On the two-component Benard problem a numerical solution

Physica 80A (1975} 76~88 ~(;;North~ttottand Publishtm¢ Co,

ON THE T W O - C O M P O N E N T BENARD P R O B L E M A NUMERICAL SOL|JTION LC. LEGROS and D. LONGREE Faeu#F des Sciem~es ~!ppliq,~&s, Service & chimie Physiq~w E . P . U~h~ersitF kiba-e & Brvxell¢,~', f050 lh~.~'seL';, th~&hm~'

af~d G. CtfAVEPEYER and J.K. PLAT"7[:N F)~euttd dcs 5kqences, Umvers;#F de F*:Tn.~o 7000 ~]'lon,v, Belgium

Received 22 December ~974

A nmnerical sotuiion {k~r the two-component Bimard p.~ob!em is pre~;e~,.:d. ~akh'}g i~to account the contribution of tbermai diffusion to the total density g~adicr~. The r~:~;~i~sare com~ pared with the approximate sottl~iOr~ obtained by the varia~ior~al tech,siqoe of ~b.elocai poter~t}a~ ir~troduced some years ago by G1ar~sdorffar~d Prigogme~ The res~}~ c~,a!c~Aa*.edb? ~he ~wo mc~.h.. ods are in agreement when the Sorer coef'~cien* is not {oo large. Bu~ when the gr~.die~:,~sbecome important, the exact numerical solution presented here shows a small divergeace from '&e varia~ tional method. The critical Rayleigh number is also compared wi~h the oqe extrapolated from ~}'~earm}y{icai ao!ution obtained for free boundary conditiot~,s.

1, Introduction in recent years, there have b e : n n u m e r o t i s rela~e;~, papers which indicated ~he infl~-.~m~ce of small mass-fraction gradients on the h y J r o d y n a m i c :;lability o f a Liquid layer healed f r o m below or f r o m above. Ti~e i m p o r t a n c e o f thermal diffusion, also called the Soret efl)?ct ~) in liquids, on the stabi!ky o i ' a system heated from below was suggested by Prigogine. It was first experime~tally reported by Legros, V a n H o o k a n d Thomaes2), and confirmed by thJrle fred Jakeman:'), a n d b2 Caldwell ~*). A mass-fraction distribution is established l]*c.m the h o m o g e n e o u s syst,-:m a n d the density is nov, a f u n c t i o n o f t e m p e r a t u r e (T) nd mass fraction (N~) =

(I

-

c,A T + 7AN,).

0)

76

) ~, .,J......., s: ,.,~, ~ .~-:'., ~

Pf~,t)BLF/4

,,

~r4tialiy d@~ idea f~ad {o k,e de~.eX:@cffk:ier~v becaue< t~e " "" -.'~ .......r;.+-,,,,.,, ~.~ gradien{ d,:a~ i5 *'~. tt;.ml-e~aturv, g.r.a. .d. i. e. .r ~ r ~'........ ~4t........ ~r,o, ~o: inh.ia~.c .~tm~e~t~c>~; .... carrer~b:, de~)ends, o n the e x t e a d o f the d>~zrmai dTiYu:4or~. B m tee c~a>ic;,J de[crminalio~.~ o~:"the c~idcal Ray{eigh m:{~.~ber b3 m e a s u r M g {{ae v a r [ a { i o s of Lea[ f]ox at [he o~.se~ o f corr,:cciion f3iis. Wi,h D+D po~.:ii:ive ~' 1he c o n v v c t h e mo~,cmea~{> are of oo..,o.m-~p<~e-m. B@~ard prob!em v, ith b~.~' t d'R:c~.. S.uhuck~{er.. i>rigogmc ,am! }b:t~mT:5) ,:cd; :;~.m,.)dd hi v, Mch {he amplific~:~{on fac:~of is rea{ aitho~,~gh {he f.,,robMn has '4 r~e}f'-adjein~ probiexn, A new. !brrnu~Mior~ ,aa5 pr(q>o,..,ed b, 1.oegros. P}atten a n d Poty4),. a n d gcgros, o , L,:>,a,:~c:, ~o .-,~,,~<,v i~-.,<,.. ~, ~..ad~o, ~,t probk: m v,a.~ ,r,o!~ed by mcarr, o l d i e k,~-~aDpoi.:r~tJal techr~i(iue o f (Jh~nsdorffand PrigogMe~). }t uric a~d Jakem;xa")qtudicd cxa.K:tiy {he >amc problem, aht~{>~.@a some .@~~pliiying ;!ssumptions >c~e m a d e }r, d:,6r anM}>;i> (a.}~ich ,here m,t m~.de m rd':< 4 ;:rod 6). T h e aim o ! fi'4:, paper i:, 1,a ~:i=vc a~ exact su.m~erica{ ,.,o~utio~ {,s thi> problem ~,~ithout any >Jmp{ifI:,ing a'~,~ump~ion t~;>,,ir~g the ,4rai?..h~i:orv~ard Runge Kutta pnvcedurc.. A o;>rnp:.wi>~ o f per,~.urbadon amplitude> {xpacc=,:!epem.!cn!~ a~d we check the overstab}lhy cc4'}diliort Of Pi;:;h::r; a~d (-ha_,vepe}cr'*L

2. F,armulation o f the pr~blem

,'~, ,,s,eg [ . . . . . . . .

~.,4>c, ~.,<(., I .t:~i eBe~:~:}, o f ~,qol.>}e~~{ttgl a,lK~

i~o.tiz:oula!., v~i{h a thick~eas J_ 7~qtia!iy ir~ mechanical equilibrim> bu~ subjec!ed m a gradiem of ~emperature v, hich induces a mass-frac{ior~ gradient. /:7'

?C', 7 -

(:t

?T

/ ~',j

+ t*~(,C -: .............. : - - . ¢, ~g ! ,,v.f

(2)

(3) i:t

c>,;j

{2c

70

.~x,.

d,;v]

"~:' D ' / D is positive by deSnition when ~.De denser componem migrates towards the cold plate,. here this effect is destabilizing.

7g

LC. LEGROS, D. LONGREE, G. CHAVEPEYER AND J.K. PLATTEN

~t

i):,:j

(4)

gxs

" gX; /

?&

with the incompressibility condition

For the study of binary liquids we keep the classical Bovssinesq approximation and we neglect the Du Four effect. Moreover, we shall restrict ourselves to dilute solutions (N, <{ t) bu,t: the generalization to concentrated solutions is straighttbrward6). We study small fluctuations around the steady state and we adopt for the perturbations the following expression: (6)

a = A(z) exp (ik~x + ik~,y) exp ( - a l ) ,

v.ith k = (k{ + a';~), the wavenumber. The amplitude of :he components of the velocity pertmbations are chosen to satisfy eq. (5). With ~q. (6) and after pressure climination the linearized and dimensionless perturba{ion equations are given by'~): (D'---k 2 + aSc-57'D)z (D:

+ 5~ [.N;(D z _., k e) + ,~D] O = (Sc/Pr) SfW, (7)

~- k 2) ( D 2 -- k 2 q- cr) IV = R a

(I) 2 - k2 + a p r ) O . . . .

W,

k20

-

Rthk2z,

(8)

(9)

where ("~)q D"

=

- - -

i'z"

and P r = ~,/K is the Prandtl numbe~', S c = v/D die Schmidt number, 5 # = ( D ' / D ) . ! I T ~he Sorer namber, Ra = (g~AT/Kr)d a the Rayleigh number, Ra~ = (gTN~/K,,)d a the Rayleigh number of thermal diffusion. O, W; Z are respectively the amplhudes of the perturbations of temperature, velocity and mass fraction. With the boundary conditions at z = 0 and z = 1 O ..... W = D W = O

(ll)

and D Z -~,,, 5/' X + 5CNi~D(9 = O,

,he differential homogeneous system of equations (7), (8), (9) defi,~¢s an eigenvMue problem. For given values of the parameters Ra, R,h, Sc, Pr, 5;: and k there exist

ON T H E TWO-COMPONENT BENARD PROBLEM

79

nontrivial solutions only for given values of <~. Con{rary to the cases o f pure !iq~Jids and o f free surfaces, this system is no{ self,-adjoin~, and the eigeni:uncfions have no symmetry properties.

3. N u m e r i c a l

solution

"We choose for the stationary sta~e a Iinear mass-fraction d i s t r i b u t i o n given in dimensionless form by

(J 31) This is a good approxim ,tion because D ' / D is smat~ (gone;ally ~3ot greater than 10-~). We use a numerical m e t h o d simi!ar to the o~e which permitted the exact solution o f the O o r - S o m m e r f e l d problem ~°). We transform the system (7), (8) and (9) o f three differential equations into a system of eight first-order diftk:rential equations. To do this we write 7. = u~ = ,'t + i~'2,

114}

D Z =: u2 = .% + it,,,,

(15)

W =: ua = v5 + iv(,,

(!6)

DW D2W

= u4 = v7 + iVs,

= us == v9 ~- h'~o,

(17) (18)

O = u7 = v~3 4- iv:,~,

(20)

D O = u,s = l'ts + h,;~,,

f21)

After this trans!k,.rmation, the system t7). (8), (9), may be writte~ as ~)lIows: Du~ = u:,~

(22)

Due = (lc2 -- ~ Sc) u~ + og~uz + (~'V;N~ 4. J~ Sc/P~) u~ 4. 5~N~o- Pr uv - 5V~us,

(23)

Du3 =: u,~,

(24)

Du4 = Us,

(25~

Dus = u~,,

(26)

J,(?. LE(iROS, D, L ) N G R E E , G° CHAVEPEYER AND J.K. PLATTEN

80

Dl*.6 =

-?I C t' U s

-- k2lg3

"

61l 5 -r' ~ f f k 2 l t 3

-

lc2 Re.iul -;-' k 2 Ra l'-~7~

D u v = us,

!)u~ ........... u.~ Tl~e

(27)

(128) + k2)~v .... (* Pr u - ,

(29"1

',actoi ' • " ,v

is generally complex (~ = % +. i%). As we study the ,~etm'al stabiiily curve whose e q u a t i o n is ~)~ = 0 w e replace <~in the above system by i%, W e c o n s l r u c t a general solution as a linear corn bination o f eight particular l i n e a r t y : h M e p e n d e n t solutions of' ibis e i g h l h - o r d e r problem. a r t , , t f,.l i l i c a i i o r i

+ c,,tl, ~, (:) + c,,,~ <'' (z) + c . : 4 " (:) +
(3o)

A~bi~rari! 3, we take 'a)~'!()) ..... h~.j

( j --:: I, 2 . . . . . 8).

(31)

:~s h'~iti;d values (at :7 := 0). ~vl~ere the superscripI j corresponds to the linearl$ .... ir~d
()~ ib,~' oiker L~a:md.eq. (12) becomes ~'+::(0)

.....

:i'.~

(0)

+

<"/:Ni~,'s t O )

= O.

OY

V. . . c~.7'{0) ,:. ...... -- : i ,va ':..,?J~ ~o) + ? : ~


g ,- t

(357

.j = t

a~vJ ih~mks to ecb (3t) c , -. ;:;;,% ~-- ~..~~(t

--

,'-/','~),,, +o cs. =

0.

(36)

ON THE TWO-COMPONENT BENARD PROBi.EM

81

W e thus convert the b o u n d a r y - v a l u e p r o b l e m to an initial-vMue p r o b l e m with a co~straint such that: - ,S

!

0

&' )(1)

,,-9" (I -- . ~ / 2 )

0

.

.

.

.

. I/~ ~( t )

•

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@>(

90/2,,,,

+ Su~2> (I)

,,,)

a-, ,'~u,~-" ~s,. (1)

,,,)

+ .>ua" (~> (I)

-'-, ->~-~s" ~) (t)

-,, 0,

(39)

where S = , 9 ( t -.l- (5<"2) to ensure ~i~at l:he b o u n d a r y cor.,ditio,,~s a~, ,: = I are -;atisfied. The sys{em o f diffi-,remiat eqtiatkms D~t, .... ./i(u~),

,:40}

i = I. 2 . . . . . 8.

must be s o N e d for five differenl initial conditions, namely d = l, 2, 5, 6, 8 in eq. (31). T o this end, we used the M e r s o n r o m i n e From the C o m p u t e c Cenlre o f l.he University o f Brussels~ which is a f o u r f l > o r d e r R u n g e - K u t t a m e t h o d , ar>i we adjus!ed the p a r a m e t e r s o f the p r o b l e m to sati:>J}/eq. {39).

4. R e s u l t s

The results are presented in tables I, tl and [II. T h e y are c o m p a r e d with those o b t a i n e d by the local-potential technique a) and with those o b l a i u e d for free-. surface conditions ~2). In this last case, it is possib!e to find an analytical soh..,~ion : Ra ~ = 657.5 -- R , h J ~

Pr + Sc

when

c~ = 0

(41)

P~ and Ra ':'~ .

657.5 (t + S c ) ( S o .+ Pr) . . . Sc ~

Pr R , ~ 4 F ..............

when


(4.2)

Pr 4- I

The numerical constant 657.5 is fhe critical value o f the Rayteigh n u m b e r for a pure c o m p o u n d with fl-ee-surPace conditions, To c o m p a r e with the exact results we replace this value by I708 which c o r r e s p o n d s to rigid boundaries. O n the o t h e r hand, the physical meaning o f R a ~'~ as a function o1: J ; is very difficult to understand, because b o t h R a and 47' are lhnctions o f .IT. Th~.t is why after {t-~e corn-

82

J.C. LEGROS, D. LONGREE, G. CHAVEPEYER AND J.K. PLATTEN

putation was done, we switched to a ~epresentation in which Ra ~-~is a function o f :::, the contribution o f the thermal diffusion to the density gradient relative to that resulting t'rom the temperature gradient. ,/

in

:2 =: 7 (1: :D) N~

R.y

(43)

This tram&:rmation can be done by a simple comparison o f triangles~2), The correspondir~g ~alues of,.:~ are given in each tables, In table I, the thermal-diffusion effect is des tab/gizing (D'/D > 0) and we know'*) that in this case the principle of" exchange of stability is valid. There is a good agreement between the numerical and the t ?ria~ional results; there exists in the Ra~-:Y plane a horizontal asymptote as eq. (4!'~ can be rewritten as l~.a~*~ --~

657.5

,

Ra ':~ --, 0

when

~' -, + c~.

(44)

I + ;/: [(Pr + Sc)/Prl

The extr~poia|ion from exact f'ree boundary solutions, replacing in the last formula 657.5 by 1708, give also results not too different fl'om the exact ones. Moreover {he numerical and the variational methods predict that 1,'~ tends to zero for large thermal-diffusion contributions. In table IL we set out results for a stabilized system (D'/D < 0) ~br which ;~;e impose ch := 0. in this case, free-surthce solutions predict that there exists a vertical asymptote ~@,ose equation is J ...... Pr/(Pr + Sc) as follows from eq. (44)° One can see in this table that the local-potemial technique provides large finite values of Ra ~* for d)~ < ---Pr/(Pr + Sc) but it is difficult to say from t!~ese results whether a vertical asymptote exists or not~ because the precision of the variational m e t h o d f'or large values of ~5;:is not known. On the contrary, the numerical solutions show Iha! there is no asymptote and tha~ fbr some values o t ' d ~ there corresponds two values of R a i l Results for D'/D < 0 and o-~ ¢ 0 are reported in table III. This case, the m o r e interesting one, is also the more difficult to solve. If the variatiom~t m e t h o d and Jhe free-boundary case give values o f o'~ corresponding to a value o f Ra% in the ntm~ericai method, it is a parameter which must be adjusted, and the determinant (39) is very sensitive to smatt variations o f this parameter. Relative variations ofa~ equal to 10 -6 change the value o f the determi~qant (39) of several orders o f magnitude when :5':' is large. Mo:eover in the vicinity o f the critical point, the variations of(39) are extremely rapid and it is impossible to use an iterative m e t h o d ot" ~he Newton--Raphson type. Results were, impossible to find ~or ~q' > t5 x 10-~'l; the method became numerically unstable with respect to t h step size used in tee numerical integration scheme. A smaller step size couM m~t be used due to the prol~ibitive CP time our C D C 6400 computer. In thi,~ ~ase, the flee b o u n d a r y

ON THE TWO-COMPONENT BENARD PROBLEM

83 t

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ON THE TWO-COMPONENT BENARD PROBLEM

R~

soluti ms predict also the existence o f a vertical a~ympto~e whose equation is .¢~ := - ( P r + l)/Pr~ indeed eq. (42) can be rev,' rltteh ~- as . ka ~ =

657.5 (i + Sc) (Sc + Pr) Sc 2 [t + 5¢~ Pr/(Pr + 1)l

(45)

~L~s means that there exists a regio: between ;~ 1 and .J~ . . . . . (Pr + i)/Pr where the total density is decreasing t)om the bottom to lhe top of the layer aiid thus Favours stability. But there exists a critical Rayleigh number at which the %'s~em becomes trustable. The situatio'~ shows clearly thal, a simple mech~mical / n t e p r e t a t i o n of the instability based on tb_e state o f the system, in other words that: a causal description is no longer possible. This conclusion was also verified when the system is heated from a b o v e ~ ) . These unexpected results were rmt confirmed by the local potential and were impossible to :study by the numerical m e t h o d because .g~ was too !arge. However, it must be emphasized that we have ~sed in the local potential very simple triat functions with only o~e variational parameter and this may be responsible of the disagreement between ihe resuhs from the Iocat potential and the free-surface extrapotatio~ al. / / < --3 × 10A m o r e realie~tic treatment with the local-potential technique is needed al~d catculations using Tchebiehev polynomial expansions and several variafiorta} p::~rameters is nov.,, in progress.

S. Perturbation amplitudes We develop Z = u,. !..I/= z*a and 6) = ur in accordance ~i,h eq, <3()} With t~le b o u n d a r y condnions sati'.~fied, it is possible to find Z(=), W{:} aad O(_-~ &-,., i~. is possible to find not only the eigenvaiues ,:~ {see previous section, s) b:r aiso the eigenvectors ot the problems defined by eqs. (7), (8) and (9). Resu!ts for ,5f =.: 10are plotted in figs. 1, 2 al~d 3 i=or Pr = 10, Sc =: i000 and R,~. ..... ~0000. If ~~: compare this result with t}-~- !fiat f'unctions we used in ~he variational approach, v,e see that W and O are qualitatively the same. but Z has a completel}, differer~ shape. We plot in fig. 4, the Z := 7~(z) that we used in ref. 4. This caa explain the discrepancy observed between the two methods whe~l the concen~,ration gradiem is large.

6. OverstaNlity conditions It "~sgenerally Found that, overstable motions prevail when/7:' is ~.~g,m~.~; ~ Steady motions are only possible for small ~,atdes of the Sorer number. This !est~It has been fiound first by extra.l.~o!ation ..... of the variational restfits ~*) a~d after that b~'

86

J. C', I J L G R O $ , {), LONGREtI:], G. C I I t A V f ! P E Y E R A N f ) J= K. P [ , A T T E N

Ltl

=

1t

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, ?

I

= I

13.4

0'I

1.1! !i

!

f f,2 ¢,,'~ IS; I I

i.l

Fig. I

Fig, 2

F;g. 3

Fig. I, An'~{'yti~de of the tempera{ure fluc~t~a{io~v; g-;' as a iu~;c~io_r~ .cd :: for -~/ :: !0 -~. Pr =- I 0 5c ..... 10(~), Rm : : ~OiX~0 i~ art'Atrary tmi~5;. Fig, 2. A.mt>{itude of' lhe vertical vek~ci'~y fluctuations W as a fi.mction o f : for .7 :: ~0 -:~, Pr = 10, Sc = t000, R ~ = I O N N in a r b i t r a r y units. Fig, 3, A m p l i t u d e o f the concemratim~ fluctuations X as a f~mction c f :." for : / = i 0 - a Pr = 10, Sc = {(XD, Rta :: } 0 ( ~ in a r b i t r a r y u n i ~ .

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1-~% ~ A m p I i t u d e o f the c o n c e n t r a t i o n i?uctuations Z as a f u n c t i o n o f z used in the ~a~'iatio::~a[ appre, act~ .7 =-= ]()-:~ Pr =-- i0, Sc = 1000, R;~, ~= 100(NI i~3 a r b i t r a r y kmi*,s

Plattm~ and Cbavepcyer'0. Using the toca] potential, they estabiished that ae~ overstaNe solut:on exists il" t(/'i > .52"*~

with

~9+* = K

P r ( P r + 1) Sc 2 R~,~

2/:* depends on the same physical parameters as for free boun "ary conditions for uhich it is possible to have an analytical expression~-~). The ~ aly difference is the

ON THE TWO*COMPONENT FJENA R,D P&OBLE M

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J.C. LEGROS, D, LONGREE~ Go CHAVEPEYER. AND J~.K. PLATTEN

~,alue o f the " c o n s l a n t " K, The ~,uttmrs o f ref, 9 {).~tmd I);,r rigid botmdarie~ K 554,4 inslead o~" 2*7=4/4, =;-

We calculated the critical value of the Raylelgh rmmt:~:;:rsass~Jming the e×cha~ge of ;(ability validity (Ra~,~) i):~r a set o f va}ties o f the pmameters Sc, Pr and li~, and for three difi~erent values o f ,yf, s!ight!y greater (Ban ,~""% We repeated the same calculations for the overstable motion and f\~tind other critical Rayteigh numbers (P,a;~}, By graphical extrapob,~iom ~.~e I}nd ti:e vah,,~e 5/'* o f J " such that Ra~2 := Rao,, . We do this for the fotar sels oF value,5 o f I:q, ;5c a~?{.i Ra, ~hicI? are presented in table tV. We found a larger dispersion o f the results than expected from the pre ::ision o f the method, perhaps indicating tha~ K is not realty a constant bt!l t[~at it varies very sh>wty with ,the par~tmcters. We i'~mm.t a meant valt~c ~! K :-:-- 555.0 + 3.0, . er

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Acknowledgments TI:c authors ~ish to thank Professors G t a n s d o r f f a n d Prigogi,~.e for ~heh~ imeres{ and stipulating comments during the course o f this work, They are also i~deMed ~_o Profi::ssor Thomaes .~b- making hhnself' availaMe to discuss tile work v.iti~ ~.hem. Two of us (I).L. and G+C.) are indebted to 1.R.S.I.A. (tnstimt pore" i'Er~co~.~ragemen( de la Recherche Scienlifique dans Ftndustrie e, t'Agric~.:t~.ureo Belgique} Ib~ a gra~(.

iReferenees 1) S,R.de Groot and P. Mazur, Non-EquiJibrium Thermodynamics (North-Hollat~d PubI. Comp., Amsterdam t962). 2} J.C, Legros~ A,van Hook ar,d G.Thomaes, Chem. Phys~ Letters 1 (!968) 69(~; 2 (t9(:~.;,~} 249: 4 (1970) 632. 3} D..T.J,l-lurle arm E,Jakeman, Phys. t:tMds 12 (~969) 276L 4) J,C.Legro~:, J. K.Pla~te~ and P, Poty, Phys, Fh~ids 15 (1972} I383. 5} R.S.Schecbter, I, P-igogine and J. Harem, Phys. Fttxids 15 (1972} 379. 6} J.C. Legros, P. Poty and G,Thomaes, Ph,,/sica 64 (t973) 48l, 7) P, Giansdorff a~,d I. Prigogine, Thermodynamic Theory of S~r~ct~.~re,S{abiiily a~d Vh~ctua~ (ions (Wiiey ]nlersciet~ce, New York, t97t). 8} D.T.J,Hurle and F,Jakemar~, J. Fluid Mech. 47 (197t) 667. 9) J.K. Ptatten and G,Chavepeyer, J, Fk ~d Mech, 60 (1973) 3~5. 10) D.P. Chock and R.S. Schechter, Phys. Fiuids I6 (1973) 329. | 1} D.R. Catdwell, J. phys. Chem, 77 (1973) 20t)4. !21 J.C. Legros, Bullo C!asse Sci. Acad. Roy. Belg. ~;9 (1973) 382, 13} J. K. P~atler~, private communication. t4) J.K. Pla,tten and G.Chavepeyer, Phys. Fluids 15 (1972) 1555. 15) J.K.P!atten, Bt'll, Ctasse Sci. Acad. Roy. Belg. 87 (1971) 6{,9. 16} R, S, Scnechter, M. Velarde and J. K. Platten, Adv. in Chem, Phys. 2:~ (i 974) 255,