Nonlinear theory of traveling wave convection in binary mixtures

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Nonlinear theory of traveling wave convection in binary

mixtures

D. Bensimon, A. Pumir, B.I. Shraiman

To cite this version:

Nonlinear

_theory

of

_traveling

wave

convection

in

_binary

mixtures

D. Bensimon

_(1),

A. Pumir

_(1,3)

and B. I. Shraiman

₍₂₎

(1)

LPS,

Ecole Normale

_Supérieure,

24 rue _Lhomond, Paris

_75005,

France

(2) AT

& T Bell Laboratories, 600 Mountain Ave.,

Murray

Hill, NJ07974, U.S.A.

(3)

SPhT, CEN

_Saclay,

F-91191 Gif-sur-Yvette, France

(Reçu

le 27

_février

1989, révisé le 25 mai _1989,

_accepté

le 15

_juin

₁₉₈₉₎

Résumé. 2014 Les ondes

propagatives

dans les fluides binaires en convection sont _{étudiées par} un

développement

perturbatif

en

_puissance

du _rapport de

_séparation

_{(| 03C8 |~}

_{1 )}

à

_partir

de l’état convectif d’un _{fluide pur}

₍

_|03C8|

_{= 0).}

Pour des conditions aux limites libres-libres et

_perméables

la transition de l’état conductif vers l’état de convection

_propagative

est

_{supercritique.}

Pour les conditions aux limites

_plus

réalistes

_{expérimentalement}

(rigide-rigide-imperméable)

avec

03C8 ~ 2014

L2,

on obtient la courbe

_{sous-critique}

décrivant la transition de l’état conductif vers l’état de convection

_propagative

et on évalue les

_profils

de concentration. La transition secondaire de l’état de convection

_propagative

vers la convection stationnaire est

_surcritique

et la valeur du nombre de

_Rayleigh

à

_laquelle

elle

_apparaît

est calculée

_{analytiquement}

et

_{numériquement.}

Ces résultats sont

_comparés

avec les observations

_{expérimentales}

et d’autres

_expériences

sont

suggérées.

Abstract. 2014

Traveling

wave

(TW)

convection in

_binary

mixtures is studied

_by

an

_expansion

in the

séparation

ratio

_{(|03C8| ~ 1)}

_{around the pure fluid convective} state

_{(03C8 = 0 ).}

For

free-free-permeable boundary

conditions the bifurcation from the

_conducting

state to TW convection is critical. For the

_{experimentally}

relevant

_{rigid-rigid-impervious}

_boundary

conditions with

03C8 ~ 2014

_{L 2,}

the sub-critical bifurcation curve from the

_conducting

state to TW convection is obtained and the concentration

_profiles

are evaluated. The transition from

_traveling

waves to

steady

convection is

_predicted

to be critical and the value of the

_Rayleigh

number at which it

happens

is calculated

_analytically

and

_numerically.

_Comparison

with available data and

suggestion

for further

_experiments

are discussed. Classification

Physics

Abstracts

45.25Q

Introduction.

Thermal convection in a

_binary

fluid mixture is an

interesting experimental

[1-3]

and theoretical

_[4-8]

model

_system,

for which the destabilization of the

_{quiescent (conducting)}

steady

state occurs at finite

_{frequency through}

a

_Hopf

bifurcation

_[4].

This leads. to a rich

dynamical

behavior near onset

_[1-3].

The

_conducting

state is observed to

_undergo

a transition to a

_steady

state of uniform

_travelling

convective rools

_{[2a, la].}

Then upon

increasing

the

temperature

difference across the

_cell,

this state of

_traveling

waves

_undergoes

a

_hysteretic

transition to uniform

_{steady overturning}

convection

_{[2a, la].}

The

_spatial

structure can be

quite complex involving

the coexistence of left and

_{right traveling}

waves

_(TW),

fast and slow TW and TW and

_conducting

state

_[1-3].

The linear

_instability

of the

_binary

mixture

_conducting

state is

_by

now well understood

_[6].

However the

_present

non-linear

_analysis

_[5,

7,

8]

is

_only

valid near the so called

codimension-two

_point

_{[5, 8],}

where

_weakly

non-linear studies

_[9]

are

_justified.

As most

_experiments

are

done very far from this

point,

the

_agreement

between the

_theory

and the observations is at

best

_qualitative.

_{The purpose of this work is} to extend the theoretical

_analysis,

to describe

finite

_amplitude

convection and to

_provide

some

_quantitative

results to _{compare with the}

experimental

data. In the

_following

we are

going

to review

_briefly

the mechanisms

_reponsible

for convection in pure fluids and in mixtures. We will introduce the basic

equations describing

convection in mixtures. We will then solve them for

_{free-free-permeable}

and

rigid-rigid-impermeable boundary

conditions. We will solve

_analytically

when the

_traveling

wave

velocity

vo is much

larger

than the convective

velocity

u (vo > 1 u

and when

vo « 1 u|

1

and

numerically

between those limits.

Convection _{in pure fluids results from the}

_competition

between the

_stabilizing

effect of heat diffusion and

_viscosity

_{[10] (on}

time _{scales of Td} =

d2/ K

and _{T v} =

d2/ v

respectively)

and the destabilization

_by

the

_buoyancy

forces

_resulting

from the unstable thermal stratification on a

time scale Tb =

(

a8dTg ) 1/2,

, where g

is the

_{gravitational}

acceleration,

a the coefficient of

a&Tg )

thermal

_expansion,

K the heat

diffusivity, v

the kinematic

viscosity,

8 T the

_imposed

temperature

difference across the cell and d its

_height.

When Tv Td :>

Tb,

temperature

and

vorticity

fluctuations do not

_damp

fast

_enough

and convection sets in. In terms of the dimensionless

_Rayleigh

number :

the convection

_instability

occurs when R exceeds some critical value

_Rc,

the

_precise

value of

which

_depends

on the thermal and

_{hydrodynamic boundary}

conditions

[10].

In

_binary

mixtures,

a second

_stabilizing

_{mechanism may exist due} to the

_coupling

between

temperature

and concentration fluctuations

_[4a].

This

_coupling

is

_{parametrized by}

the

separation

ratio t/J = -

(a’1a ) ST co(l - co),

where cl ’ is the coefficient of solutal

_expansion,

ST

the Soret coefficient and co the average concentration.

When Vi

0,

the

_coupling

is such that the

_lighter

fluid

_migrates

to the colder

_region,

thus

_establishing

a

_stabilizing

concentration

stratification,

which opposes the

destabilizing

thermal stratification. A linear

_{stability analysis}

of the

_conducting

state

_[6]

_correctly

_predicts

the existence of a

_Hopf

bifurcation at the critical value :

characterised

_by

a

_{frequency w ~}

_{(-03C8 / (1}

_{+ 03C8 )}

)1/2

and hence

_{corresponding}

to convective rolls

_traveling

with a

_velocity

_{Vo =}

_{103C8 I}

11/2.

The linear

_theory

_[6]

also

_{correctly predicts}

the

dependence

of the rate of

_growth

and the

_frequency

of the unstable mode as a function of its

wavenumber

_[2b].

However the

_experiments

show that the unstable mode fails to saturate at

small

_amplitude

even

_arbitrarily

close to onset

_[1-3].

Thus the bifurcation is backward and the transition

_hysteretic

_[1-3].

This can be understood if one observes that in the

experiments

the

Lewis number L =

D 1 K

is very small

(L =

10- 2)

and _{as a} result in the presence of a

is obliterated and the

_conducting

state with its linear

_profile

of concentration ceases to be a

good approximation.

This limits the

_validity

of the linear and non-linear

_analysis

done around

the

_conducting

state, to a

_regime

where convective motions are not

_destroying

the concentration stratification. The destruction of this concentration stratification

_happens

close

to the codimension-two

_point

_[5]

where the

_mixing

time due to

_convection,

is of the order of the molecular diffusion time _Tmol =

d21 D,

needed to establish a concentration

_gradient

(D

is the molecular diffusion

_constant).

This

corresponds

to

_separation

ratios :

Since L =

_{10- 2,}

such a

_{regime requires}

_Il

= - 10- 4,

_{which is very difficult} to achieve

experimentally.

Most

_experiments

_[1-3]

are done for values

of t/1

« -

L 2.

The concentration

gradient

is therefore

_{effectively destroyed by}

the convection flow and in order to

_explain

the existence of the observed

_traveling

waves one must understand the influence of the Soret effect in the

_{boundary layers,}

where a concentration stratification may still

_persist.

The purpose of this paper is to

_develop

a

_{theory correctly accounting}

for the effect of

convective

_mixing

of concentration and to

_present

results relevant to the

_experiments

done at

values

_{of Ji}

« -

L 2.

In

_particular,

we would like to _{understand the very}

_rapid

decrease of the TW

_period

_(by

almost 2 orders of

_magnitude)

_along

the

_hysteretic

branch when the control

parameter

changes

[2d]

by only

10 %. We would like to know the form of the concentration

profiles

in the TW

_regime

and understand the observed transition from TW to

_steady

overturning

convection

_{[2a, la].}

The

_analysis

we are

_going

to

_present

is done in an infinite

medium,

but its results are valid for

_{experiments performed}

with

_{periodic boundary}

conditions,

i.e. in an annulus

_[2d].

The basic idea behind our method is to treat the

_separation

ratio Il

as a small

parameter,

and

analyse

the convection in

_binary

mixtures

_by

an

_expansion

around the pure fluid convective state and not, as is

_usually

done

_[4-8],

_{by expanding}

around the

_binary

mixture

_conducting

state. Thus the

_temperature

and

_velocity

field have the same form as in a _pure

_fluid,

_though

maybe

not the same

_amplitude.

Moreover we shall take into account the fact that the Lewis number L satisfies : L

B/201303C8,

, and that the Péclet number :

p =

1 u 1 IL > 1.

More

precisely,

we introduce two small

_parameters,

B/2013 03C8

and

_L/

B/2013 03C8,

thus

_taking

into account

the ratios between the control

_parameters

most

_commonly

used in

_experiments.

It

_happens

that the

_{limit L}

B/201303C8 ~

_{0 is very}

_singular

and must be dealt with

_carefully.

On the

_contrary,

the limit

_{B/2013 03C8 ~}

0 is well

behaved,

and it is

_possible

to do a

_{systematic expansion}

of the

equations

in powers of

B/- 03C8.

We will elaborate these ideas in turn. The

_equation

for the concentration field can be solved

_analytically

in two limits :

_{vo > 1 u}

₁

_by

an

_expansion

in

1 u 1 lvo «

1,

and

_{vo «}

_{1 u 1}

_by

a

boundary layer analysis.

We have determined

numerically

the

concentration field

_by

a Galerkin method in between those values. The

_velocity

of the

_TW,

vo, and their

_amplitude

is obtained as

_solvability

conditions on our

_equations.

Below we will first describe the basic

_equations,

which will be solved for the case of

free-free-permeable boundary

conditions where many

analytical

results can be obtained. In

particular,

one can show that the

_Hopf

bifurcation in this case is forward. We will then deal

with the

_{experimentally}

relevant

_{rigid-rigid-impermeable boundary}

conditions. Close to onset

(vo > 1 DI),

one can show that the

_Hopf

bifurcation is inverted. One can also

study

analytically

the transition from TW to

_{steady overturning}

convection

_{(vo > 1 u 1)}

and find the critical value Pc =

_{1 uc 1 /L at}

which the transition

_happens.

In between those

values,

our

equations

are solved

_numerically

and checked in the relevant limits

against

the

analytical

Basic

equations.

In the limit of infinite Prandtl

number,

which

_maybe

the relevant limit for

_binary

fluids like water-ethanol

_{[1, 2]}

but not

He3 - He4 mixtures

_[3d],

the

_equations

_describing

convection are

[4] :

a)

the Navier-Stokes

_equations

in the

_{Boussinesq approximation :}

where w is the vertical

_velocity

_component,

T the

temperature,

y the

concentration,

R the

_Rayleigh

number

and If

the

_separation

ratio. In this paper the unit of

length

is

d,

of time

_d2/K,

of

_temperature

6 T and of concentration

_{8 T ST co(l - co ).}

b)

the heat and concentration diffusion

_{equations :}

where

_Dt

is a convective derivative :

Dt

=

_at

₊ u . V. Notice that

_{equation (lc)}

can be

rewritten as

with

_{y = y’+ L TI (1 - L ).}

Since L 1 and T is of

₀₍₁₎

and

_nonsingular,

we will

approximate

y = y’.

Let us introduce the

temperature

fluctuation 0 defined

_{by :}

T = - z + 0 and stream

_{function 0 :}

u =

(-

a,o,

0,

ax). Equations (la,

b,

d)

become :

We also need to

_specify

the

_boundary

conditions on the fields

_{0, 0}

and y. We will consider

a) free-free-permeable

boundary

conditions :

b) rigid-rigid-impermeable

boundary

conditions on z =

0,1 :

We look for

_traveling

wave solutions :

contrast with

_previous

theoretical

_analyses

of this

_problem

which assumed

_[4-8]

yo = _{z, we} do not

_expand

the concentration field y around the linear concentration field

existing

in the

conducting

state. We are thus able to

_correctly

describe the concentration

_{boundary layers}

existing

when convective

_mixing

is efficient. However for the

_expansion

_{around the pure} fluid

convective state

_(Eq.(3))

to be

valid, 03B5

must be small

_{( s « 1 ).}

Since from the linear

_stability

analysis

and from

_experimental

data we know that in the domain of existence of

TW,

B =

0(B/2013 03C8),

we

require 1 03C81

1. This allows us to treat the

_coupling

between the concentration and

_velocity

fields

_(the

term

_R03C8raxy

in

_{Eq. (2a))}

_{perturbatively.}

Let us define

Inserting

the ansatz

_equation

₍₃₎

into

_{equations (2a,}

_b,

_c)

_yields

to order

_{0(03B5) :}

with the concentration diffusion

_{equation :}

where po =

vo/L.

This form is

appropriate

close to onset where it can be solved

by

an

expansion

in powers

_{of a - 1}

|u|

1. Far from onset,

_{(a - 11}

_{|u1| >}

1 )

we shall use

_boundary

layer

techniques

to solve

_{equation (5c)}

rewritten as :

where ul =

Aûl,

A is the

amplitude

of the convection field

(see below),

_p =

EAIL

is the

Péclet number and

_{g - 1 =}

«A -1=

_{vo/ EA}

₍

_{1 )}

is the ratio between the TW

_velocity

and the

convective

_{velocity amplitude.}

It

_{is important}

to notice that in

_{equations (5c, d)}

both

Po

1 and

_{p -1}

are small,

of order

_{LI -.p.}

However the limit

_Po

1,

P 1-+

0 is

_singular

and

must be treated

_carefully.

To order

_{0 (e2) :}

To order

_{0 (e3):}

The zero modes of the

_adjoint

are

_{(01, - 01)}

and

_{(a"ol, - a"ol).}

At

_{0 (E2),}

the

_solvability

condition is

_{(1) :}

where the

_{brackets (... )}

define the inner

_{product :}

_{(h, f )}

-

dx dz

_{h f .}

Due to

_symmetry

under reflection

_{(x -+ - x)}

the last term on the left hand side is zéro.

_Hence,

This

_solvability

condition determines the TW

_velocity.

At

_0(e3)

the

_solvability

condition is

(1) :

Which

_yields

the

_equation

for the

_amplitude

_{of the TW. We will argue later that when} convective

_mixing

is efficient

_(far

from

_onset),

concentration

_gradients

exist

_only

in

_boundary

layers

and therefore from

_{equation (8)}

the TW

_velocity

vo is small

_{(vo «.c 1 u}

_).

In this case the

right

hand side of

_equation

₍₉₎

is also small and its left hand side

_yields

the same

_amplitude

equation

as for a _{pure fluid. We} will thus conclude that away from the onset of convection in

binary

mixtures,

one

_expects

slow

_traveling

waves with an

_amplitude

_equal

to the

_amplitude

of

stationary

convection in a _{pure fluid} at the same

_Rayleigh

number.

In the

_following

we are

_going

to solve

_equations

_(5-9)

for two cases of interest :

free-free-permeable

and

_{rigid-rigid-impermeable boundary}

conditions.

Free-free-permeable

boundary

conditions.

This case is

_interesting

not because of its

_experimental

_relevance,

but because it is tractable

analytically

and instructive. The solution of

_equations

_{(5a, b)}

with

b.c ;

(2d)

is

_{[10] :}

with

_{corresponding R, =}

_{(k2 +}

_{lr2 )’Ik2}

and

_choosing

k =

kc =

ir

/

/2.

This choice of

k is

_{suggested by}

our

_expectation

that the

_velocity

field is close to the

_velocity

field for

Rayleigh-Bénard

convection near onset. The

_equation

_{for the concentration yo,}

_equation

_(5c)

then becomes :

with g =- a -’A

= EA

/vo =e u1

/vo

and

_{Û1 ==}

A-1

_U1 =

(-

_ir sin kx _{cos 7TZ,}

0, k

_cos kx sin _7rz

).

Let us look for a solution :

(1)

At each order

_{orthogonality}

to the second mode of the

_adjoint

induces a small

_{O (L2/1/1 )}

_phase

shift between the

_velocity

and concentration fields. This can be taken into account, but

complicates

the

satisfying

the

_boundary

condition

_{1’00 1 0}

=

0,

yoo 1 1 = 1 ;

l’on

_10,1

=

0,

for _n

= 1, 2,

... One

obtains a set of

_equations

which can be solved to _{any order. It has} to be

_emphasized

that,

although

po 1

is

small,

one cannot

_neglect

_it,

because of the

_singular

nature of the

_{limit Po 1 -+}

0. One has

_{immediately :}

yoo = z. The solution can be written as :

with

mn * 1.

For

m = 0,

ymn

is

_{0 (p - 1)}

and thus

_negligible.

_{yR is}

_orthogonal

to

cp

1 and thus does not contribute to the

solvability

condition at next order

(Eq. (8)).

The

normalisation factor

_{f (g, po)}

can be evaluated

_analytically

to _{any desired order. To lowest}

order

in g

it is :

We have also solved

_equation

₍₁₁₎

_{numerically, by writing}

a Galerkin

_expansion

of

yo and

inverting

the

(truncated)

system

of linear

_equations.

Numerical results are

_presented

in

_figure

_1,

_{and agree very well with}

_{equation (14)}

at _{small values of g. The}

_solvability

condition

_determining

the TW

_{velocity equation}

₍₈₎

_{yields :}

Fig.

1. - The function

f (g, po) in

the case of the

_{free-free-permeable}

conditions as a function of g for po = 50. Curves

corresponding

_to various values of p0 > 10 _are

we recover the result of the linear

_{stability analysis}

_[6].

As

_f

decreases with

_increasing

g

(see

Fig. 1),

the TW

_velocity

_vo

_{decreases away} from onset. Notice however

that,

since g -

eAlv

o, to

actually

calculate vo one has to solve for the

amplitude

of convection

A.

In order to determine the

_amplitude

_equation

for

_A,

we need to find the

_velocity,

temperature

and concentration fields at order

_O(e2),

_equation

_{(6a, b,c).}

These are

(see

appendix

A) :

with . The

expressions

for OR, OR

and

y e)

are

_given

in

_{appendix A.}

At this order the fields can thus be written as :

Notice the

_{phase lag by}

_{vol (k2 +}

_{11’ 2)}

between the

temperature

and

_velocity

fields

_[7].

Close

to onset

_{(g 1 1 ),}

this

_phase

_lag,

due to the

_coupling

between the

_temperature

and concentration

fields,

is

_responsible

for the

_instability

to TW of the

_binary

mixture

_conducting

state

_[7].

The second

_solvability

condition,

equation

(9),

yields

the static

_amplitude

_{equation :}

with

_bi

=

kB2/8 (k 2+ 7r 2)

and

i (g, po) -

f (g, po ) (see Appendix A). Equations (15)

and

(18)

form a set of two

_coupled

non-linear

_equations

in two unknowns _vo and A. In

_{general they}

must be solved

_numerically,

however in two

_{limits g «}

1

_{and g >}

1,

one _may

_actually

obtain

analytical

results. Thus away from onset

_{(g > 1 ),}

i.e. for slow

_travelling

waves,

_{vo «}

_{[ u 1}

and

(as

one verifies

_{numerically) f (g,po)}

1,

the

_amplitude

of convection eA in a

_binary

mixture is identical to the one in a _{pure fluid} at the same

_Rayleigh

number

_{[11] :}

A =

/8 (k 2+

11’2)lk2

= J24.

However close to onset

(g « 1 )

_as

f (g, po) -

1,

this is _no

longer

true.

_{Combining equations}

_{(15, 18)}

one obtains :

And from the

_expression

for

_{f(g,po), equation (14),}

near onset one derives :

with

_fi

=

(ir 2

₊

k2)/8

and

f2

=

(k2

+

11’2)(k2

+ 9

_{ff2 )/64.}

The bifurcation from the

_conducting

_{state (g =}

₀₎

to a convective state of

_traveling

wave

subcritical if al

b1 - f,

0. For

_{free-free-permeable boundary}

conditions,

al

bl - f 1=

0,

the

bifurcation is

_{supercritical}

with the

_amplitude

of convection eA

_increasing

as the

_quartic

root

of the deviation from onset

_{[12, 8c],}

_{e2 + 03C8 :}

This result

_implies

that for

_{free-free-permeable}

_boundary

_conditions,

the usual

_perturbation

expansion

around the conduction state

_[6-8]

is valid.

Rigid-rigid-impermeable

boundary

conditions.

This is the

_{experimentally}

relevant case. The solution of

_equations

_{(5a, b)}

with b.c.

_(2e)

is

[10] :

:

with £ =

z - 1/2,

Re

= 1707, k

=

3.117,

_qo =

3.973,

q = 5.195 ₊ i

2.126,

_a =

-1

₂

_[0.0615

_{+ i}

_{0.1038 ].}

The solution for the concentration

field,

equation

q

(5c),

(5c

)

proceeds along

P

Fig. 1.

Fig. 2.

Fig.

2. -

The reduced

amplitude

of convection eA as a function of the reduced temperature

e2/ 1 03C8

1

for L

= 10-2 and 4r

= - 1/4, in the _case of

rigid-impermeable boundary

conditions. The lower

branch

_corresponds

to fast

_{(and unstable)}

TW solutions, the upper one to slow and stable one.

Fig.

3. - The

phase velocity

of the

_travelling

waves

_vo/

/- 03C8

as a function of the reduced _temperature

e2/1 t/J

1for L

=10-2 and 03C8 = - 1/4,

in the case of

_{rigid-impermeable boundary}

conditions. The inset is

Fig.

4. -

Isoconcentration patterns for various

_(unstable)

solutions for L = 10-

2,

and 03C8 = 2013 1/4

(a)

e2/03C8!

[

= 0.700 and

_(b)

_{E 2/ 1 p}

_]

= 0.850. The difference between two isoconcentration lines is 6 x

10-2

for

_figure

4a, and 8 x

10- 2 for figure

4b.

the same lines as for the

_previous

case

_{equations (11-13),}

_although

the

_algebra

is more

cumbersome. Close to onset,

_{(g « 1 )}

one obtains

with

_f,

₌ 7.856. From

_equation

₍₂₂₎

one calculates the value of al in the

equation

for the TW

velocity, equation

(15) :

_{al = (axq,1)2) 1 (âx6l)2)}

= 40.344. From Manneville and

Piquem-al

_[11],

one also knows the value of the coefficient

_bl

in the

_{amplitude equation,}

equation

(18) : bi

= 0.0612. One thus verifies

that,

since al bl - f

0,

the bifurcation from

the

_conducting

state to the TW state is subcritical

_{(i.e. hysteretic).}

To make further progress,

one has to

_compute

_{f (g, po)}

_numerically.

Then

_{assuming equation}

₍₁₈₎

to hold to all g

(it

is known to hold

_{for g}

1

_{and g >}

_1),

one can solve

_{numerically equations}

₍₁₅₎

and

₍₁₈₎

(see

Appendix

B)

and determine the

_amplitude

of convection : eA

_{(Fig. 2)}

and the TW

velocity

vu

(Fig. 3)

as a function of the reduced

temperature £2

=

(R -

Rc)IRc

for various

values of the

_separation

ratio

03C8,

and Lewis number L. The concentration

_profile

yo(x - vo t, z)

is determined

simultaneously

(Figs. 4, 5).

Thus close to onset and on the

lower

_(unstable)

branch of the

_{hysteretic loop,}

the concentration

_profile

deviates

_{only slightly}

from the linear

_{gradient existing}

in the

_conducting

state

_{(Fig. 4)}

and the TW

_velocity

is fast

(0(B/03C8| )).

On the upper

(stable)

branch, however,

the concentration

_profile

exhibits clear

boundary

layers

(Fig. 5)

and the TW

_velocity

is slow

_{(vo «}

B/|03C8| ).

When g >

1,

the

_amplitude

of convection is

_equal

to the

_amplitude

of convection in a _pure

Fig.

5. - Isoconcentration

pattems for various

_(stable)

solutions for L

_{=10- 2,}

and 4, = - 1/4,

(a)

E 2/ 1 03C8|

1

= 0.700

(b) e 2/ |

1 03C8 1

= 0.850. The difference between _two isoconcentration lines is 3 _x

10-2

for

fluid at the same

_Rayleigh

_number,

and the TW

_velocity

can be evaluated

_analytically

from

equation

(15)

by

a

_{boundary layer}

calculation

[13]

for the concentration field which is

described next.

Slow

_traveling

waves and concentration

_{boundary layers.}

The characteristic scale for the

_velocity

of convective flow is set

_by

the thermal

_diffusivity.

Since the later is much

_larger

than the molecular

_diffusivity

_{(L ~ D/K 1 )}

the Péclet

number,

p

₌

_{1 u IlL (2),}

based on the convection

_velocity

is

_large

and the

_mixing

due to the flow has a much

_stronger

effect on the concentration

field,

than on the

_temperature.

While

the uniform vertical

_temperature

_gradient

established in the

_quiescent

_conducting

_regime

is

slightly

perturbed

and modulated

_by

the

flow,

the uniform vertical concentration

_gradient

is

completely

destroyed (in

the limit L -

_0).

Instead,

the concentration

_gradients,

_imposed

_by

the

_coupling

to the

_temperature

field

_(via

the Soret

_effect),

are confined to the

_boundary

layers

near the walls

_(where

the flow

_velocity

_vanishes),

and to the free

_{boundary layers along}

the vertical

_separatrices

of the flow

_(see

_Fig.

_6).

This

_picture

suggests

an alternative

_approach

to

_study

the TW convection :

instead

of

_expanding

about the

_conducting

state we start with the limit of

_large

Péclet

number,

i.e. finite

_amplitude

convection and L - 0. In that limit the

system

is in a state of

_stationary

convection _{but as p goes down}

_(with

the

_Rayleigh

number which is controlled in the

_experiment)

it

_undergoes

a transition to a

_traveling

wave state. This transition can be

_thought

of as an

_instability

of the concentration

_{boundary layers}

in the

convective flow.

The

_strategy

of the calculation is as follows. First we determine the distribution of a

_passive,

1/1

=

0,

impurity

in the convective flow with Soret

coupling

to the

_temperature

field. For this

purpose we assume a

_single

mode

_{rigid boundary}

convection and use the

_{boundary layer}

theory applicable for p >

1 to determine the concentration field.

Next,

we calculate the correction to the flow

_{(and temperature)}

field due to the

_{finite 1/1}

_{perturbatively}

in

1 If 1

1. The

_{perturbation theory}

involves a

_solvability

condition

_relating

the

_asymmetric

distortion of the concentration

_{boundary layer}

with the translational mode of the cellular

pattern.

We will find that such

_{asymmetric boundary layer}

solution

_{corresponding}

to

non-vanishing

TW

_velocity

can occur

_{for p pc (and}

hence for R _

_R’)

_{with p,}

_(|03C8

_{11/2/L )8 /7}

Thus,

we seek a

_steady

state solution of

_equation

_(5d)

with

_g-1=

0

_{(vo = 0 ) :}

where p

=

sA /L

is the Péclet number and the flow field

(ul

=

Aûl)

is defined

by

the

stream-function 01

1 of

equation

(22)

corresponding

to the

single

mode

rigid boundary

convection.

The

_boundary

conditions for the concentration field are :

The

_general

solution of

_equation

₍₂₄₎

can be written as _yo =

yh +

_yp

where

_yp

satisfies

equation

(24

a,

b)

and yh satisfies

equation

(24a)

with

_{homogeneous boundary}

conditions :

âzl’h1z = 0,

1 = 0. The

inhomogeneous boundary layer

concentration

yp

is

generated by

the

simultaneous diffusion and advection

_[14]

of the concentration

_gradient

due to the Soret effect

existing

near the walls :

_equation

_(24b).

Since at a

_{distance e}

« 1 from the

wall,

~l = A’ 2 sin kx,

balancing

the diffusion term

(p-1 a§yo)

with the advection terms

(ûl . oyo)

in

_equation

_(24a)

_yields

an

inhomogenous

boundary layer

width

_scaling

as

Fig.

6. -

Steady

flow _pattern

_{(vo = 0 ).}

The notation used coincides with the one in the text.

p-l/3.

This

_boundary

_layer

detaches at

_points

A

_{(A’ )}

_giving

birth to a free

_{boundary layer}

along

the

_separatrix

AB

_(A’B’).

The latter in turn extends

_along

the streamlines into the wall

regions

forming

the

_top

_(bottom)

_{homogeneous boundary layers}

_{(Fig. 6).}

Since

_along

the

separatrix çb 1 -- kAw (z) x,

balancing

the diffusion across streamlines

p -1 ax yh

with the

advection

_along

streamlines

_yields

a

_{boundary layer}

width

_scaling

as

p - 1/2.

As the

AB (A’B’ ) boundary

layer

extends

_along

the streamlines into the

_top

_(bottom)

wall

_regions,

the

_expansion

of the streamlines will

_change

the width of the

_layer.

The concentration

y h in

the

_{boundary layer}

varies with p 1/2

0,

and

since 0 ~

z 2,

we infer that the concentration

scales as

p1l4

z and

_hence,

the

_{homogeneous boundary layer}

width near the wall scales

[13a]

as

p-1I4.

The effect of diffusion

_{(p-1 a2 z Y h)}

across this broadened

_layer

is of 0

(p - 1/2)

and can

thus be

_neglected

in

_comparison

with advection

_parallel

to the wall

_(azo

_{axyh ’--.}

_zaxyh)

which is of order 0

_(P-1/4).

Thus the

_{homogeneous boundary layers along}

the walls are dominated

by

advection and the variation of yh

along

streamlines in these

_regions

can be

_neglected.

It is

_important

to notice that the

_{homogeneous boundary layer}

_(of

width

_p-

_1/4)

is much thicker than the

_{inhomogeneous boundary layer}

_(of

width

_p-1/3).

_Thus,

the

_inhomogenous

boundary layer

can be accounted for

by including

a source of

_strength

_{F(- T )}

at the

stagnation point

A (A’ )

in the

_equation

for yh.

By

conservation of flux the

_strength

of this

source is F=

_7Tk-1p1l2

which is obtained

_{by integrating}

_az

_"Yplz=l,O

_along

_AB’(A’B)

and

rescaling

with

_pl/2

as

_appropriate

for the

AB (A’B’)

_{boundary layer.}

We now solve for the concentration field

_along

the

_{separatrices. Introducing}

streamlines

coordinates

_{[13] :}

with ~

1 = kx

w (z )

one derives

_[13]

from

_equation

_(24a)

the

_equation

for the AB

_boundary

layer

whose solution is :

To obtain an

_equation

for

_{Yh(o-, 0)}

we follow yh around the closed

_loop

ABA’B’ and

remember that diffusion has no effect on _{yh in the wall}

_regions

BA’,

B’A. We find :

where _{To =}

_{T (1 ).}

_Introducing

the Fourier transform

and fourier

_{transforming equation}

₍₂₉₎

_{yields :}

Substituting

this into

_equation

₍₂₇₎

_{yields :}

which back in real space has the form :

with :

_TI;

=

2(T(z)

₊

nTo)/p.

Having

found the concentration

_profile

for the «

_{passive » impurity}

we can turn back the

coupling

to the

_{flow, l.p 1 =F}

0,

and calculate the corrections to the flow

_{perturbatively.}

As has been shown

_earlier,

the

_{perturbation theory}

will involve the

_solvability

condition :

equation

(8).

Provided that

_{y (x, z )}

is

_symmetric

under x - - x, the

solvability

condition is

trivially

satisfied.

_However,

if the

_{boundary layers}

were

_asymmetric,

the

_solvability

condition

would

_require

a

_{non-vanishing}

translational

_velocity

of the rolls -

that the TW

_velocity

Vo #

0. We can look for such a non-trivial self consistent solution

_{simply by replacing}

the

streamfunction ~1

in

_equation

₍₂₆₎

_{by ~1 ~ g- 1 Z +}

₁ which

_corresponds

to the TW

convective flow as seen in the

_comoving

frame

_(with

_g-1~

_vol~A

_being

the rescaled TW

velocity).

As

_long

as

g- 1

is

small,

i.e. smaller than all other small

_parameters

in the

_problem,

our

_analysis

for the

_stationary

convection

_applies

without

_change

and the concentration field is

_{given by equation}

₍₃₂₎

_{with ~1 replacing ~1.}

The self consistent value of

_g-1

is determined

by

the

_solvability

condition of

_equation

_{(8) :}

with : à p

_{A - 1 0 1.}

The inner

_{product: ôx’Yo, dxcÎ>}

=

2 (ôx’Y AB’ ÔxcÎ>

(the

factor of 2

comes from the contribution of the A’B’

_separatrix),

can be evaluated from

_equation

₍₃₂₎

with _rc

_{= 16.98}

a

_positive

constant. The value we deduced from our numerical solution of

equations

(19, 15)

is K = 4.5. The

discrepancy

is due to the

_{approximations}

in the evaluations of the

_integrals,

as discussed in

appendix

C. The nontrivial

(g- 1 ~

0)

solution appears for

p pc with Pc

(g -

~

0 as p ~ pc from

below) :

The p

= pc

point

corresponds

to the transition from

_stationary

to the TW convection

which,

in

terms of the reduced

_Rayleigh

number is therefore

_predicted

at

£TW = Pc L = 1 03C8/114n.

Equation

(35)

is confirmed

_by

a numerical solution of

equations

(15, 19) (see

_Fig.

7),

which

also shows the bifurcation from

_steady

to TW convection to be critical

_{(Fig. 3).}

Notice that the convection

_amplitude

is smooth

_through

this transition. This is in

_partial

_agreement

with the

experimental

results

_{[2a, la],}

where a continuous transition from TW to

_steady

convection

was observed on

heating.

On

cooling though,

the transition from the

steady

convective state to TW convection was

_hysteretic.

However the

_experiments

were done in a

_rectangular

geometry

where the influence of the lateral walls is known

_{[1-3, 9]}

to be

_important.

These walls may stabilize the

steady

convective state

_against

a TW

_{perturbation,}

thus

_leading

to

hysteretic

behaviour.

Fig.

7. - The critical Péclet

number Pc

at the transition from TW to

_stationary

convection as a function of 1 =

L/

B/2013 03C8.

The

slope

of the _curve is - 1.045, with an error of 0.005. The difference with the

analytic

result

_{(- 8/7)}

_may come from the fact that the

asymptotic regime

has not _yet been reached

(the

values of 1 are not small

_enough).

Discussion.

For

_{free-free-permeable}

_boundary

_conditions,

we have recovered

_previously

known

_{[8, 12]}

results and showed that the bifurcation from the conduction state to a state of TW convection is

forward,

with the

_amplitude

of convection

_scaling

as

_{(R -}

Rco)1/4.

convection is

subcritical,

i.e.

_hysteretic,

in

_agreement

with the

_experiments

_[1-3].

In addition

we have been able to evaluate

_analytically

the critical

_point

where the

secondary

bifurcation from TW to

_steady

convection

_happens.

This

system

is one of the few

_examples

where such an

analysis

can be done.

We have calculated the full bifurcation

_diagram.

_{Its upper branch}

_corresponds

to the observed uniform state of slow TW. In

_agreement

with the

_experimental

observations

_[1-3],

the

_amplitude

of the TW convection on the upper branch is found to be

_equal

to the

_amplitude

of

_steady

convection in a _{pure fluid with similar}

_properties.

Our results exhibit a fast decrease

of the TW

_velocity

_(by

almost two orders of

_magnitude)

as 03B5 is increased

_by

a mere 10 %. This

is in

_qualitative

_agreement

with the

_experimental

data

_[2d].

The

_{analysis predicts}

a critical

(forward)

transition from TW convection to

_steady

convection. This however is at odds with the

_experiments

_{[2a, la]}

which find the transition to be

_{weakly hysteretic.}

This

_discrepancy

could be due to _{the presence of lateral walls. An}

_experiment

in an annular

_geometry

is

_clearly

needed. The

_steady

convection state becomes unstable to TW convection at a value of the Péclet

_{number Pc =}

_{1 UcIlL}

_{given by equation}

_(35).

It would be

_interesting

to

_study

this transition as a function of the

_separation

_{ratio t/J}

_{experimentally}

and make a

_quantitative

comparison

with our

_prediction.

It would also be of interest to

_study

the concentration

profiles directly

and

_verify

the existence of concentration

_{boundary layers. Experimental}

comparison

should be

_attempted

in a one-dimensional annular

_geometry

_[2d],

since the influence of the lateral walls in a

_rectangular

_cell,

which are not taken into consideration in

our

_analysis,

is

_{experimentally}

relevant

_{[le, 9].}

On the lower

_(unstable)

branch the TW

_velocity

is fast

_(of

0(V- 03C8)),

and the concentration

_profile

is close to the linear concentration

_{profile existing}

in the

_conducting

state

_(see

_Fig.

_4).

_Although

such a state

_(with

_infinite

_extent)

is

unstable,

one wonders if it could not exist as a stable state of

_finite

_length.

Due to the existence of fast

TW,

infinitesimal disturbances are advected and for a convective

_region

of finite

_{length, they}

_may not

_develop

to destabilize the structure. In other words the fast TW

_existing

_{in the lower branch may be}

only convectively

unstable,

not

_absolutely

_[9].

_{This may}

_explain

the

_stability

of the observed confined states

_(always

characterised

_by

fast

_TW),

which should then be understood as solitonic structures

_[15].

To check for the

_possibility

that the fast TW in a confined state

correspond

to the fast TW on the lower

branch,

one can think of various measurements. One could measure the concentration

_profile

in a confined state and check whether it is as

predicted

for the lower

branch,

i.e. close to the concentration

_profile

in the

_conducting

state.

One could also measure the TW

_velocity

and

_amplitude

in the confined states as a function of

e and compare with the theoretical

_predictions

(Figs.

_2,

3).

Let us notice that the

_analysis

presented

here can be extended to values

_{of tp}

and L different from the ones we chose.

Appendix

A.

We seek a solution in the form :

Inserting

this ansatz into

_equation

_(Al)

_{yields :}

The

_arbitrary

_parameter

À

_corresponds

to a translation of the solution and doesn’t contribute to the

_solvability

condition at next order. We thus set it to zero. Notice that the fields

02

and

₆₂

can be written as :

The

_equation

_{for the concentration field y} 1 is now :

the solution to which can be written as :

where

_l’Á1)

satisfies :

The solution

_’YÁ1)

can be

_sought

as a fourier

_expansion.

The

_solvability

condition at

Notice that :

Where we

_{identify :}

Thus the

_solvability

condition

equation

(A8)

yields :

with

_{f ~}

_f

+

_{h R.}

It can be

_readily

seen that

_hR

_{is very small. Near} onset

_{(g « 1)}

one has

hR -

0

_(g6)

_{f =}

0

_{(1 ),}

and far from onset

_{(g > 1 )}

due to the existence of

_{boundary layers}

f , hR --> 0.

We shall therefore use as an

_approximate

_{amplitude equation, equation (A.10)}

with

_{f =}

_f.

Appendix

B.

This

_appendix

is devoted to the numerical solution of

_equations

₍₁₅₎

and

_(19).

We will calculate the

_amplitude

of convection

EA,

and the

_velocity

_vo of the

_traveling

waves in the case

of

_{rigid-impermeable boundary}

conditions,

as a function of the reduced

temperature

2

R - Rc

E2

=

Re,

, the other

physical

parameters

L

being

fixed.

Rc

p Y p

(03C8,L ) bein

g

Both

_{equations (15)}

and

₍₁₉₎

involve the function

_{f (g, po).}

Our first task therefore consists in

_computing

this function.

_Using

the definition

_{(Eq. (13)) :}

where yo is

the solution of

_{equation (11) :}

In order to solve

_numerically

equation (B.2)

one uses a Fourier series

_{decomposition}

of the concentration field :

which satisfies

_obviously

the

_boundary

conditions

_{Ôz 1’01 z = 0,1}

= 1.

_Likewise,

the

streamfunc-tion,

which has to

_satisfy

the

_boundary

_{conditions CP1 = dzCP1 = 0 on z = 0, 1,}

can be written

as a series of sines in z :

which satisfies

_{obviously 0}

₁

_{z = 0, 1}

₌ 0 or as a series of cosines in z :

Which satisfies

_trivially

_{âzcf> 11 z = 0,1 =}

0.

_Only

odd

_(even)

terms contribute in the

decomposi-tion

_{(B.5a) (B.2b).}

The coefficients of the

_{expansion 01}

_{1 and t/J}

can be

_{computed analytically}

from

_equation

₍₂₂₎

and

_expressed

in terms of

_elementary

functions. Thus

_equation

_(B.2)

can

be rewritten as :

By using

the Fourier series

_(B.5a)

in the evaluation of

_û,

on the left hand side of

equation (B.6)

and the

_{decomposition}

_(B.5b)

on its

right

hand

side,

one

_obtains,

after

straightforward manipulations,

an infinite set of

_equations

for the

_{Am, n}

and

_{Bm, n’s.}

A

peculiarity

of the solution of this

_system

is that all the coefficients

_{Am, n}

and

_Bm,

_n’s,

such that

(m

+

_{n )}

is odd are zero. A truncation of the

_system

to a finite number of modes

_yields

a finite set of

_equations.

The

_resulting

_system

has been solved

_by

an IMSL inversion routine up to 30 modes in x and 20 modes in z.

Of course, this truncation is valid as

_long

as the

gradients

that build up in

boundary layers

(see

the

_{boundary layer}

_calculation)

are well resolved. We have

_{systematically}

checked the resolution

_{by making}

sure that the Fourier

_spectrum

of the solution decreases fast

_enough.

In

practice, problems

arise

when g

becomes

_larger

than

1,

and the Péclet number p =

gpo

exceeds a certain

value,

for a

_given

number of modes.

Once y has been

obtained,

it is

_{straightforward}

to

_compute

the scalar

_product

ôxYo, ÔxcÎ>m

hence the function

_{f (g, po).}

In order to obtain the

_amplitude

of convection A and the

_velocity

_vo of the

_traveling

waves as a function of the reduced

_temperature

e2,

one first writes

_{in a sli tly}

different way

This

system

can be solved

_efficiently

_{in g}

_{and po by}

a standard Newton’s method. Once the solution has been

obtained,

the values of the

_{physical quantities}

vo and A can be extracted :

The values we have chosen are L

=10- 2 ;

and

w

=-1/4;

tp

= - 1/16; 03C8 = - 1/36 ;

03C8

= - 1/64

and w

= - 1/100.

Appendix

C.

In this

_appendix

we will show how to evaluate the inner

_{product :}

To evaluate

_equation

_(C.1)

one notices that from

_equation

_{(22) : ~}

= sin kxw

(z).

Near the

walls at z =

0, 1

w(z)

=

K,2

where 03B6

is the distance from the wall

( 03B6 = z

_or 1 - z and

rc =

15.45).

Introducing § m

kx and

6,, (z) =-

Tln(z)/w(z)

we rewrite

_equation

_{(C.l) :}

The main contribution to the

_integrals

comes from the

_{region along}

the

_separatrix

AB (z.j. « z « zmax) for

which

_{8n (z ) 1,}

and not from the

_stagnation

zones near

A (0 «

z «

_zm;n)

or B

(zmax « z « 1 ).

Since

_8n

_1,

we _may

_approximate

_{sin e -- e}

and set the limits of

_integration

_{on g}

to ± oo . The

_{gaussian integration}

is

_{straightforward, resulting}

in :

Where the constant Notice that

_by

setting

the limits of

_integration

in

_equation

_(C.3)

to _{zm;n, Zmax} we are

_{overestimating}

the value

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