Onset of convection in binary liquid mixtures: improved Galerkin approximations

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Onset of convection in binary liquid mixtures: improved Galerkin approximations

O. Lhost, S. Linz, H. Müller

To cite this version:

O. Lhost, S. Linz, H. Müller. Onset of convection in binary liquid mixtures: improved Galerkin ap- proximations. Journal de Physique II, EDP Sciences, 1991, 1 (3), pp.279-287. �10.1051/jp2:1991104�.

�jpa-00247518�

ClassificaUon

Physics

Abstracts

47 20 47 10 47 25

Onset of convection in binary liquid mixtures

_:

improved

Galerkin approximations

O Lhost

(~),

^{S J. Linz} _(~>

~)

and H. W. Mfiller

(2)

(~) Unlversitd de Mons-Halnaut 21, Avenue Malstnau, B 7000 Mons,

Be1glum

f)

Institut fur Theoretische

Physik,

Umversitfit des Saarlandes, D-6600

Saarbrucken,

F R G (3)

Department

of

Englneenng

Sciences and

Applied

MathemaUcs, Northwestem

Unlversity,

Evanston, IL 60201-3125, U-S A.

(Received

4

July

1990, accepted _m

final form

23 October

1990)

R4snmk. Deux moddles

approchds ~l'un

valable

lorsque

le nombre de Lewis L _est thffkrent de

zbro,

l'autre

lorsque

L

=

0)

sont

prksentks

_{en vue} de dkcnre les conditions

d'appantion

de la

convection dans _un

mdlange

binaire lorsque les bords honzontaux sont

ngides

et

imperrndables.

Les semis de stabihtk _{ainsi quo} la

frkquence

de

Hopf

sont dkterrnlnbs

analytiquement

Les deux moddles sont

comparbs

_avec les approximaUons

prdckdentes

et _avec la soluUon _exacte obtenue

nurnknquement

Abstract. Two approximate mode truncations

(one

for _nonzero and _one for _zero Lewis number) _are

presented

_m order to describe the _onset of convection of _a

binary

mixture between realistic _no

slip

impermeable honzontal boundanes Stabihty thresholds _as well _as the Hopf

frequency

_are determined analytically The outcomes are compared with earlier approximations

and the exact result obtained

numencally

1. lntToduction.

In the last few _years much effort has been spent

by expenmentahsts [1-5]

and theoreticians

[6- l3]

to understand linear and nonlinear

properties

of convection in a horizontal

layer

of

binary liquid

mixtures dnven

by

_a vertical

temperature gradient.

One

interesting aspect

m thJs field _is the

question,

how well the

boundary

conditions

(b.c.)

at the

plates enclosing

the

layer

_on

_top

and bottom have to be modelled to get

qualitative

_{or even} quantitative agreement with the

experiments. Th~s question

is

already important

^{for the} onset of convection

[6-8].

In contrast to

Rayleigh-Bbnard

convection

[14]

^for an

one-component fluid,

where the _use of idealized free

slip

b-c

only changes

the numencal value of the cntical

Rayleigh

number and the cntical

wave

number, things

_are different _m

binary

mixtures Here also the

boundaTy

conditions for the concentration field have _to be taken into account In

early

theoretical works

[9, 10]

idealized, permeable

bc. have been used which

prohJbit

_any convective concentration fluctuations at the

plates. Only recently

Galerkm

approximations [6]

have demonstrated that the

impermeability

b-c- _is important ^{for the} onset of convection _as well _as

weakly

nonlinear

convection. Later _on numencaJ

stability

calculations

[7,

8] assuming

impermeability

of the

plates

and realistic _no

slip boundary

conditions have

provided

^the exact treatment for _a lateral

280 JOURNAL DE

PHYSIQUE

II M 3

infinite _or

penodic layer

But for these b.c. the _exact

stability thresholds,

the cntical _wave

numbers,

and the

Hopf frequency

_{are no}

longer

accessible

analytically.

Since these

quantities

vary with the three fluid

properties

Prandtl

number,

Lewis

number,

and

separation

ratio, we

feel that

(a)

from _an

expenmentahst's point

^of view there _is need _to

provide simple approximative analytical

expressions for the

stability quantities

^{whJch take realistic} no

slip, impermeable

b.c.

~IQSI)

into account and differ

by

_a few

percent only

from the exact

results,

(b)

from the theoretical

point

^of view it is instructive _{to compare} the different kinds of b _c.

used _m the literature to give some more

insights

into their effects _on the

stability

properties.

These two _reasons motivate the

investigation presented

m this work

2. The system.

The

experimental

_{setup we} consider consists of _a

binary

fluid

layer

of

height

d and _mean concentration

C~

enclosed between two

horizontal, parallel, perfectly

heat

conducting

and

impervious plates

at z=

-d/2

and

z=+d/2 Perpendicular

to the

plates

acts the

gravitational

field _g

= -ge~. ^The upper

plate

is _on

temperature

_To the lower _{one on}

To + AT with AT

being

the

extemally applied

vertical

temperature .gradient.

If

lengths

_are nondimensionalized

by layer height d,

times

by d2/x (where

_x the thermal

diffusivity), temperatures T(x,

_z, t

) by

_xv

la gd~ (where

_v is the kinematic

viscosity

and _a the thermal _expansion

coefficient)

and concentrations

C(x,

_z,

t) by

_x

pip gd~ (where fl

is the solutal _expansion

coefficient),

the conductive

profiles

of temperature ^and concentration read

[11]

Tcond(Z) T0

" R _Z &~d

ccond(Z) Cm

"

R#Z (1)

Here _we have introduced the

Rayleigh

number R

= a

gd~ AT/x

_v

being

the non-dimensionah- zed temperature difference between the

plates

and the

separation

ratio

#

^{which is}

proportional

to the thermodiffusion ratio

k~ depending basically

_on the _mean temperature and _mean concentration m the mixture To facilitate the

handling

of the

impermeable

b.c. _we

replace

the concentration field C

by

(=C-#T. (2)

Up

to _a

prefactor

equation

(2)

_is the

potential

field of the _mass diffusion current, J~ =

L

V(.

The linearized

equations

^for the vertical

velocity

field _w, the deviation from the conductive

temperature profile,

= T

T~~~~(z),

and the

(-field

read _m Oberbeck-Boussm- esq

approximation

(3,

_tr

V~) ^V~w

= tr

3jj(1

₊

#)

₊

(1 (3a)

(3, V~)

= Rw

(3b)

(3,

L

V~) f

=

#

V~@

(3c)

Here _«

= VIx denotes the Prandtl number and L

=

DIM

the Lewis number where D is the concentration

diffusivity.

Since _m

(3)

the Dufour effect has been

ignored,

tills set of

equations

m

only

appropriate

[15]

to

study

the onset of _convection in

liquid

mixtures ltke 3He-~He

[I]

_or

ethanol/water [2-5].

3. The

hottndary

conditions.

In the literature _on convection _m

binary

mixtures

[6-13]

three

types

^{of horizontal b-c-} are

discussed :

(1)

^free

slip, permeable (FSP).

w ₌

3)

w _{= =}

(

= 0 at z ₌ ±

1/2 (4a)

(ii)

free

slip, impermeable (FSI)

_{: w}

=

3)

_w

= = La

~(

= 0 at _{z =} ±

1/2, (4b)

(iii )

_no

slip, impermeable (NSI)

_w

= 3~ w _{= =} La

~(

= 0 at _z

= ±

1/2. (4c)

As _an aside _we note that the

equations (3) together

with the

impermeable

b-c

(4b)

,_or

(4c)

are invariant with respect to a constant shJft of the

(-field, ((x, t)

-

((x, t)

₊ a. This

invanance is broken for

permeable

b _c

Although

NSI _are the most realistic b-c- with

respect

to the

expenments

the idealized FSI and _even FSP b.c. have led to _some important

insights

into

stability

of the conductive state

(FSP b.c.,

show

qualitatively

the

ngllt global stability

behaviour

[10]

:

stationary instability

for R

~ 0 and

#

~

0, oscillatory instability

for R

~ 0 and

enough

_negative

#,

FSI b-c- treated withJn _a Galerkm approximation

[6] qualitatively predict

the

nght

behaviour at- the

codimension two

(CT~ point (where stationary

^and

oscillatory instability meet),

like _wave number

splitting

and

frequency

_jump, and the

vanishJng

of the cntical _wave number for the onset of

stationary convection).

The _«

disadvantage

_» of

impermeable

b-c- _is that the exact

stability analysis

of the conductive _{state can no}

longer

be carried out

analytically.

_one has to

perform

numencal

analyses,

_{i e}

solving

_a

complex eight

order

polynominal

of the

underlying eigenvalue problem [8]

or using a

shooting

method

II 2, 16].

As _an alternative _we

suggest

here

simple

Galerkm

approximations [6]

^{for NSI b-c}

leading

to

stability

thresholds _m

good

agreement ^{with the}

numencally

determined exact ones.-

4. Mode

expansion

of the fields.

The first step ^to construct a Galerkm

approximation

_is to

expand

the

spatial dependences

of the basic fields in

appropnate

^mode systems ^which

sattsfy

the

boundary

conditions. For the _x

dependence

of _{w, @,} and ( a lateral Fourier expansion m terms of

e'~~ (where

k _is the _wave number and _n is _an

integer)

_is suitable _since the

system

is assumed to be

laterally penodic

_or

unbounded To model the _z

dependence

^for no

slip, impermeable boundary

conditions

(4c)

one needs different orthonormal systems

reflecting

the different

boundary

conditions for w, @, and

(

at

z=±1/2.

An

appropriate

orthonormal system ^{for the} no

slip

bc

w = 3~ w ₌ 0 at _z

= ±

j2

^{are the} Chandrasekhar functions

[17].

To

satisfy

=0 _at

z = ±

1/2

_{one can use} 2 _cos

farz, /

sin marz with

positive

odd

f

_{and even}

m For the

impermeability

cond1tlon L 3~

(

=

0 the system

(I, /

_sin

marz,

/

cos

farz

_{; with positive}

even

f

_and odd

m)

seems to be

adequate

_as

long

_as L _~ 0.

5. The three mode

approxbnation

for L _~ 0.

We approximate ^the

underlying hydrodynamic

fields

by

_using

exclusively

the basic modes of the above mentioned _{expansions :}

w(x,

_{z, t}

)

=

[@(t)

e~~~ _{+ c c}

Cj (z) (5a)

(x,

_z, t

=

[I(t) e'~~

_{+ c c}

/

cos arz

(5b)

(

(X,

_z,

t)

"

ii(t) e'~~

₊

c-c-

(5c)

282 JOURNAL DE

PHYSIQUE

II M 3

Here

cosh

(A

j

z)

_cos

(A

j

z) Cj(z)

=

(6)

cosh

(A j/2)

_cos

(A j/2)

denotes the first Chandrasekhar function with A

j =

4.73.

Inserting

these Galerkm truncations into equations

(3a-c)

and

projecting

these

equations

onto w, and

( respectively

leads to _a set of

ordinary

differdntial

equations

for the time evolution of the

amplitudes @(t), I(t) and

f(t) _Performing

standkrd linear

stability analysis

of the conductive state, @

=

I=

f

=

0, yields

the

following

results _:

(i)

stationary

instability

the conductive state becomes

linearly

unstable towards a'

monotonous

growth

of small initial

perturbations

with _wave number

k

_{at the reduced}

stationary stability

threshold

ml<(t)

=

4~U/#) ₍₇₎

+

ill

₊

(a/fl)(i/L)j

(ii) oscillatory instability

the conductive state becomes

linearly

unstable towards _an

oscillatory growth

of small initial

perturbations

with

frequency

_w~ and _wave number

k

_{at the reduced}

oscillatory stability

^threshold

row(()

=

4~~f/#)

^{~~ ~}

~~~l'~ ^~~~~~

^~

^~~~'~ ₍₈₎

(1

₊

fY)(I

₊

#) al provided

that the square of the

HOPI frequency

_{w~, given}

by

~ ~

j

~ .~

a(i

₊

fli)u&

+

pi)

T

WH(k)

=

4 fl

L +

#

,

(9) (1+1')(1

₊

#)

_a #

is pos1tlve.

Writing

down equations

(7)-(9)

_we have introduced the

following

reductions

k=k/ko; _r=R/Ro

where

ko

₌ ^{3.097 and}

Ro

=1728 38 _are the critical _wave number and cntical

Rayleigh

number for

#

= 0 Moreover _we have introduced

4~(I)

= 1<~ +

arjili/ii

₊

ar

iii] _(10a)

#(I)

=

k2(1 y)/(12 _{y) (lob)}

/(I)

=

#(I)j14

₂

<2

_{y + I}

jij(1

2 _{y +} I

4212j (toe)

and

I(k)

=

(P/@~) L/3,

Y

= 1.944 _{«, T}

=

I/(k(+ ar~)

=

0.0514 The constants y =

282,

=

5.437,

_a

=

0.758,

and

fl

= 1.479 _arise from

projections

of the field equations.

Let _us note that

setting

#

= 0

reproduces

the results of Niederlinder _et al.

[14]

for

Rayleigh-

Bbnard convection in _a pure fluid There the

approximate

cntical _wave number

ko

^{and the} critical

Rayleigh

number

Ro

_agree within 0 8 §b and 2

§b, respectively,

with the exact results

[17].

As in the earlier FSI

approximation _[6]

_our

stability

thresholds and the

Hopf frequency

_are

explicitely

_wave number

dependent.

To get ^{the critical reduced}

Rayleigh

numbers _one has to

minimize

r~,~,(k)

and

r~~~(k)

^with respect ^to

k

Let _{us now} discuss _our

approximative

^{results for} two

typical liquid

mixtures used _m recent expenments, ethanol-water with _«

=

10,

L

= 0.01 and

3He-4He

_with

« = 0

6,

L

= 0.03. We compare with the

corresponding

results of the FSI

approximation [6]

and the exact results obtained

numencally

_m the _{same way as m} reference

[8]

In all

figures

_we mark the exact values

by

full

lines,

the NSI

approximation by

dashed lines and the FSI

approximation by

dotted lines.

In

figure

I _we present ^{the cntical} stationary

Rayleigh

numbers r(~~, ^for

positive #

and the

corresponding

cntical _wave numbers

k[m, Figure

^la and b

represent

^ethanol water

mixtures, figure

lc and d the helium mixture. For both mixtures _one finds that the FSI

approximation

underestimates the onset of

stationary

convection

by

_a factor of

1/2,

whereas _our NSI three mode

approximation

deviates from the exact result less than 8 §b The

stationary

cntical _wave

number tends _m all three _cases to zero, but the _crossing point

#o

with the

#

_axis

significantly

differs. In all three _{cases one can} denve _an

analytical

_expression

~° f-L

where

f=

0.26 for the _exact solution

[8], f=

0.37 _{m our} NSI

approximation

^and

f

=1.62 in the FSI

approximation [6].

In contrast to the FSI approximation our NSI

approximation

shows the

right topological

behaviour of _k[~~~.

Cl 08

al

~~~~t

~ 8

nisi F51

O

oG

Q~ ~

off

o o

''

oo

bi di

08 08

~j ~j

~~oc ^,

,,~~

^,~~

oc~~

oo

"

'~

oo

ooo oo2 ooc ooo oos olo 015

4 4

Fig

I. Cntical stationary Rayleigh numbers r(~~ ^and critical stationary wave numbers

([m

_as function of the separaUon ratio # for ethanol-water mixtures a) and

b)

with _a

=

10, L

= 0 01 and for 3He-~He mixtures

c)

and d) with _a

= 0 6 and L

= 0 03 Full lines exact numencal solution for NSI b c, dashed lines _our NSI approximation, dotted lines FSI approximation

In

figure

2 _we present ^{the cntical}

oscillatory Rayleigh

numbers

r(~,

the

corresponding

critical _wave numbers k(~~ ^{and the}

Hopf frequencie3

_w

~ = w

~(k[~)

for the two

liquids (a-c

for

ethanol-water,

d-f for helium

mixtures).

In that _case the

oscillatory stability

threshold for the FSI

approximation

deviates about 20 9b from the exact

result,

whereas _our NSI approximation less than 3 §b We find

qualitative change

due to the NS b _c. for the critical _wave number _m

284 JOURNAL DE

PHYSIQUE

II M 3

5 3

~ 12

o)

3

°j

g

~ ', ll

2 ^°

o

bi ei

ioff ⁱoff

o o

o g

,

~- Off

<r~ 102 ',

~'~,-

102<+

~,_ --_

II _I-T

098 098

30

,

,,,~ ⁷⁵

f

20 ,,,

')"'

₅₀

_f

10

~~~

~,

'~~"'

25

0 00

-08 -06 -0C -02 00 -02 -01 00

4

4

Fig

2 Cntical

oscillatory Rayleigh

numbers r(~~, critical oscillatory wave numbers

([~,

and

Hopf frequencies

_wH as function of the separation ratio #

a)-c)

for ethanol-water mixtures,

d)-o

for helium

mixtures _a, L and

meaning

of the line types as in

figure1

contrast to the FSI approximation where

k[~

_is

_# independent

there _is a

slight growth

of k[~~ ^{for NSI b-c} as in the _exact result AJso here the accuracy is of the order of 2-3 §b The

Hopf frequency

_{is worse}

approximated

as one can see e.g. for ethanol-water mixtures in

figure

2c _wH differs for the FSI

approximation by

³⁸

9b,

^but for the NSI

approximation by

18 §b.

Due to the weak

#-dependence

of the cntical _wave number

k[~

it _is reasonable to

approximate

_{r(~~ and w}

~

by evaluating (8), (9)

at the fixed _wave number

k

= Thls

simplifies

equations

(10) remarkably

_since

@~(k

₌

I)

=

/(k

= I

)

=

#(I

=

I

=

I,

and

I(k

=

I)

=

L/3

One finds

~j

_~~

⁽ⁱ

⁺

flL)(i

+

w)(i

₊

gild)

~°~ " " ~~~

(i

₊

a)(i

₊

j) «j

^'

~

~~j

_~~

~~ ^a(i

⁺

^flL)(&

⁺

^pi)

~ "~ ^~

~

~~

(i

₊

&)(i

₊

j) a#

These

expr(ions

_for

k

=

I _can not be

distinguished

from the

corresponding

dashed _{curves in}

figure

2 where

k

=

k[~~(#).

Since

a m the denominator of

(11)

and

(12)

_anses from the

projection procedure,

_one has to

expect

^that the _errors in _our

r~~~ and _{w~ increase} whenever

the absolute value of I

al (I

₊

Y) #

is of the order I. Thus the

larger

^the Prandtl number and the smaller the absolute value of the

separation

ratio

#,

the better

(11)

and

(12)

work.

The _{error increases} for

«-#-combinations

where the nominator _m

(I1), (12)

becomes small. In companson ^to

r~~~(k

₌ ^{I for FSI} b _c equation

(I I)

differs

by

the

prefactor

1.479 _m front of L and

by

_a

prefactor

1944 _m front of the Prandtl number _«. Furthermore the FSI-factor

81ar~

₌ ^0.811 m the nominator is

exchanged by

_a

=

0.758. In comparison to the calculation

[6]

^of

r~,~,(k

₌ ^{I with} FSI b _c

only

the

prefactor

_m front

off

in

(7)

is diminished

by

a factor

of 0 632. Thus for fixed _wave number

k

= I and not to

large

negative

#

FSI and NSI

approximation

_are

formally

connected _{via a}

simple scaling

of the

physical parameters

^{L and} «

But the

scaling

of L differs for the two types of instabilities

To _summarize The

assumption

^{of free}

slip

bc underestimates _m

general

the

stability

thresholds,

the critical _wave numbers and the

Hopf frequency, partly

_up to a factor

1/2.

We

emphasize

that all cntical

Rayleigh

numbers discussed here _are reduced

quantities

scaled

by

their

corresponding

values at

#

=

0 Thus the unscaled

Rayleigh

numbers for FSI and NSI b-c- differ

by

a factor of about 5.

The

simplicity

of _our mode truncation ignores the

boundary layer

behaviour of the

marginal

stable concentration field found _m the numencal solution for

is

-0.2. But from the accuracy of _our

approximation

_we infer that this detail _{seems to} have

only

_a minor influence

on the

stability

thresholds.

6. Three mode

approximation

for L

= 0.

The Lewis number of

typical liquid binary

mixtures is of the order 10-2 _ThJs suggests to discuss _{as a} further reasonable

approximation

the limit L

= 0. Since the stationary

insta§ihty

is quite sensitive to L

=

0

(one

finds r~,~, =

0 for

# ~0,

and r~m, = I for

#

₌

0,

and for

#

_< ⁰ there _{is a}

stationary instability

at r~,~~ = 0 when the

layer

_is heated from

above)

this approximation is useful for the

oscillatory instability only

_as

already

noticed _m reference

[7]

But when

setting

^L ₌ ⁰ one has to remember that the

impermeability

L

3~(

₌ 0 at the

plates

is

guaranteed independently

of the mode truncation of

(

Thus with

regard

to

equation (3c), 3,(

₌

#

V~@, _a

spatial (-field

_{expansion m} the _{same manner as} for the b-field should be

more

appropnate

^{than the} one used _m

chapter

4. Thus for L

= 0 also

(

= 0 at the

plates

has to hold

implying

that there _{is no} distinction between

permeability

b.c. and

impermeability

b _c.

Substituting (5c) by

( (x,

_z, t

)

=

[f(t)

_e~~~

+ c c.

/

cos arz

(13)

leads after

performing

standard

stability analysis

and setting again

k

= I to

rose =

~~~ (14)

For the

squared Hopf frequency

_one obtains

~u 2

~

~

#

where

~ l ₊

c# (15)

a = c = I

pj(1

₊

&) (16a)

b

=

'P/i(I

₊

_{@) T~i} ('6b)

and _Y

= 1.944 _{«, p}

=

vi (2 /),

_{and I}

_IT

= 19A61. In

particular

for _«

= 10 _we find

a =

0 946 and b

= 399 ThJs _{is m}

good

agreement ^{with the} exact numencal result

[16]

for

286 JOURNAL DE

PHYSIQUE

II M 3

L

=

0,

_«

= 10 and not too

large negative #

which _can be fitted to _a

=

0.937,

b

= 373 and

c = 0.968. Here the _error of _our

approximative expression

for _{r~~~ is} about

(10~

^~

# ) fb,

the accuracy of _{w~ is} about 3.3 §b.

Semiquantitative

formulas

[18]

for r~~ and

ml,

_used

_by

_the

expenmentalists [4, 5],

can also be

brought

into the form of

(14)

and

(15)

For

« =

10

they yield

_a

= c ₌ 0.91 and b

= 407

being

_a little bit _worse than _our result.

Setting Lk0

m our

L#0 approximation, equations (11)

and

(12),

leads to a= 0.94 and b

= 273 In

particular

the

slope

of

ml

differs

significantly (by

about 26

9b)

^{from the} exact

result,

which reflects the fact that the mode truncation

leading

to

(11)

and

(12)

is _not appropnate in the limit L

= 0.

Let _us

finally

note that the L

= 0 result coincides within the

pencil's

width with the

numerically

determined exact results for L

= 0.01

(water ethanol)

shown in

figures

2a and 2c

Obviously

for -small Lewis number

(eg.

water ethanol

mixture)

_one has to

prefer

the L

= 0 formulas

(14), (15)

rather than the L, ⁰

approximation

of

Chapter

5. However if _one

is interested _m the critical wavenumber

dependence

_or the

stationary instability

_one has to use

(7)-(9).

Why

_a L

=

0

approximation

works _very well also for small _nonzero Lewis numbers _seems to be caused

by

the basic mechanism

producing

the

oscillatory instability

: Stern

[19]

has

argued

that

opposite

conductive

temperature

^and concentration

gradients

and _a fast relaxation of temperature fluctuations _{m companson} to concentration

fluctuations,

_i-e

x »D

bi~'L I _{I, %n}

_{lead to}

an

oscillatory

_nsing and

falling

of small fluid volumes for

appropriate

r and

#.

This _{scenano seems} to be

quite

insensitive to the limit D -0 _or L

- 0.

7.

Sunnnary

and discussion.

We have given a

simple approximate analytical

treatment of the

stability

properties of the conductive state _m

binary

fluid mixtures m the

Rayleigh-Bknard _geometry

with

regard

to the realistic _no

slip impermeable _~IQSI) boundary

conditions at the

plates.

The cntical

stationary

and

oscillatory

instabilities of _our L # 0 Galerkin approximation agree well with the exact numencal _{ones as} demonstrated for water-ethanol and

3He-~He

_mixtures. The cntical _wave numbers and the

Hopf frequency

show the

nght qualitative

behaviour and approximate much better the exact numerical results than _an earlier free

slip, impermeable (FSI) approximation.

We have not discussed the immediate vicinity of the CT

point

_since there the Oberbeck-

Boussinesq

approximation fails

[20].

As _an altemative to the NSI

approximation

for L ~ 0 _we have also

presented

_a NSI mode

approximation

for L

= 0 whJch _{is a} reasonable way to estimate the behaviour _on the

oscillatory

branch for low Lewis number

liquids

To

improve

further the

approximative stability

_{properties one} has to extend the mode

expansion

for the

z-dependence

of the

hydrodynamic

fields To find out which mode _seems to be'the next

important

one we remember that the z-component ^of the

velocity

field

w and the

temperature field each fulfill _two

boundary

conditions at each

plate

_w

= 3~w =

0 and

= 3)@ =0 at z=

±1/2 _(3)@

=0 follows

directly

from

(3b) by

_{using w=} =0 _at

z ₌ ±

1/2).

However the

(

field fulfills

only-one boundary

condition

3~(

= 0. A second _one

can

easily

be calculated from

(3c)

and _3)@

=

0 _{at z}

=

±1/2.

One finds that

3)t

_{has to be}

proportional

to

tat

the

plates.

To fulfill this condition _one should extend the

(

^field truncation m

(5c) by

an additional _term ~cos2 _wz

Carrying

_out the

calculation

of the

stability

properties [21] (which

_we do _not want to present ^here since the additional mode blows _up

considerably

the

analytical formulas)

shows that there _is indead _no

appreciable

improvement of the

stability

thresholds but the critical _wave numbers and the

Hopf frequency

_are better

approximated Roughly speaking,

there the differences between exact numencal _{results and} three mode

approximation

_are halved _m the extended _version.

Based _{on our} approximate truncation for the critical modes _a Galerkin model similar to the

one given m

[6, 11]

can be constructed. For details _we refer to reference

[21]

Acknowledglnents.

One of _us

(OL) acknowledges

F-N-R-S-

(Fonds

National de la Recherche

Scientifique, Bruxelles)

for _a grant. The authors thank Professor Dr. M Lucke and J. Niederlinder for

helpful

comments, and W Hort for

providing

_us the data used _m

Chapter

6 to calculate the constants a, b and _c of the full numerical solution.

Special

thanks to Mrs G

Krug

for

typing

the

manuscript.

References

ill

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Phys

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1373,

SULLIVjItT

S and AHLERS G,

Phys

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SURKO C M., PASSNER A and WALDEN R W

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Rev, Lett, 56

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2621,

SURKO S M and KOLODNER P,

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Phys

Rev A 38

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3147,

Phys

Rev A 40 (1989) 4552,

Phys

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KOLODNER P

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Rev A 35

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Phys

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2548,

KNOBLOCH E,

Phys

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Phys.

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AHLERS G and LUCKE M,

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Phys.

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Europhys

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Phys

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JOURNAL DE PHYSIQUE U -T I,M3, MARS lwl j4