Hydrodynamics instabilities of cholesterics under a thermal gradient

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Hydrodynamics instabilities of cholesterics under a thermal gradient

E. Dubois-Violette

To cite this version:

E. Dubois-Violette. Hydrodynamics instabilities of cholesterics under a thermal gradient. Journal de

Physique, 1973, 34 (1), pp.107-113. �10.1051/jphys:01973003401010700�. �jpa-00207348�

HYDRODYNAMICS INSTABILITIES

OF CHOLESTERICS UNDER A THERMAL GRADIENT

E. DUBOIS-VIOLETTE

Laboratoire de

Physique

des Solides

(*),

Université

Paris-Sud,

^Centre

d’Orsay, 91, Orsay (Reçu

^{le 28}

juin 1972,

révisé le 19

septembre 1972)

Résumé. ²⁰¹⁴ Nous étudions les instabilites

hydrodynamiques

^de

cholestériques

^{soumis à un}

gradient

^de

température.

^On

envisage

^{le cas} de distorsions lentes à l’échelle du pas P mais

rapides

à l’échelle de

l’épaisseur

^{d. Nous utilisons une}

description macroscopique

^{« en couches » du cho-}

lestérique.

^On

prédit

^un comportement ^du

cholestérique,

^{soumis à un}

gradient thermique, analogue

à celui observé sous

champ électrique

alternatif

(basse fréquence)

^ou

champ magnétique, parallèles

à l’axe. La

périodicité

^des

domaines,

^au

seuil,

^{varie en}

(Pd)1/2

^{et le}

gradient

^{seuil en}

(Pd)-2.

^On

s’attend à observer des instabilités en chauffant le

cholestérique

par ^en

bas,

^{dans le cas d’une}

anisotropie

de conductivité

thermique positive (ka

>

0).

^{Dans le cas}

contraire,

ka

0,

^{on s’attend}

à observer ces instabilités en chauffant le

cholestérique

par ^en haut.

Abstract. ²⁰¹⁴ We present ^a theoretical

study

^of convective instabilities of cholesteric

liquid crystal

under a thermal

gradient.

^We consider the case of distortions smooth in

comparison

^{with the}

pitch

^P but fast in

comparison

^{with the}

sample

^{thickness d. A}

macroscopic

^{« in}

layers » description

of cholesterics is used. The behavior of cholesterics under thermal

gradient

is found similar to the

one under a. c. electric field

(low frequency)

^or

magnetic field, applied parallel

^to the helical axis.

The

spatial periodicity

^of the distortion at the threshold varies as

(Pd)1/2,

the threshold

gradient

as

(Pd)-2.

Convective instabilities are

expected by heating

^the

sample

^{from the} top ⁱⁿ cholesteric with a

negative

^thermal

conductivity anisotropy

^Ka

(K~ , ^{K~ being}

^the

conductivity parallel

^and

perpendicular

^{to the}

molecules).

^On contrary ^{for Ka} > 0 one expect convective instabilities

by heating

^the

sample

from the underside.

Classification Physics abstracts :

03.10

1. Introduction. ^- The instabilities of cholesteric

liquid crystal

^{under the}

application

^of

magnetic [1], [2]

^{or a. c.} electric fields

parallel

^to the helical axis are

experimentally [3]

^and

theoretically [5], [6]

well under- stood. The behavior of the cholesteric structure is

quite

similar to the nematic one. In the case of a cholesteric structure, ^{there is a}

coupling

between the local axis tilt

angle

and the local twist

angle.

^{The threshold field}

varies as

(Pd)-1/2

^{and the}

spatial periodicity

^{of the}

distortion as

(Pd)1/2 (where

^{P is the}

pitch

^{of the helix}

and d is the

sample thickness).

Very

^similar

patterns

^to those of Rondelez and Arnould

[1] have

been observed

by

Gerritsma and Van

Zanten,

^{when a}

plane

^{texture of} cholesteric was cooled down

[4].

These

hydrodynamic

instabilities similar to the Bénard

[7]

ones, also appear in nematic

liquid crystals

submitted to a thermal

gradient.

^{It has} been shown

[8]

that,

^for

nematics,

^the

instability

^{was dominated}

by

^a

mechanism due to the

conductivity anisotropy

^{of the}

liquid crystal.

The threshold thermal

gradient

^{is then}

much lower in the

anisotropic phase

^{that in}

ordinary liquid.

Such convective instabilities have been observed in nematic

samples

^{submitted to a}

temperature

^diffe-

rence, between

plates,

^of

only

^few

degrees [21].

In this paper, ^we shall

give

^an extension of the model

proposed

for nematics and consider cholesteric

samples

for which : P « d.

2. Initial

configuration.

^{- The}

coupling

^{between a}

thermal

gradient

^{and the helical structure of choles-}

terics has been first

reported by

^Lehmann

[9].

^{A choles-}

teric

droplet,

^{when submitted to a thermal}

gradient parallel

^to its helical

axis,

^{shows a} uniform rotation of the local molecular axis. If the cholesteric

layers

^are

pinned

^{on the}

plates

^the presence of a thermal

gradient

modifies the

pitch

of the helix inside the

sample [10].

These

effects,

^{first discussed}

by

^Oseen

[11], may be

understood in terms of Leslie’s constitutive _eq.

[12].

In what

follows,

^{we shall}

essentially

be concerned with distortion

wavelengths

^much

greater

^{than the}

pitch

of the helix and so use a

macroscopic

^{« in}

layers » description

^{of the} cholesteric state as introduced

by

several authors

[10], [13].

^{In the}

appendix,

^{it shall be}

shown that similar results may be obtained

by using

Leslie’s

microscopic

eq.

[12].

(*) Laboratoire associé au CNRS.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003401010700

108

The free _energy

F(T)

^{at the}

temperature

^T may be written as

(1) :

where the

« phase » cp(r)

^{will be defined}

later, p

^{is the}

density

^{at the}

temperature T,

so is the

entropy

^{and C} is the

specific

^heat.

The elastic constants

L22

^{and K} may be

expressed

in terms of the

microscopic

^{elastic constants}

K22

and

K33 [6], [10] :

where _qo

= 2 nlp,

^{is the wave vector} of the helix.

L23

⁼

L22 a Log Plô Log

^T characterizes the tem-

perature dependence

^{of the}

pitch.

We shall now define the

« phase » cp(r).

^Let

m(r)

label the

layers. m(r)

is related to the

pitch

^{of the}

structure

p(r)

^{and to the «}

phase » cp(r) :

The

equilibrium configuration

^{at the}

temperature To

is

planar

^{and defined}

by

Suppose Voo(r) in

the oz-direction

(Po

⁼

z)

^and

consider a small deviation

àç(r)

^{from this}

equilibrium configuration

^induced

by

the presence of ^a thermal

gradient.

The

entropy

^{source TI} may be written as :

where

Jg

^{is the}

entropy

flux and CU’ the viscous dissi-

pation :

where the

« permeation

^{force »}

(1) ^{We use the notation} f,i ⁼ bflbxi and the convention of sommation on the identical indices çJ,; rp,i ⁼

(V

^ço)2.

is here

expressed

^{as :}

The

phenomenological equations relating

^{the fluxes}

and the forces _may be

written, using Onsager’s

relations

[23],

^{as :}

Jq

⁼

^TJ,

^{is the heat flux. A =}

yqo@

^where y is a

viscosity.

ⁿ is the unit vector

parallel

^{to the local}

cholesteric axis.

K1

^and

K2 depend

^{on the molecular}

thermal conductivities

K.l (perpendicular

^{to the mole-}

cules)

^and

Kjj (parallel

^{to the}

molecules) :

qo, 1on i? n2, n3 ^are viscosities. If the

liquid is suppos-

ed

incompressible

110 ⁼ n01 ⁼ 0.

Let us now calculate the distortion of the choles- teric structure under the

following

conditions :

i)

the Z-axis is

parallel

^{to the}

unperturbed

^choles-

teric axis

(Fig. 1) ;

FIG. 1. ^- A cholesteric sample ^{with a} z-helical axis is submitted to a thermal gradient ^VzT =- -,8o. ^T > To (Bo> 0) .

ii)

^{a thermal}

gradient (V z

^T

= - Bo)

^is

applied

^to

the

sample ;

iii)

^{one surface}

(Z

⁼

0)

^{of the}

sample

^at

tempera-

ture T is anchored : on this surface the direction of the cholesteric molecules is

fixed,

^which means, in ^a

macroscopic description,

^{tbat the}

« phase »

^is

fixed, and ;

iiii)

^{the surface Z = d at}

temperature To

^{is a free}

surface : there is no constraint on the direction of the cholesteric molecules

(T

>

To corresponds

^to

Bo > 0).

The

equation

^for

p(z), defining

^the

planar configu- ration,

^{is :}

with the

boundary

conditions :

oc is the volume

expansion

coefficient.

The first term is

dissipative,

^it

depends

^{on the}

presence of ^a thermal

gradient.

The second one is reactive and

gives

^the variation of the

pitch

^{as a}

function of the

temperature.

One obtains as a

stationary

^{solution :}

where the distortion of the

pitch p

^{= 2}

x/q

^is

given by :

This distortion is weak and _may be

neglected

^if

Bd « 1.

An evaluation of the

dissipative

^term

Po _{To L22} ^{Aî d}

^is

obtained

by

^a

comparison

with Leslie’s

microscopic equations

^rewritten

using Onsager’s

^relations

[14].

P. G. de Gennes

suggests [15]

^that

k3

⁼

xK2

qo where

x is ^a dimensionless numerical coefficient of order

10-2.

This leads to  ~

10-’

and

As

long

^{as we} shall be concerned with small total

temperature

differences

(AT

⁼

Bo d),

the distortion o

the cholesteric

pitch

^{can be}

neglected. (Typically

AT ~ 10 OC leads to x

PO d L-- ^T0 1

^{300 . In order to}

estimate the reactive term, ^{we need to know the}

density

and

temperature dependence

^{of the}

pitch.

From

experimental ^data,

^{the ratio}

varies over a very

large

range

(typically 10- 2 -102), depending

^{on the}

samples

^used.

Up

^{to now, we lack}

experimental

^{data on the} variations of the

pitch

^{as a}

function of the

density

^{but it seems} reasonable to

neglect

this variation

compared

^{to the}

preceding

^one.

As a consequence of this

approximation,

^{we shall}

assume that the initial

configuration

^{of the} cholesteric under a thermal

gradient

^{is a}

planar

^{one with a well}

defined

pitch

po.

3. Convective instabilities. ^- 3.1 MECHANISM. - A

description

of the mechanism

leading

^{to convective}

instabilities has been

given [8]

^{in the case of nematic}

liquid crystals.

^{This mechanism can} be extended to cholesterics and summarized as follows

(Fig. 2).

Suppose

^the cholesteric

layers

^{in the}

xy-plane

submitted to a thermal

gradient VzT = - fl along

the z-direction and consider a fluctuation

£5cp(r)

^{of the}

FIG. 2. ^- The cholesteric is sandwiched between plates ^main-

tained at temperature Ti ^and T2 (T2 > Ti). ^{Due to the aniso-} tropy of the thermal conductivity, ^there appears ^warmer (1) and cooler (2) regions. ^The gravity forces induce opposite

hydrodynamic velocities in regions (1) ^and (2).

layers.

^{Due to the}

conductivity anisotropy, (Kt - K2)

of the

layers,

the heat fluxes are deviated :

i) along

^the

layers (KII

>

Kl)

^{for cases}

a)

^and

c)

ⁱⁿ

figure 2 ;

ii) perpendicular to

^the

layers (K_L

>

Kjj )

^{for cases}

^b)

and

d).

There appear ^warmer

(2)

and cooler

(1) regions.

The

gravity

forces induce

opposite hydrodynamic

’velocities in

regions (1)

^and

(2).

These forces stabilize the

system

^{in case}

b)

^and

c)

^and destabilize it in case

a)

and

d).

^{So we}

expect

convective instabilities in case

a) for fi

> 0 and case

d) for fl

^0.

3.2 THERMOHYDRODYNAMIC EQUATIONS. ^- In order to

study

^the

stability

of the initial

configuration

^defi-

ned

by :

consider a small fluctuation

bcp(x, z)

^{of the} cholesteric axis and

expand

it in its Fourier

components

We assume

that,

^{at the}

walls,

the cholesteric

layers

are

pinned.

^This

imposes

the relation :

Then the most favourable wave-vector

(which

minimizes the elastic

free-energy)

^{is :}

110

Just as

previously,

^{we consider}

only

very smooth distortions

(compared

^{to the}

pitch)

^{and assume :}

This orientation fluctuation induces a

temperature

fluctuation

ôT(x, z),

^a

density

fluctuation

tJp(x, z)

^and

hydrodynamic

velocities

vx(x, z), Vz(x, z)

^{which can be}

taken,

with the above

assumption,

^{as :}

In our

description,

^{we shall :}

i) Only keep

first order terms in fluctuations.

ii) Neglect, just

^{as in the}

Boussinesq [16] approxima- tion,

^the

density

variations

except

^{in the}

gravity

^force.

(This

^is

justified by

^the weak value of the volume

expansion

coefhcient : a ^N

10-3-10-4.)

iii)

^{Not take into account the} thermal variations of the different coefficients

appearing

^{in eq.}

(II.5) (which

^are weak for the AT’ s under

consideration).

The z

dependence

^{of all}

quantities,

in view of the

inequality (III.3),

may be

neglected

^{in all}

equations except

in eq.

(II. 5b)

^{where the two terms}

L22

qJ,33 ^and

KqJ ,xxxX’ ^appearing

^{in the}

expression

^of

g/,

^{are of the}

same order of

magnitude.

Eq. (II. 5b)

may be

rewritten,

^{with use}

of eq. (II.4)

and relation

(II.2) :

the 1. h. s. member of this

equation

takes into account the

permeation (2).

3. 3. MASS CONSERVATION EQUATION ^- Let :

be the initial

temperature.

^The

density p(T’)

^at

tempe-

rature T’ is related to the

density p(T)

^at

temperature

^T

by :

Then,

^{one has :}

The mass conservation

equation

written to first order in fluctuations leads to :

where

(2) ^In absence of permeation the molecules are driven without rotation. The permeation corresponds ^{to a} rotation of the molecules associated with ^a translation.

Using

eq.

(III .4),

^this

gives :

The two terms

Dlêt (b p)

^and

V.V(Pi)

^{are both}

linear in oc and in the fluctuations :

they

^{can be}

neglec-

ted. This leads to :

and

justifies

^{the fact}

that,

^{in the}

following equations,

we shall

only keep

^the

Vz component

^{of the}

velocity.

3.4

EQUATION

OF THE HEAT CONDUCTION. - The conservation

equation

^{of the}

entropy s :

is

written, using

eq.

(II . 2, Il , 3),

^{as :}

The

dissipation V,

which includes

only

second order terms in fluctuations

gives

^no contribution to eq.

(III. 5).

Then, using

eq.

(II. 5c), (II. 3),

^{one obtains :}

with

The term Â

a b ôz

^{has been}

neglected :

^{g it can be}

shown, using

^the threshold law for

k,

^that

K2 Bk2 àç > Âqz àg/ ,

which in an

a posteriori justification.

3.5 ACCELERATION EQUATION. ^- The

equation

^of

motion is

(3) :

The external force is :

pF ext

⁼

pgabT.

6’ is the viscous ^tensor defined in _eq.

(II. 5a).

6e is the elastic tensor.

(3) ^{We use the} following ^notations [17] : ^{VV is a} dyadic ^with

the components (VV)ij = Yi Yj ; ^div (T) ^{is a vector} with compo- nents (div T)i =

E

Tji,j ; ( ),i ^{i means} partial derivation with

j respect ^{to xi.}

The last term is second order in fluctuations. Then

one

finds,

^from eq.

(III. 8)

where the

viscosity 11

^{is :}

4. Threshold

gradient.

^{- Let us first} recall the equa- tions

describing

^the

thermohydrodynamic

^{behavior :}

and _suppose

that, just

^{as in}

nematics,

^the

instability

^is

cellular

(the

^{solutions are} of the form est with s

real).

Then,

^{for a}

given k,

^the

instability

^threshold

B(k)

^is

obtained for s ⁼ 0 :

where

In presence of a

magnetic field H (lloz)

^this

expression

is valid

if f (k)

^{is now defined as :}

where _X. ⁼

x

Il

- xi

(x

^{Il and xi are the}

susceptibility parallel

^and

perpendicular

^{to the}

molecules).

The true threshold

gradient Bc

is obtained

for f (k)

minimum. This

gives-

the threshold distortion wave

vector kc :

For wave

lengths

^under

consideration, q. « ^{k «}

qo, the

inequality :

is satisfied.

for

typical

values in MBBA :

K22 ~ ^10-7 CGS, K33 ~ ^10-6 CGS, k2 N ^10-4 cm2/s,

^{y ~}

^10-1

^Poise

[18].

Then, using

^relations

(II. 6)

^and

(III. 7),

the threshold

gradient

may be written as :

This

expression

^is

quite

^{similar to the one obtained}

in nematics. The

significant

difference is in the wave

vector

dependence

^of

Bc. (Bc ~ q02 qz2

instead of

q4z .)

This later

point

results from the

adjustment

^{of the}

twist to the bend distortion and is

implicitly

^contained

in the

expression

of the elastic

part

^{of the} free energy.

This

coupling

^{between these two}

degrees

^{of freedom is}

explicitly expressed

^{in the}

appendix.

For an

isotropic liquid,

^the

instability

^sets up

only

^if

B > 0,

^{that is}

by heating

^the

sample

from below.

The

inequality (IV. 5)

^{shows that in} cholesterics :

- The

instability

^{is dominated}

by

^{the mechanism}

depending

^{on the}

anisotropy (k2)

^{of the}

phase.

- The threshold

gradient

^{in case of} cholesteric with

large pitch

is much lower than in

ordinary liquid.

Furthermore,

^{as shown}

by expression (IV. 6),

^we

may

expect

convective instabilities :

-

by heating

^the

sample

from below if

fl

>

0,

i. e.

if k

^Il

> k 1. ;

-

by heating

^the

sample

^{from above if}

fl 0,

i. e.

if k

^Il

^{k 1..}

This later effect is a

surprising specific property

^of

anisotropic

^{fluids. It is a direct} consequence of the

negative anisotropy

^{of the thermal}

conductivity.

^It

could not occur in

isotropic

^fluids.

The

temperature

difference AT between

plates is,

^at

the threshold

where A ~

10-5 (the

estimation has been done with

typical

^{MBBA values}

[22]

of the thermal conductivities

ka/k ~

^{0.4 and k N 5 x}

^10-4 cm2/s).

In order to observe instabilities with reasonable AT

values, compatible

^{with the}

Boussinesq approximation (AT ~

^few

degress)

^we need thick

samples (d ~

¹

cm)

and

large

cholesteric

pitches (po

^{in the} range

10-100 Il

leads to OT N

10-1-10 °C).

The

instability

^{threshold could be lowered}

by applying

^a

destabilizing magnetic

^field

H (// to oz)

^close

to the critical field

Hc (Hc2

⁼

^1/Xa (6K33 ^K22

^qo

qz)

of the Helfrich-Hurault transition

[5], [6].

^{The thres-}

112

hold field

Bc(H), in

presence of ^a

magnetic field,

^is

obtained from _eq.

(IV. 2)

^{with use} of the expres- sion

(IV. 3)

^for

f (k).

Then

If the critical field is

approached

^{within a few} per- cents

(x ~

^{5 x}

10-2),

^one

expects

^{to observe convec-} tive instabilities in cholesteric

samples

^with

relatively

small

pitches (AT ~

^{4 OC}

for Po ~

⁵

p).

Validity of

^{the model. - In the} constitutive _equa-

tions,

^{we have}

supposed

^{all the constants to be}

tempera-

ture

independent.

^In

fact,

^elastic constants,

viscosities, conductivities, susceptibilities depend

^on

temperature.

This

dependence

^{can be}

neglected

^{if the mean}

tempe-

rature of the

sample

^{is not too close to a transition}

temperature.

^These

considerations,

^{and the use of the}

Boussinesq approximation,

^{limit the total}

temperature

difference between

plates, AT,

^{to a few}

degrees.

^This

is coherent with our initial

approximation

^{of an unper-}

turbed

planar configuration,

ⁱⁿ

spite

^{of a non uniform}

temperature. Finally,

^our conclusion is that this model should

apply

^to

samples presenting

^{a weak}

^tempera-

ture

dependence

^{of the}

pitch, provided

that the total

temperature

différence between

plates

^{does not exceed a}

few

degrees.

^The

coupling (À

^{term in eq.}

(11.5))

^bet-

ween the helical structure and the thermal

gradient

has been in fact

neglected. So,

this model of convective instabilities is valid for smectic

samples

^{where the}

pitch

^{is now defined} as p ⁼ 2

nid (d

^{= distance}

between

planes)

and where the

layer

thermal conduc- tivities

K,

^and

K2

^{must be redefined} in function of the molecular conductivities. Nevertheless in the ^case of smectics non linear effects can be

important

^{and as a}

result this

instability

^{could have a «}

ghost

^{» character.}

Acknowledgments.

^{- 1 am} very

grateful

^{to 0.}

Parodi for the communication of the

macroscopic equations

^{and for many}

helpful

discussions. Stimula-

ting

discussions with P. G. de Gennes and J. P. Hurault

are

gratefully acknowledged.

APPENDIX

We

give

^the

microscopic

derivation of the threshold

Pc.

^{Let us first} recall the

expressions

^{of :}

- the free _energy

density

- the viscous stress tensor

(we

^{use Leslie’s}

^[12]

notations)

where A is the

symmetric

^tensor

and

- the viscous

torque F,

ⁱⁿ presence of a thermal

gradient

rv = - n A (Yi N + Y2 An + Y3 n A VT) (in

^{fact as} before the

dissipative

term 73 ^{n A} V Twill be

neglected)

and the heat flux :

Let ⁿ be the director as defined

by

^Hurault

[6] :

and consider small distortions of the cholesteric of the form :

with,

that induce a

temperature

fluctuation £5T and a

hydro- dynamic velocity v. :

The momentum conservation law leads to the

equation :

where 1

where

y,

⁼

y1 /2

^and

ày-/ôO

^{means the mean value}

over a

pitch.

The force

equation

^{is :}

The

entropy production

may be written as :

ÔJ--lô-c depends

on 0 and then _eq.

(A.l) give

^the

coupling

between the twist and the bend.

After calcu-

lations one obtains

and then :

The threshold

gradient Pc

^{is obtained} from eq.

(A. 2), (A. 3), (A. 4).

^Let

us just

remark that

then

where

References

[1]

^RONDELEZ F., ^ARNOULD H., ^{C. R.} Acad. Sci. Paris 273B

(1971)

^549.

[2]

^{HULIN J.} P., ^RONDELEZ F., ^{Solid State Commun.}

(to

^be

published).

[3]

^GERRITSMA C., VAN ZANTEN P., Molecular

Crystals.

[4]

^GERRITSMA C., ^{VAN ZANTEN} P.,

Phys.

^{Lett. 37A}

(1971)

47.

[5]

^HELFRICH W.,

Appl. Phys.

^{Lett. 17}

(1970)

^531.

HELFRICH W., ^J. Chem.

Phys.

⁵⁵

(1971)

^839.

[6]

HURAULT J. P., ^to be

published.

[7]

^BÉNARD M., Rev. Gén. des Sciences _pures et

appliquées

11

(1900)

1261 ; ¹¹

(1900)

^1309. Ann de Chim.

Phys.

²³

(1901)

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[8]

DUBOIS-VIOLETTE E., ^{C. R.} Acad. Sci. Paris 273B

(1971)

^923. Thèse de doctorat ès Sciences

Orsay.

[9]

^LEHMANN

O.,

^{Ann. de}

Phys.

²

(4) (1900)

^649.

[10]

^DE GENNES P.

G.,

^{MEYER R.} B.,

Liquid Crystal Physics (Book

^{to be}

published).

[11]

^{OSEEN C.} W., ^Trans.

Faraday

^{Soc. 29}

(1933)

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[12]

LESLIE F. M., ^{Proc. R. Soc. 307A}

(1968)

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Arch. Rat. Mech. Anal. 28

(4) (1968)

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(to

^be

published).

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Phys.

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G., private

communication.

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BOUSSINESQ J., ^Théorie

analytique

^{de la}

chaleur,

2, 172,

Gauthier-Villars,

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(1903).

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J.

Physique

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publish-

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Physique.

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(1971)

255.

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published.

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(1972)

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Sci. Paris 273B

(1971)

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O., private

communication.