independent of x and can be regarded as constant dunng further analysis Equations (34) and (35) will be solved subJect to the boundary condlttons

(31) The non-dImensIonal

A

(39)

number defined as

y,=O

A further resmctlon arlses when one apphes the ddferenttlal form of the eqn (34) at the edge of the momentum boundary layer It follows stralghtforwardly = 0, we have 8 = 0 at the edge that smce (auMay&,-a, of the momentum boundary layer However, 8 = 0 at the edge of the thermal boundary layer by the very defimtlon of the thermal boundary layer We thus conclude that the thermal boundary layer cannot extend beyond the

momentum boundary layer or else drop e(8,) # 0 Hence we shall consider only the sltuatlon where

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

&

--

(40)

s, -cl 8,

w

L.aminarnatural convectIon heat transfer to a vlscoclasuc fluid By assuming polynomial forms temperature profiles of the form

for the velocity

and

and on substttutlon of eqn (55) m eqns (SO)and (Sl), we get 0 =

and

&

B,x,“+~

+ [sq NBT) =

n3

7 6,q;

(47)

(s

-

$

-

l)t,

xl”(-)

+

IW,

~x,--(--1

(56)

and

(where v = (YJ&) and VT = (y,/&,)) It can be readily shown that for satisfying all the boundary and compatlbdlty condltlons except for eqn (44), we need at least I = 6 for U,(T) and I = 4 for @VT) Application of all the boundary and compatlbdity condltrons makes It possible to determine a, and b, and the results

can be shown

to reduce to

and

8(m) = (1 f TJT)(l -m-J3

(49)

The coefficient C IS as yet unspecdied and wdl be determined by usmg compatlbdlty condition (44)

p+r=n(q-r)=sq-(s-l)r-1

Equations (34) and (35) are now solved with the substitution of eqns (48) and (49) whereupon with some re-arrangement we obtain

(59)

solving for r, q and 3 we obtam r=t=z q=

and

s-

&

3n+I+2p 3n+l

WJ)

@+1)(3n+l) (3P + 1)

(n--IY(2(n+lMn+2))

[&Cf(a)l

=& Grx

pr,

(51)

where a=&18 f(a)=

(58)

and r+q-1=--r

sOINTION

SEARCH OF A S-

The condrtions for a “smulanty” solution to exist are deduced from eqns (56) and (57) It readdy follows that r = t and furthermore

(52)

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

and (54) For solvmg eqns (SO)and (51). we assume ST, = B,x,’ 81 = Bzx,’

We now examme the reahstlc values of s, n, p and q for which a solutmn wtll extst Apparently the power-law mdlces and the normal stress dtierence function mduzes strongly Influence the development of the boundary layer thicknesses and the velocrty field, a sltuatron not uncommon in the forced convectton flows of vtscoelastlc flulds [33] Case 1 Exammmon

of the shear stress shear rate and the normal stress difference shear rate functions mdtcates that m the low shear regon (O(l) set-‘), we have s = n + 1 Large amount of expenmental data exists m this regon (see, e g 3 and 20) In this case then we obtain from eqn (60) p = n and consequently r = t = 0 A speclalcaseofs =n+lextstswheos=2aodn=l,whlch IS the so called *‘second order flurd” In this case then P = 1 IO other words, the gravrty field should be of the type g(x) = gx1

and C = Bs,~

(55)

(61)

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

774

A

V SHEN~V and R A MASHELKAR

We thus conclude that for a second order fluld. the stagnation repon of a horizontal cylinder provides a physically reahstlc solution It IS Important to note here that smce the shear rates m the natural convection flows are likely to be generally quite low, the second order flow behavlour LS hkely to be closely approached and consequently the solution IS even more meanmgful We shall hence discuss this solution In some detad

(64) where wl, = &

&S(Tw

Case 3 We now examine whether or not a similarity solution exists for the case of a vertical plate maintained at constant wall temperature In this case p = 0 Conslderatlon of eqn (60) gves s = 3n + 1 No thuds are known to exist for which the relation IS known to hold and consequenily It must be concluded that no reallstlc solution exists m this case As a trivial consequence of this observation, we can also conclude that for the case of a second order fluid, no similarity solution exists for the isothermal vertical hot plate

. II4

980a2 Bt =

(65)

I

Solving (62). (63). (64). we have

Case 2

In the intermediate shear range (0(50-X10) see-‘), we have s - 2n (see 3, 14 and 29) Substltutmg s = 2n m eqn r=t= obtain p = (n + 1)/(3n - 1) and (60) we (1 - n)/(l - 3n) For the physical situaUon examined by us p 2 0 and furthermore since the boundary layer thickness can either remam constant or mcrease with x,, we have r = f 2 0 only It can be readdy shown that this ISimpossible to satisfy simultaneously unless R = 1, s = 2 and p = 1, which IS the case already considered Hence no physically realistic solution exists in this region

- Tm) 2n

K2

P

pr-

114

(66)

(297 Q - 501fk)

(67)

(68) The relationshIp between (I and Wd can be obtained from the following expression which emerges out by proper arrangement of the above equations 245f(a) (297a - 50)

495a(3a - 1) 2 _ Wls2 (297a - 50) I Pr

(69)

Now the local Nusselt number 1s defined as

(7Oa)

(70b)

Case 4 = f ,3rx’14X,“i4 (7Oc) As a special case, we consider a purely VISCOUS fluid, I for which WI = 0 and s = 0 We then have for the well studied case of vertical isothermal plate @ = 0), r = t = For the stagnation region of a constant temperature n/(3n + 1) This variation of boundary layer thickness LS heated honzontal cylinder we have p = 1 and L, = R quite m order with the theoretical predictions and the (radius of the cylinder), thus mvtng expertmental observations, and represents a reahstlc physical situation (297a - 5O)f(a) “4Gr 1,4pr1,4 X “4 Iv&=2 Thus, from the foregoing four cases It can be 980a2 E concluded that, for a vlscoelastic fluid, slmllarlty solution (71) exists only for the special case of a second order fluid In the stagnation region of a constant temperature heated The average Nusselt number can now be easily obtained homontal cylmder as For this particular case of s = 2, n = 1, p = 1, q = 3/2 and r = r = 0, eqns (56) and (57) can then be slmphfied to (297a - SO)f(a) “4 GrR,,4Prl,4 Nu,,~ = 2 (72) give 980a ’ I

1 x

3 B3 O=i$X-B+2

2 10 WI,% 99 B2

(62)

where Gm = p2R”+2M3(Tw - T-)1*-”

Bn2Bf(a) = & The third equation for the solution of B,, B2 and Bs can be obtamed from eqn (44) which alone was not satisfied m makmg the choice for the velocity and temperature profiles Thus, we have

075

KZ

(73) (74)

The results of the above analysis are borne out by Figs 2 and 3

Lammar

031 OD

natural convectIon

I 0.04

002

I 006

heat transfer

1 006

I 01 (Wr,

to a vlscoelasuc fluid

1 0 12

I 0 14

775

I 018

0 16

, 02

‘ 022

)VPr

FIN 2 Variation of the ratlo of the boundary

layer thrcknesses

with vlscoelastrclty

,r

f

, -I 1 -

0495

0485

04=oo

I

002

I

I

004

006

Fig 3 Vanatlon

I

I

of the average

Nusselt

DISCUsslON

mterestrng to note that the influence of VISCOelastlclty on Nusselt numbers depends upon the magmtude of the Welssenberg number At small Welssenberg numbers, the Nusselt number appears to go through a margmd enhancement, however, at larger Welssenberg numbers, there 1s a marked reduction Based on the materml parameter data for viscoelastic fluids and the natural convection process parameters, it would appear that the range of 10e3 < WI < 10 IS of Interest It would thus seem that the net influence wdl largely depend upon the combmatlon of process and material parameters as qven m eqn (65) Expenmental data on natural convectlon heat transfer from a horizontal cyhnder provided by Lyons et al 1191 to moderately elastic drag reducmg polyethylene oxide solutions (10&1000ppm) indeed show that with mcreased polymer concentration (Increased elastlclty and It

IS

I

01 0 12 ( Wls)2/Pr

008

number

I

I

014

016

I

Ol6

1

02

I

022

with vlscoelastlcrty

Welssenberg number), there IS a decrease m Nusselt numbers m comparison to the Newtoruan value Our theoretical predictlons, thus appear to be borne out by the experImental data Unfortunately, no quantitative comparison can be made due to the fact that no material parameter data (such as relaxation trmes) have been obtained by Lyons et al [19] and furthermore, our analysis IS pertment only to the stagnatlon ree;lon of the horizontal cylinder It IS mterestmg to note also, that Amato and TEn[2] also observed that Nusselt numbers for vlscoelastlc fluids either increased or decreased depending upon the value of a vlscoelastlclty number Although the observation IS parallel to the one m this work, due to the hmltatlons mdlcated m the “Introduction” this may be consldered as a mere comcidence financial support of the Bntlsh Gas the tenure of ttus work IS greatly appreciated

Acknowlrdgemenr-The Corporation

dunng

A V

776

SHHNOY

and R A MASHELKAR

MYrATIoN

nth &vhn-Enckson acceleratton tensor spectic heat per untt mass body force term parallel to x-dwctron body force term parallel to ydtrectron acceleratron due to gravrty component of the acceleration due to gravrty m the x-duectlon g” conuugate metrrc tensor Gr, characterrstlc generahsed Grashof number defined by eqn (29) G~R generahsed Grashof number based on the radtus of the cyhnder defined by eqn (73) generahsed local Grashof number defined by eqn (38) thermal conductrvrty matenal constant charactenstrc length matertal constant exponent m shear stress power-law local Nusselt number defined by eqn (7Oa) average Nusselt number pressure Prandtl number defined by eqn (74) charactenstrc generahsed Prandtl number dehrted by eqn (28) generahsed local Prandtl number defined by eqn (39) exponent m normal stress power-law temperature temperature of the sohd surface temperature of the bulk of the Rued veloctty component along x-co-ordmate dlmenslonless veloctty component deCned by eqn (32) charactenstlc velocrty velocity component along y-co-ordmate drmenslonless velocity component defined by eqn (32) velocny vector drstance along the curved surface dunenstonless dtstance defined by eqn (32) distance normal to the curved surface dunensionless drstance defined by cqn (32) generahsed Werssenberg number defined by eqn (40) Wetssenberg number for a second order fltnd defined by eqn (65) Greek

symbols rat10

of the thermal boundarv laver thuzkness to the momentum boundary layer tinckitess expanston coefficient of the Butd defined by eqn (IS) momentum boundary layer thtckness dtmenstonless momentum boundary layer thtckness defined m eqn (36) thermal boundary layer thwzkness dunenstonless thermal boundary layer thickness defined m eqn (37) stmdanty vartable defined by eqn (47) sundartty vartabk defined by eqn (47) dtmensronloss temperahue ddbrence de6ned m eqn (32) matenal functton of fl vtscostty of tho second order Btnd materml function of R denstiy of the fluid at temperature T densuy of the flmd at temperature T,, normal stress m the x-due&on shearmgstressmthex-ydtrectton normal stress m the ydnectron

7” 0 fi

devtatonc stress tensor matenal function of Ii second utvarlant of Si\,

REFERENCES

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