Non linearities close to the thermal threshold in a planar nematic liquid crystal

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Non linearities close to the thermal threshold in a planar nematic liquid crystal

E. Dubois-Violette, F. Rothen

To cite this version:

E. Dubois-Violette, F. Rothen. Non linearities close to the thermal threshold in a planar nematic liquid crystal. Journal de Physique, 1979, 40 (10), pp.1013-1023. �10.1051/jphys:0197900400100101300�.

�jpa-00209178�

Non linearities close to the thermal threshold in ^a planar

nematic liquid crystal

E. Dubois-Violette

Laboratoire de Physique ^des Solides, Université Paris-Sud, Centre d’Orsay, ⁹¹⁴⁰⁵ Orsay, ^France

and F. Rothen

Institut de Physique Expérimentale, Université de Lausanne, CH 1015 Lausanne Dorigny, Switzerland

(Reçu ^{le 12 avril} 1979, accepté ^{le 29} juin 1979)

Résumé.

On étudie les non-linéarités dans l’instabilité thermique ^{induite dans un} échantillon nématique planaire

chauffé _par en bas. On caractérise d’abord le comportement critique ^{en utilisant un modèle avec} des conditions

aux limites libre-libre. On étudie ensuite les propriétés ^{au-dessus du seuil en utilisant un} développement ^en pertur- bation. On montre que le premier coefficient R(1) du développement ^{du nombre} critique ^{est nul. On calcule} l’ampli-

tude des perturbations au-dessus du seuil et on trouve que pour la plupart ^des nématiques la bifurcation est normale.

Abstract.

We study ^non linearities in the thermal instability ^induced by heating ^{a nematic} liquid crystal ^from

below. First a model with free-free boundary conditions is studied. Critical behaviour of the free-free model is

analysed. Properties above the threshold are determined with use of perturbation analysis. ^One shows that the first expansion coefficient R(1) of the critical number vanishes. One determines in an approximate ^{manner the} amplitude of the disturbances above the threshold and finds that for most nematics the bifurcation is normal.

Classification

Physics ^A6stract,s

61.30

47.20

1. Introduction.

In previous ^works [1], [2], [3],

one of us (E.D.V.) has studied the thermal stability

of a planar ^or homeotropic ^nematic sample ^submitted

to a thermal gradient. ^{Here we shall only consider}

the stationary instability ^which appears in ^a planar sample ^{heated from} below with a periodic pattern

corresponding ^{to rolls with a} wavelength ^{of the}

order of the sample thickness d. We shall consider neither the stationary instability ^{which occurs in a} homeotropic sample ^{heated from above nor the} oscillatory ^{one when} heated from below [4]. ^{A standard}

normal mode linear analysis has allowed to determine the threshold [5]. Experimentally ^one gets ^results above the threshold which remain unexploited ^{in a}

linear analysis.

A study ^{of non} linearities has been performed

elsewhere [10] ^{in a} liquid crystal ^{for the case of an} instability ^induced by ^a simple ^{shear where a homo-}

geneous distortion of the nematic appears. In this paper ^we shall give ^an approximate ^{method in order}

to estimate the disturbance amplitude ^{close to the}

threshold and study ^{the nature} of the bifurcation.

If the threshold corresponds ^{to a} normal bifurcation (see Fig. 2b) ^the amplitude of the disturbances decays slowly ^{as one} approaches the threshold. In the vicinity

of the bifurcation, perturbation ^{methods can be used}

to describe the system ^{and to} determine the amplitude

of the disturbances. Such a description, ^{valid for}

small disturbances, ^{has been} sketched for an isotropic liquid by ^Landau [7] and Stuart [8] and is related in a

review _paper [9].

Let A be the amplitude ^{of a} disturbance (for example

the fluid velocity) that becomes unstable at the

Fig. ^1.

^A planar ^nematic sample is submitted to a thermal _gra- dient. The temperature ^{of the lower} plate ^is Tr ^{+ AT and the tem-} perature ^{of the} upper ^one is Tr. ç is the angle ^{in the} plane ^{xOz of} the molecules with the x direction.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197900400100101300

threshold. In our case the threshold will be characte- rized by ^the temperature difference AT between the two plates ^{of the} sample (Fig. 1). ^As long ^{as we are}

concerned with the stationary ^{case where no oscilla-}

tions occur (exchange ^of stability) ^{one can consider}

the amplitude ^A and the time evolution rate 0’ as real.

Close to the bifurcation one assumes that the evo-

lution equation ^{of A is of the} following ^{form :}

0’ characterizes the slowing down of the disturbances close to the transition and is of the form [11]

R is some dimensionless parameter. R - Rc ^cha°rac-

terizes the departure from the threshold Rc. ^{In our}

case one can define R ⁼ Ô.T/Ô.Tc.

For AT > 0394Tc, 03C3 ^is positive and fluctuations _grow

exponentiàlly. ^{On the} contrary, ^{for 0394T} AT,,, u ⁰

and there is a damping of the disturbances.

’t 0 is a positive quantity determined by ^{the linear} theory, which defines the linear growth ^{rate of the}

disturbances near the threshold.

kc ^{is the wave vector of the} marginally ^{stable state}

at R ⁼ Rc. ^{Above the} threshold, perturbations ^with k # kc ^{can occur for an} infinite horizontal layer :

this is described by the second term of the right ^hand

side of _eq. B(1.2). The influence of lateral boundaries will depend ^{on a} characteristic length 03BE ^{defined in} [11] :

where 03BEo ^is determined by the linear theory.

The _presence of a quadratic ^term in eq. (1.1) depends ^{on the} symmetry properties ^{of the convection} pattern corresponding ^{to the} change ^{A ~- A.}

If a # ⁰ the bifurcation is not normal : deviations

can be amplified under the linear threshold (Fig. 2a).

There is hysteresis ^between Ro and Rc.

The situation is similar to a first order phase ^tran-

sition when a cubic term appears in the expansion ^of

the free energy. In section 4 we shall demonstrate that in our case no such quadratic ^{term is} present ^and for similar symmetry properties every ^even power of A vanishes _{in eq.} (1.1) (a ^{= c =} 0...). One is left with a discussion on the sign of the coefficient b :

-

For b 0 one can reach a stationary ^stable

solution

which decreases as (R - Rc)1/2 ^{close to} the threshold.

This case corresponds ^{to a} direct bifurcation and reminds a second order phase transition behaviour where the order parameter would be the velocity.

The behaviour is compatible ^{with mean field} theory

where the critical exponent fl ⁼ 1/2. ^{Here we shall}

Fig. ^2.

Different bifurcations. a) The bifurcation is inverse due to the non equivalence ^of fluctuations A and - A in the system.

b) The bifurcation is normal. The amplitude goes continuously ^to

zero as one approaches the threshold. c) The bifurcation is inverse.

A and - A are equivalent, ^{but the} system is unstable against ^distur-

bances of finite amplitude. The doted lines correspond ^{to an unstable}

branch. The full line corresponds ^{to a} stable branch.

not consider the thermal fluctuations close to the threshold as introduced by ^{several authors in} isotropic liquids [12], [13]. An estimation of these fluctuations close to the threshold of an instability ^induced by ^a

shear in a nematic liquid crystal indicates that they

are likely ^not observable [10].

-For b>0

Eq. (1.1) ^{has two} kinds of solutions. One solution goes ^{to zero as Q} tends to zero and corresponds ^{to an}

unstable branch as indicated with a doted line on

figure ^2c.

For R > Ro ⁼ R, - b’12 yro ’, ^one gets ^finite

amplitude instabilities corresponding ^{to a stable}

branch (full line on ^{figure 2c). At 6 = 0 the amplitude}

is A ⁼ ± Jbjy ^{(y is supposed to be} positive). ^This

case corresponds ^{to an} inverse bifurcation and hyste-

resis is expected ^between Ro ^and R,,. This situation may be compared ^{to a} first order phase transition.

An important point ^{in order to} determine the nature

of the transition is the sign of the b coefficient. The

condition a ⁼ 0, ^b 0 is necessary ^to get ^{a normal}

bifurcation. This condition will be the right ^{one if one}

proves that the threshold determined by the linear

theory ^{is the correct one. A} criterium of global stability

such as the _energy method [6] ^will give ^{such a} proof.

But as long ^{as the} experimental ^results agree with those determined by the linear theory ^{we shall assume this}

last point ^{to be true. Then the nature} of the bifurcation will be determined by ^the sign ^{of the} coefficient b.

Let us now indicate how we shall proceed ^{for such a}

determination. From a mathematical point ofview, ^the

resolution of the problem depends strongly ^{on the} boundary conditions. For isotropic liquids ^{a stress-}

free boundary ^model (we shall refer as free-free case)

is _very popular since it leads to very simple ^mathema-

tical solutions [9]. ^{In nematics the} free free ^{case also}

leads to simple analytical solutions. In section 2 we

shall develop ^the analysis ^{of the} free-free ^{case and} give

the solutions of the linear problem. ^{We will also cha-}

racterize the critical behaviour and determine the two

quantities ro and 03BEo. ^{In section 3 we} present ^the general features of the non linear formulation. In section 4 we develop ^a perturbation analysis ^{in order}

to demonstrate that R(l) ⁼ 0. Then we present ^an approached method in order to compute ^{the dis-} turbance amplitudes ^{and to} determine the nature of the bifurcation.

2. Critical behaviour of the linear régime.

^From

a formal point ^{of view, the} hydrodynamic equations governing ^a disturbance _away from the basic flow

can be written under the form

where £ is a linear differential operator ^{and NU} represents ^{the non} linearities. U is a vector notation.

In our case the U components (p represent 0 ^the

(03B8 ~ vz) ^vz

temperature disturbance 0, ^the angle (p between the director and the x direction (Fig.1 ) ^{and the} velocity v,,.

As we are concemed with a stationary instability, DUIat ^{= 0.}

Linearized hydrodynamic equations governing ^the stability of the conductive state have been given ⁱⁿ

ref. [5]. ^{Let us} define the temperature ^{at the lower} plate

as

and at the _upper plate ^as

AIT > 0 (or AT 0) ^{will l} correspond ^{to a} sample

heated from below (or ^from above). ^The temperature

gradient ^at the threshold is

Using eqs. (2.1), (2.2), (2.3), (2.4) and notations of ref. [5] ^{one can} explicitly write the linear problem

£U(1) = 0 as :

with

The operator 04 has been obtained after elimination of the _pressure from the force components fx ^and fz.

11 p is defined by ^{the above} expression. ^{The transverse} velocity v,, also disappears ^{with use of the} continuity equation

where the derivate a/ax has been used in order to eliminate _vx.

The second line corresponds ^{to the heat} equation.

The third one reads or yi ax ^where fy ^{is the total torque}

(viscous ^{+ elastic} + ...) ^{exerted on} the molecules for which one assumes an isotropic elasticity ^cha-

racterized by ^an elastic-constant K. At threshold a

periodic distortion _appears and one looks for solutions which are periodic ^{in x with wave vector qx. For the} planar ^case, the x direction is defined by ^{the molecule}

orientation at the _upper and lower plates ^{of the} sample.

Solutions of system (2.4) depend ^{on the} boundary

conditions. Exact solutions have been given ^{in ref.} [5].

In order to get ^{tractable non linear} calculations, ^we shall not use the exact solutions but consider some

simplified ^model. Simple ^{solutions (one mode in z)}

are obtained if one considers a stress-free model where all real boundary conditions are taken into account except ^{on the} transverse velocity vx. This model usually calledfree-free ^{case has been} largely

considered in isotropic liquids [14], [15]. ^{The corres-} ponding boundary conditions are :

An exact condition on vx should be :

Such a free-free model largely simplifies ^the compu- tations, since one easily ^obtains simple analytical

solutions. This model defined by eq. (2.6) ^is quite

reasonable for the planar ^{case where the transverse} velocity ^{does not} play ^a very important role, ^since the dominant viscous torque acting ^on the molecules is a avZ. ^{On the} contrary ^{in the} homeotropic ^{case the}

~x

dominant viscous torque ^q is ll2 ² az. ^{~z Viscous effects}

are not well described in a model where the z depen-

dence of the _vx velocity profile ^is badly ^{taken into}

account. Then this model is not appropriate ^{to describe}

the homeotropic configuration. ^{In that case one should}

solve the exact boundary ^condition (2. 7) ^on Vx.

Solutions of _eqs. (2.4) ^with boundary ^conditions (2.6) ^are, apart ^{from a} normalization constant,

where

The continuity eq. (2. 5) imposes

2. 1 THRESHOLD. - The compatibility condition of system (2.4) ^{leads to the} marginal curve as a function of the parameters x ⁼ qx/qz

where

The result (2.9) ^{is an} approximation valid when

ka ~ > K (~ ^{is a} typical viscosity coefficient). ^This

last condition is realized for all real systems.

A typical marginal ^{curve is drawn on} figure ^3.

Computation ^{has been} done for MBBA where nume-

rical constants are heat diffusivities

elastic constant A ⁼ 6 x 10-’ CGS and viscosities

a6 ^is determined by the Parodi relation [18]

Fig. ^3.

Theoretical marginal ^curve of the thermal instability ^{in a}

nematic sample ⁱⁿ a free-free ^{model. àT} > 0 (or 0) corresponds

to a sample heated from below (or above). ^{Point B} corresponds ^to

the threshold with circular rolls..x ⁼ qx/qz.

Point B corresponds ^to the threshold minimum value of ~T of the simplified ^model

It only gives ^a crude estimation of the exact threshold

given ⁱⁿ [5] ^which corresponds ^to (~Tc d 3) = ^{2.6 x 10- 3} and Xc ^{= 0.95. The} discrepancy of ^{the threshold comes}

from the fact that a little difference in the estimation

of the qz ^{wave vector modifies} strongly (- q4z) ^the

threshold value. For small values of x, expression (2.9) changes ^its sign ^with ax, then point ^{A would} correspond

to an instability ^induced by heating from above (DT 0), ^{with a structure of} rolls, deeply ^flattened

in the x direction. Such a situation has also been

reported ^{in an} approximate resolution of the linear threshold with use of a Galerkin method written to the lowest order of the expansion [16]. ^{In such flattened}

rolls the velocity gradient ~vxl~z ^is largely ^enhanced.

Such a situation cannot be reasonably ^considered

in a simplified model where the _v,, velocity ^is very

badly sketched. Then in the following ^we shall restrict

our attention to circular rolls (À-x 1’-1 ^{2 d)} correspond- ing ^{to the B} point ^{of the} diagram and omit the A point

which should be studied in a model taking ^{into account}

a more realistic boundary ^{condition on} v,,.

2.2 CRITICAL SLOWING DOWN. - Near the threshold the intensity of the scattered light depends

on fluctuations of the director curvature. It is then easy ^{to measure} the decay ^or growth ^{rate of such}

disturbances. Near the threshold one expects ^{a linear}

growth ^{rate of} disturbances Q behaving as R - Rc.

Let us write them as :

where 03B8(1), ~(l) ^and v(1) z ^are the solutions given ⁱⁿ eqs. (2.8).

0’ is the smallest root of the equation

(one looks here for ^Q real).

Eq. (2.11) ^is of the third order in 0’, but the root

width the smallest 1 a is easily ^determined

Using ^the inequality k~/ k >> ^{1 and the} expansion

of the marginal ^{curve close to} the threshold :

where dqx = qx - q, (q,, ^is the critical wave vector

corresponding ^to the critical value Xc ⁼ qc/qz), ^one

obtains

To, which defines the critical slowing ^{down, is obtained} by identification with _eq. (2.12)

where close to the threshold the last term inside the brakets reads

For circular rolls where x - 1 the last term is

negligible. ^{The sum of the two first terms} of eq. (2.14)

is normally positive. ’t 0 is ^{scalled by the time}

characteristic of the orientation relaxation.

For the critical value xc ^{= 0.795,} 7:o/T ^{= 0.35.}

2. 3 INFLUENCE LENGTH.

The stability marginal

curve defines the interval of wave vectors qx close to qc where the system ^is linearly ^unstable (Fig. 3). ^{At the} threshold Rc, ^{for an infinite} (without ^lateral walls) box, ^{the wave vector is} well defined and is equal ^to qc.

For a finite box the amplitude ^of fluctuations is not any ^{more constant} in an horizontal box. It needs to

adjust ^{itself to lateral} boundary conditions. The influence of lateral boundaries can be characterized

by ^{an influence} length 03BE which behaves like [11]

where Ço ^{results from the expansion of the} marginal

curve close to qc (eq. (2.13)).

After some algebra ^{one obtains :}

with

This leads for x _{= Xc} to

3. Perturbation approach ^{to non} linearities.

In the case of stationary convection ^0’ is real. The steady

state corresponds ^to the solution of

At the threshold Rc ^{the solutions U(1) of the linear} problem ^{are defined} by :

Above the threshold, ^{at a} value defined by ^{R, U(l) is}

neither solution of the linear problem :

nor solution of the non linear one (eq. (3 .1 )). ^The

non linear terms in _eq. (3.1) ^are very complicated, particularly ⁱⁿ liquid crystals. So instead of solving exactly eq. (3.1) ^{we look for} approximate ^solutions.

Different methods can be used. We shall indicate the basic line of two of them we shall use in the follow-

ing. ^A perturbative ^{method used} very often [14], [15]

has allowed to solve the non linear Rayleigh-Bénard

case. It is sketched as follows : one searches solutions U at a given ^{value R,} developed ^{as a series of a small}

parameter 8

Eq. ^{(3.1) has to} be solved at each order of the

expansion. ^To the first order this reduces to eq. (3.2) and U(l) is the solution of the linear problem. ^To

higher order, ^a solubility condition of _eq. (3.1 ) reads at order n :

where U(l) is the solution of the adjoint problem :

g(n) denotes the nth order ^term of _g.

In the following ^we shall consider a roll system with the axis parallel ^to the y direction and then all

quantities ^will be y independent.

The scalar product is then defined as :

Ax ^{is the} wavelength ^{in an} horizontal plane, d ^{is the} sample thickness. In the following ^{we shall use the}

notations :

Solutions 0(1) of the adjoint ^{t + are defined with the} following boundary conditions :

and solutions of system (3.6) ^with boundary ^condi-

tions (3.5) read :

where

A complete iterative resolution of the perturbative problem ^will proceed ^{as follows : to} the jth ^{order the} solvability ^condition (3.4) determines R(j-l). The

resolution, ^at the jth ^{order of} eq. (3.1) gives ^the

solutions U(j) and so on. In the next section we shall

only ^{write the} second order term of eq. (3.4) ^{in order}

to show how the symmetry properties of the convective flow allow to set R(l) ⁼ 0 and then justify the absence of a quadratic ^term in eq. (1.1). ^{In order to determine} the leading ^{term of} the disturbances written as

R-R R(2) C ^{R(2) one should solve, , to the second order, , eq. q (3.1) ( )}

in order to determine U(2 ) and report ^{the solutions} in eq. (3.1) ^{written to the} third order in order to calculate R(2). This calls for an exact solution of eq. (3.1). ^Such perturbative analysis ^{will be} developed

in another paper. Here ^we shall restrict our attention

to a much simpler method which only ^{needs the} knowledge of the solutions U(l) and allows to deter-

mine ^{in an} approximate ^{manner the} disturbance

amplitude and indicates the nature of the bifurcation.

This approximation follows Stuart’s method [8] ^as exposed ⁱⁿ Chandrasekhar’s book [17]. Instead of

solving exactly eq. (3.1) ^{one searches} approximate

solutions U. One assumes that, above the threshold, the motions and the distortions of the liquid crystal ^are

similar to those appearing ^{at the} threshold, ^{as is} experimentally observed. Then disturbances will be taken of the form

U = AU(l).

Furthermore, ^one takes into account the modifica- tions of the mean > vertical thermal gradient ^due

to non linearities. This last point ^{will be} exposed ⁱⁿ

detail in the next section. But the principal ^{feature is}

the following : just ^at the threshold the mean

vertical thermal gradient ^{is constant. Above the}

threshold this gradient ^{is no more constant in the} sample thickness but is reduced by convective effects.

In an isotropic liquid this modification depends ^{on the}

ratio of the thermal diffusion time Tth ^{and of the}

convective time Tconv ^{and reads} [17] :

where g(z) ^{is a} function of z depending ^{on the fluc-}

tuations profiles ^at the threshold. A similar modi- fication will also _{appear in} nematics. But we shall

see that the essential non linearities will concern the orientation disturbance which is the slowest relaxing

variable.

4. Non linear régime.

Non linearities in liquid crystals ^{are much more} complex ^{than in} isotropic liquid. ^{First we} get ^non linearities due to convective terms as in isotropic liquid. ^{The term} v.V(v) ^{can in} fact be neglected ^{since we are} dealing ^with sample

of large Prandtl number (P - 103). The thermal convective term v.V(T) will reveal also negligible compared ^{with non linear terms} characteristic of

liquid crystals. ^{The non} linearities are due to the fact that :

-

the viscous stress tensor depends ^{on both the} velocity gradient and the director n [18],

-

the elastic stress tensor depends ^{on n,}

-

the transport properties ^are anisotropic (orien-

tation dependent).

For example ^{let us} give ^the expression of the viscous stress tensor T’ in the geometry ^{we consider :}

where

Let us now write the jth ^{line denoted as} (N) j ^{of the non linear term} N(U). Taking ^{into account the elastic}

forces, ^{one obtains}

In a same way the torque equation ^{leads to :}

Let us now recall the heat conservation equation,

written for an incompressible ^{fluid as}

Here the heat flux is defined as JQ/ pCv ^{= - k.VT}

and the components of the thermal diffusivity ^tensor

k are

where ka = k Il - k, ^and kl| ⁽¹⁾ is the heat diffusivity parallel (perpendicular) ^to the molecules. D is the

dissipation.

Non linearities in the term div (vT) ^are negligible

compare ^to those of the term div (JQ). ^{Indeed the}

first term contains terms of order 0vz ^{to be} compared

to terms of order k a ⁰ Ô9. _~x Disturbances ~ ^{are of order}

Aq v ^(see the third line of the linear operator ^corres- kq

ponding ^{to the} torque equation). ^Then

Indeed this is true as long ^as the focalization effect of the heat, characteristic of nematics, is efficient and is no more valid if kl ⁼ k il

For similar reasons the dissipation, which contains terms of order r¡q2 v2, ^is negligible compared ^{to the}

term div (JQ) ^{which contains terms of order}

Then the non linearities of the heat equation ^reduce

to those of the term div (J.). ^{Contrary to the case of}

an isotropic liquid the convective term v.V(T) ^does

not contribute in nematics neither in linear problem

where the focalization effect is the dominant one nor

in the non linear one.

4.1 R(l) DETERMINATION.

Assuming ^a develop-

ment as given by eq. (3.3) ^{one writes} eq. (3.4) ^{at the}

second order in 8.

The temperature ^is developed ^{as :}

The non linear part of the heat equation ^{reduces to}

The condition (3.4) ^{reads :}

03B8(1), ~(l) vl) all ^{behave like , f = sin q x cos qz z.}

After some integration eq. (4.9) ^{reduces to}

Let us consider for example ^{the first term of the}

above equation

1 t corresponds ^to the scalar product ^{of an even function}

of _z, af - cos q, x cos z, with an odd one

~x

(1) ~~(1) ^.2

v(l) ~~ (1) sin2 _qx x sin _qz z cos qz ^z

Vx(1) ox sin qx x sin qz ^{z cos} qz ^z and so drops ^out.

It can be shown easily ^{that for} symmetry ^reasons there is no contribution due to the force and torque

equations.

As f| ^{As B1} ~~(1)/~x> ^{ié 0 it} turns out that IJ.T(1) ⁼ 0.

This justifies the absence of a quadratic ^{term A 2}

in _eq. (1.1).

4.2 AMPLITUDE DETERMINATION.

In stead of

solving ^{all the} perturbation system, ^one supposes ^as

indicated in section 3, that above the threshold the flow pattern and the distortion are not very different from those at the threshold. Then we shall _suppose all the disturbances of the form U ⁼ A U(l) where

now we have to evaluate the amplitude ^A.

Above the threshold the mean temperature gradient aT/oz) is reduced by the flow motion. Let us now

write the temperature ^as

with

Let us point ^{out that 0 is now} different from the definition given ⁱⁿ eq. (4.8). ^{Here we} get 0 ) ^{= 0}

and corrective terms due to non linearities above the threshold are included in To which behaves like

F(z) takes deviations from a uniform gradient ^into

account. Taking ^{the mean} value > of the heat eq. (4. 7) which reduces to ( ^div (JQ) ) ^{= 0, one gets :}

One can integrate ^this equation ^{with the} boundary condition :

This leads to the expression ^{of the mean thermal} gradient

Let us point ^{out that in an} isotropic liquid the thermal gradient above the threshold is reduced by ^convective

effects due to the term v. V(T) ^{and reads}

In the nematic the non linearities affect much more the orientation which is the slowest relaxing ^{variable. Let us}

now write _eq. (3.4). ^The only contribution to the left hand side of eq. (3.4) ^comes from the heat equation (second

line of the linear operator). Using expression (4.5), (4.6) ^and (4.7) ^which give (N)1, (X)2 ^and (N)3, ^{one obtains}

after some integrations :

Scalar products ^are computed ^{with use of} expression (2.8) ^and (3. 7) for the first order functions.

The relation between the angle disturbance 9 ^and (OT - 0394Tc) ^{is :}

where

with h ⁼ ka/k 1..

The nature of the bifurcation depends ^{on the} sign

of the coefficient C. For C > 0 (C 0) ^{there is an}

indication of a direct (inverse) bifurcation. The sign

of C depends ^{on the value} of the critical value _Xc of the marginal ^{curve. For} MBBA with viscosities defined in section 2 one finds a positive coefficient

corresponding ^{to a normal} bifurcation. The a3 vis-

cosity coefficient strongly depends ^{on the} temperature.

Close to the Nematic-Smectic A transition _03B13 changes

its sign. ^{Then we have} explored ^this range of values of 03B13 in order ^to look if the bifurcation remains normal for all values of the viscosity coefficients. We have considered some hypothetical nematic where the viscosities oc4l 03B15 ^are kept fixed and a2 and 03B13 ^are allowed to vary with the constraint that Onsager

relations were not violated. The procedure is then for each value of a2 and U3 ^to determine the critical value Xc of the marginal ^curve (2.9) corresponding ^{to an} instability ^{with almost circular rolls} (point ^{B of the} figure 3), ^one finds in CGS :

So it seems that for MBBA the bifurcation is always

direct for the circular roll instability. ^Of course, for other saniples with other viscosities one should verify

this point, using ^{the above} procedure ^{for all values of}

the visoosity coefficients. Let us point ^{out that the} free free ^model correctly ^describe samples ^{in which} 1 03B13 1 « I a2 1. ^{For the other} limit 1 03B13 | 1 - I a2 1 ^one should take into account the exact boundary ^condition

on v,.

5. Conclusion.

A linear theory taking ^into

account realistic boundary conditions leads to an

exacte estimation of the threshold [5]. But in that case

disturbance profiles ^are numerically ^{and not} analy- tically determined. In this _paper we have studied non

linearities above the threshold. As non linear study implies ^some heavy algebra ^{we have} developed ^an analysis ^{of a} simple ^model. Following ^the approach

used in isotropic liquid ^where a free-free model leads to a reasonable estimation of the real phenomena,

we have performed ^an analysis ^of a free f ree ^model.

We have obtained _very simple analytical ^solutions.

This has allowed the rapid determination of the critical behaviour. The critical slowing down is characterized

by ^{a time} io, scalled by the relaxation time of the nematic orientation. We have also determined the correlation length ç 0’ characteristic of the side walls effect. This free-free model is suitable ^to describe the

instability ^{in a} planar nematic-sample ^{but not in a} homeotropic ^one. Indeed the free-free case corres-

ponds ^{to an} unrealistic boundary ^{condition on} v.,.

This is not very important ^{in the} planar ^{case with}

usual viscosities where the viscous torque ^{due to the} velocity gradient a3 oVx/OZ ^{does not} play ^an important

role. We have studied non linearities close to the

threshold but in an approximate ^{manner. Instead of}

computing the solutions above the threshold we have

used a simplified ^{method as} inspirated by ^{Stuart and}

Chandrasekhar. To this aim we have assumed that the disturbance profiles above the threshold are not very different from those at the threshold. Then we have calculated the corrections due to the non linearities to the mean vertical gradient. ^With the free-free ^model

this approximate method has led to very tractable

computations. We have obtained an estimation of the fluctuation amplitude and of the nature of the bifur- cation.

This _paper presents ^{a first} approach ^{to the} study ^of

thermal instability above the threshold the next step will be to give ^a description of these effects with a

model taking ^{into account the realistic} boundary

conditions and a complete perturbative approach.

In conclusion, ^this paper is a preliminary attempt for the comprehension ^of phenomena ^{above the}

threshold. This is a necessary step ^{to the} understanding

of the transitions beyond the linear threshold. The

analysis ^{of the} stability of such solutions would

provide ^the description ^{of the whole cascade} up ^to the turbulence.

Acknowledgments.

^{We want to thank H. Lekker-} kerker, ^P. Manneville, ^C. Normand, ^{Y. Pomeau} for fruitfull discussions.

References

[1] DUBOIS-VIOLETTE, E., C.R. Hebd. Sean. Acad. Sci. Paris 21

(T 275) (1971) ^923.

[2] DUBOIS-VIOLETTE, E., GUYON, E., Mol. Cryst. Liq. Cryst. ²⁶ (1973) 193.

[3] PIERANSKI, P., DUBOIS-VIOLETTE, ^E., GUYON, E., Phys. ^Rev.

Lett. 30 (1973) ^16.

[4] LEKKERKERKER, H., J. Physique ³⁸ (1977) ^277.

[5] DUBOIS-VIOLETTE, E., Solid State Commun. 14 (1974) ^767.

[6] JOSEPH, D., Stability of fluid ^motions (Springer ^{tracts in Natural} Philosophy, Berlin) 1976, ^{vol. 27.}

[7] LANDAU, L., LIFSHITZ, E., Fluid Mechanics (Pergamon ^Press, Oxford) ^1959.

[8] STUART, ^J. T., ^J. Fluid Mech. 4 (1958) ^1.

[9] NORMAND, C., POMEAU, Y., VELARDE, M., ^{Rev. Mod.} Phys. ⁴⁹ (1977) 581.

[10] MANNEVILLE, P., J. Physique ³⁹ (1978) ^911.

[11] WESEREID. J., POMPAU, Y., DUBOIS. M., NORMAND, C., BERGE, P., J. Physique ³⁹ (1978) ^726.

[12] GRAHAM, R., Phys. ^{Rev. Lett. 31} (1973) 1479 ; Phys. ^Rev.

Lett. 10 (1974) ^1762.

[13] HAKEN, H., ^{Rev. Mod.} Phys. ⁴⁷ (1975) ^97.

[14] MALKUS, ^W. V. R., VERONIS, G., ^J. Fluid Mech. 4 (1958)

225.

[15] BussE, ^F. H., Rep. Prog. Phys. ⁴¹ (1978) ^1929.

[16] VELARDE, M., private communication.

[17] ^{See the} appendix ^I of the book of CHANDRASEKHAR, S., Hydro- dynamic ^and Hydromagnetic stability (Clarendon Press)

1961.

[18] PARODI, O., ^J. Physique ³¹ (1970) ^581.