Steady poiseuille flow in nematics : theory of the uniform instability

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Steady poiseuille flow in nematics : theory of the uniform instability

P. Manneville, E. Dubois-Violette

To cite this version:

P. Manneville, E. Dubois-Violette. Steady poiseuille flow in nematics : theory of the uniform in- stability. Journal de Physique, 1976, 37 (10), pp.1115-1124. �10.1051/jphys:0197600370100111500�.

�jpa-00208509�

1115

STEADY POISEUILLE FLOW IN NEMATICS :

THEORY OF THE UNIFORM INSTABILITY

P. MANNEVILLE and E. DUBOIS-VIOLETTE

(*)

Service de

Physique

^{du Solide et} de Résonance

Magnétique,

Centre d’Etudes Nucléaires de

Saclay,

^BP

2, 91190 Gif sur-Yvette,

^France

Résumé. ²⁰¹⁴ Nous étudions la stabilité d’un écoulement de Poiseuille dans un nématique planaire

relativement à une distorsion uniforme du directeur. La théorie hydrodynamique linéarisée _que nous

développons permet de caractériser la

symétrie

des modes instables. On s’attend à deux types ^de solutions suivant la présence ^{ou non} d’écoulement transverse

global.

La solution exacte, ^donnée dans un cas

simplifié,

^{montre que} l’instabilité avec écoulement transverse global ^a le seuil le plus bas,

en accord avec les résultats expérimentaux.

Abstract. ^- We study ^the stability ^{of a} Poiseuille flow in a planar nematic, ^with respect ^{to a uniform} distortion of the director. We develop ^a linearized hydrodynamic theory which allows one to charac- terize the symmetry of the unstable modes. One expects ^two types of solutions according ^{to the}

presence ^{or not} of a net transverse flow. An exact solution is given ^{in a} simple ^case which shows that the instability ^with net transverse flow corresponds ^to the lowest threshold in agreement ^with

experi-

mental results.

LE JOURNAL DE PHYSIQUE ^TOME 37, ^OCTOBRE 1976,

Classification Physics ^Abstracts

6.350 ^- 7.130

1. Introduction. ^- An

experimental description

^of

the Poiseuille flow

instability

ⁱⁿ nematics has been

given

^{in the}

preceding

paper

[1] (see

also ref.

[2]).

Experiments

^are

performed

^{on a} nematic oriented in a

planar

geometry

(director parallel

^to

Ox)

^and

subjected

^{to a} pressure

gradient

^{in the}

y-direction (Fig. 1).

^Both

steady

^and

alternating

Poiseuille flows have been examined. Two

types

^of

instability

^can

develop

^{in the}

sample.

^{The first one}

corresponds

^{to a}

distortion of the director which is uniform in the

xOy plane.

^{The second one leads to the} formation of convective rolls which have their axis

parallel

^{to the}

flow direction

Oy. Furthermore,

^as

long

^as

phenomena

above the threshold are not

involved,

^{the uniform}

instability

^{is found to}

develop

^{first for a} d.c. excitation.

FIG. 1. ^- Geometry : ^a pressure gradient 0394P/L ^is applied ^{in the} y-direction ^{on a} planar ^nematic (molecules along Ox).

In this paper, ^we

give

^a theoretical

description

restricted to the uniform distortion observed at threshold in the

steady experiments

^{and we leave to a}

forthcoming

paper the case of

alternating

^flows.

In this context, ^{we want to}

emphasize

^that

experiments reported by

Pieranski and

Guyon

^{in ref.}

[1]

^and

[2]

correspond

^to observations at and above the threshold.

In

particular, they

^{have seen two} types ^{of uniform}

distortion,

^{A and}

B,

characterized

by

^{the absence}

or presence of a net transverse flow in the x-direction.

Regime A,

^which

corresponds

^{to a} metastable state, is obtained from a

highly

^{distorted flow}

by decreasing

the _pressure

gradient

^{and is} outside the _scope of this article. On the other

hand, regime

^B is obtained from the

unperturbed

Poiseuille flow

by increasing

^the

pressure

gradient.

^The

instability

^which

develops

^is

linear and

stationary [3]

^{and can be}

analyzed

^within

the framework of a linearized

hydrodynamic theory.

Regime

^B is characterized

by

^an average twist of the director and

by

^a net transverse flow in the x-direction.

This

secondary

^{flow is} different from the one we had

predicted [4]

and which has been observed

[5]

^{in the}

simple

shear flow

experiment.

^{In that case the trans-}

verse velocity 03C5x

^{was an} odd function of z such that :

$$

This

corresponds

^{to a}

long

^roll

parallel

^to

Oy, occupying

all the width of the cell with no net flow V in the x-direction. In the present ^{case, a net flow} V ~ 0 is observed at threshold. This

secondary

flow cannot be sustained all

along

^the channel and there must exist

regions

^{where it}

develops

^{in one}

(*) Laboratoire de Physique ^des Solides, Universite de Paris-Sud,

91405 Orsay, ^France.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370100111500

direction and

regions

^{where it}

develops

in the other

one. This leads to the _presence of domains in the

y-direction (in

^{a frame}

^moving

^{with the}

^primary

^flow

(see ^Fig.

^{8 of ref.}

[1]),

^the

secondary

^{flow stream lines}

have to close in the

x0y plane).

^{Such a domain struc-}

ture is

closely

connected with the finite width I of the

experimental

^{cell but not with the}

instability

^mecha-

nisms. In the

following,

^{we shall}

neglect

these lateral

boundary

^effects

(i.e.

^{take the} limit of infinite

width)

and discuss the

instability by

^{means of a}

simplified

one-dimensional model.

2. Linearized

hydrodynamic equations.

^{- Let us}

consider the cell described in

figure

^{1 and call}

G ⁼

AP/L

^the pressure

gradient applied

^{in the}

y-direction.

^{Molecules are oriented}

(through

^surface

treatment)

^{in the x} direction. When the flow is _very

slow,

^{due to elastic}

effects,

^this

alignment prevails

in the bulk. Viscous

torques

^{exerted on} the molecules

are zero and the

viscosity

measured in this geometry is _lla

[6, 7].

^{For a} cell of infinite width

I,

^the

velocity profile

^{is :}

independent

^{of x}

(the origin

of the coordinates is taken

midway

between the

plates). However,

^{in an} actual cell of finite

width, vy

^may be considered as

independent

^{of x}

only

in the central part ^{of the}

cell ;

this central

part

gets wider and wider as the width I is

increased,

^{as can be seen in}

figure

²

(calculation

^for

an

isotropic liquid).

^{For narrow channels}

(typically lld 10)

^the

boundary layers

^{on each side are wide}

and the

velocity profile

^{has no}

longer

^the

simple

form

(2 .1 ).

^{In the}

following

^{we shall}

only

^consider

wide cells and examine the

stability

^{of the}

unperturbed plane

Poiseuille flow.

FIG. 2. ^- Transverse unperturbed velocity profile ^{at z = 0 for an} isotropic liquid Poiseuille flow in a channel of finite width I. Curves

are given for different values of the aspect ^ratio r = l/d, ^{r =} 2, 5,10, ^50.

Eq. (2.1)

^{defines a shear rate}

which is

positive

in the lower

half (z 0)

^{of the}

sample

and

negative

^{in the} upper ^one. At first

sight,

^{one could}

look for a

description

^{of the actual flow in terms of}

two

superposed

average

simple

^shear flows : 7 for

z 0 and - s for z > 0.

Then,

^{one would use the}

analysis developed

^earlier

[4].

However,

though

^the

instability

mechanisms are

unchanged,

^{the situation}

is somewhat different since the shear rate cannot be considered as

piecewise

^constant.

Furthermore,

the

boundary

conditions are different : in the

simple

shear

flow ^problem,

boundaries were

rigid planes;

here, the simple

^{shear s would be}

applied

^{between a}

rigid

^{wall at z = -}

d/2

^{and a}

free boundary

^{at z =}

0,

the other one - s between the free

boundary

^at

z ⁼ 0 and the

rigid

^{one at z = +}

dl2

^{and one would}

have a

continuity

^{condition at z = 0. This}

simplified picture

^cannot

give

^{more than a}

qualitative descrip-

tion but it stresses the

similarity

^{of the}

instability mechanisms,

and the linearized

hydrodynamic

equa- tions can be

immediately

deduced from those _govern-

ing

^the

simple

shear flow

problem (1).

Let us denote bn ⁼

(0,

ny,

nz),

^{ðv =}

(vx,

vy,

vz) and p

the

orientation, velocity

^and pressure fluctuations.

We do not discuss the roll

instability (ox 1= 0)

^and

the domain structure

(Oy 1= ⁰⁾

^{so that we}

^only

^consider

the

z-dependence

of the fluctuations. In this _case,

as for the

simple

shear flow

problem,

^{one can show}

that vy ^{= vZ = p = 0}

everywhere.

Moreover,

experi-

ments suggest ^{that the}

instability

grows ^at threshold without time oscillations so that we are left

with

^a

time

independent problem. ^Then,

the linearized

equations

^{reduce to :}

Torque equations :

Force

equation :

Boundary

conditions read :

Let us

give

^{here a}

rapid analysis

^of eq.

(2.2)-(2.4) referring

the reader back to ref.

[4]

^and

[8]

^{for a more}

complete description

of the mechanisms.

The viscous

destabilizing

contributions to the total

torque

a3 sny ^{and a2} snz

correspond

^{to the}

Guyon-

Pieranski

instability

^mechanism

(2).

^{The corrective}

(1) ^The only difference with equations written in appendix ^{A of}

ref. [4] ^{comes from the} z-dependence ^of s(z) in the Ericksen-Leslie tensor term QZx which contains s(z) _ny.

(2) This mechanism works only ^when a3 0 (a2 ^is always 0).

If a3 > 0, another mechanism leads to a roll instability [4, 8]. ^In the following ^{we shall assume} a3 0.

1117

term a3

0,-,V-,

ⁱⁿ eq.

(2.2)

^{is due to} the presence of a

transverse flow _{v.,. Two} mechanisms induce this transverse flow. One related to the twist distortion

(term s aZny

^{in eq.}

(2.4))

^was

already present

^{in the}

simple

shear flow

problem.

^{The other one due to the}

variable shear rate

s z term

ⁿ

ds

is characteristic dz)

of the Poiseuille flow

[9].

From a

general point

^of

^view,

^{it is}

interesting

^to

find the symmetry

properties

^of eq.

(2.2)-(2.4)

which are

compatible

^with

boundary

conditions

(2. 5).

From the form of these relations one may guess that non-trivial solutions will have a definite

parity.

Let us consider for

example

eq.

(2.3)

^and look for the solution at two

points symmetric

^{relative to the}

xOy plane.

^Since

s( - z) = - s(z)

^we

get :

Let us define the even

(e)

^{and odd}

(o) parts

^{of a} fluctuation

w(z) (where w

stands for ny, nZ ^or

vx) through :

Combining

eq.

which expresses that the even

part

^of ny ^is

coupled

^to

the odd one

of nZ

^and

conversely.

^{A similar}

separation

can be

performed

^{on eq.}

(2.2)

^and

(2.4)

^{which leads}

to the fact that a solution of

given parity (for example

ny, nz, ^vx)

^{does not mix with a} solution of the

opposite

one

(ny, ny, ^vx).

^This

^property

^can be extended to the

case of a

general

fluctuation

depending

^{on x,} y, ^z.

It has been summarized in table I : TABLE I

Two modes are obtained. The first one

(T)

^corres-

ponds

^{to a finite} average twist

(ny ^even)

^{and a net}

transverse flow

(f v,,, dz =A 0 .

The second one

(S) corresponds

^{to a}

finite

_average

_splay (nZ even)

^{and no}

net transverse flow

(vx odd).

^In

general,

^{these two}

modes will have different threshold values.

3. Threshold determinations. ^-

Performing

^the

transformation ^z

= d

Z 2

we obtain the dimensionless

equations :

where

is the Ericksen number

[6] characterizing

^{the flow.}

E

are two dimensionless numbers related to external fields.

Integrating

eq.

(3.3)

^we get :

where K is an

integration

^constant.

Replacing D vx

ⁱⁿ eq.

(3.1) gives :

and we are

left

^{with the}

problem

^of

solving

eqs.

(3.7), (3.2), (3.6) together

^{with the}

boundary

^conditions

Ny ^{= NZ}

⁼

^vx

^{= 0 at}

^{Z =}

^{± 1. The}

^general

^case

F # F’ can

only

^{be solved}

using

^an

approximation

method. On the contrary, ^{when F = F’ the combi-} nations :

further

simplify

^the

problem and lead

^{to an}

analytical

solution in terms of

Airy

^functions

[10].

^{Let us turn}

to this

simpler

^case.

3.1 NO EXTERNAL FIELD APPLIED

(F

^{= F’ =}

0).

^-

Combining

eqs.

(3. 7)

^and

(3. 2)

^we

get :

The

parity property

^{allows one to} separate ^a solution with average twist

(T)

^{from a} solution with

average

splay (S).

^{In the first case,}

Ny ^{and Yx}

^{are even}

functions of Z so that

ZNy

^and

^DVx

^{are odd : from}

(3 . 6)

^we deduce that K ⁼

0,

^{i.e. U and V are}

decoupled

and that

vx

^is

simply given by :

V,(O)

^is determined

by

^the

boundary

^condition

On the contrary

when NZ

^{is even}

(mode S)

^{it will be}

shown that K is different from zero and

actually couples Vx

^to

Ny

^and

^NZ,

3.1.1

Average

^{twist solution}

(Ny

^{even function}

of

Z, K

⁼

0).

^{- Let X =}

ZE//3. Eqs. (3 . 8), (3. 9)

now read :

the

general

^solution may be

expressed

^{in terms of}

Airy

functions

Ai,

^Bi

[10]. Recalling

^that

and

assuming

^that

N,

^{is an even} function of X

(or Z)

we get :

Boundary

conditions

Ny ^{= NZ}

^{= 0 at Z =}

^{+ 1}

^lead

to

The solution will be the trivial one A = B ⁼ 0 unless

Xl

^{is a root} of the characteristic determinant :

and the

instability

threshold will be reached for the lowest root

Xc :A

^{0 of} eq.

(3.14).

^This

equation

may be written as :

However,

^for

large

^{X :}

so that the I.h.s. of _eq.

(3.14) rapidly

^{decreases to}

zero when

Xl

increases.

Neglecting Ai(Xl)lBi(Xl)

in eq.

(3.15)

then leads to the

approximate

^{solution :}

or

a result which _{is very} close to the exact one

The

corresponding

Ericksen number is

in

good

agreement ^with

experimental

^results

given

in

[1].

^Once

Xf

^is

known,

^{one can} deduce the fluctua- tion

profiles

^at threshold from

(3.12)

^and

(3.10)

^with

B /A

^{extracted from}

(3.13)

To the same

approximation

^{level as for}

(3.16),

^we

simply

get :

A

The aspect ^{of the} distortion at threshold is

given

within an

arbitrary

^{constant in}

figure

^{3 which}

clearly

reflects the

parity

property. ^In

particular,

^the

velocity profile vx implies

the presence of a net transverse flow

compatible

^{with the domain structure}

experi- mentally

^observed.

FIG. 3. ^- Fluctuation profile for the T-solution with mean twist and net transverse flow (no external fields applied).

3 .1. 2

Average splay

^solution

(Ny

odd function

of Z).

- In this _case, if we assume K ⁼

0,

^{we are left with}

the

previous problem

^after

interchange

^of

Ny

^and

^N.-

The critical value

Xf

^is

unchanged but Vx

^{must be an}

odd function

of Z ; then, Vx(Z

⁼

0)

^{= 0 and}

This solution would no

longer

^{fulfil the}

boundary

condition

on Yx

^{so that the}

assumption

^{K = 0 is}

wrong.

1119

The

general

solution with

Ny

^{odd and}

NZ

^{even as}

functions of Z is :

Boundary

conditions on

NY

^and

^Nz

^{or U’ and V’}

at X ⁼

± X1

^{lead to :}

so that A and B are now functions of

Xl.

^But

Ny

^and

Nz

^still

depend

^{on the}

integration

constant K

(or k).

Replacing

in eq.

(3. 6)

^we get

and the

boundary

^condition

Vx(Xl)

^{= 0 leads to the}

compatibility

condition :

in order to get ^a non-trivial solution

(k # 0).

^In

contrast to the first _case, the lowest root of _eq.

(3.18) depends explicitly

^{on a} dimensionless parameter

characterizing

^the

liquid crystal : l1b/l1a’

The critical Ericksen

number Er

^{as defined}

by

eq.

(3.4)

^is

given

as a function of

(1 - l1bll1J

ⁱⁿ

figure

^{4 for 0} llb na’

Figure

⁵

displays

^the aspect of the fluctuations at threshold for MBBA with

It may be noticed that :

a) When na

= qb, the system ^{is unstable}

against

^a

distortion with average

splay

^{at the same critical}

value as for a distortion with average twist. In fact

vx

vanishes as can be deduced

directly

^from eq.

(3.3)

and the two types of distortion become

equivalent.

FIG. 4. ^- Critical Ericksen number for the S-solution with mean

splay ^{as a} function of

(I - 1-b)

for 0 I’/b na For MBBA na

1Jb/1Ja ^{= 0.572 and} Er’ ⁼ 17.313. The critical Ericksen number as

defined by eq. (3.4) ^is independent ^of nb/na ^{in the T-case.}

FIG. 5. ^- Fluctuation profile ^for the S-solution in zero external field.

b)

^The threshold value increases with 1 ^-

1Jbl1Ja’

In order to account for this variation one has to consi- der the contribution of the transverse flow to the

destabilizing

mechanisms. In ref.

[4],

^{we discussed}

these mechanisms in terms of effective viscous torques. ^{Written in a} dimensionless

form,

^these viscous torques ^{read :}

and the

mechanism

is

locally destabilizing

^{if :}

Consider first a distortion with _average twist. Then K ⁼ 0 and the shear rate

D Vx

^is

proportional

^to

ZNy.

The condition

(*)

is fulfilled

everywhere

^{in the}

sample.

On the other

hand,

when K # ⁰

(average splay distortion)

and the

sign

^{of the} contribution of the shear flow D

Vx

depends

^{on the}

position

^Z.

Looking

^{at the}

profile

^of

Ny

ⁱⁿ

^figure

^{5 as} obtained from the exact

calculation,

one can guess that

which has the

right parity

and satisfies the

boundary conditions,

^{will lead to a}

good approximate descrip-

tion.

Then,

and from the

boundary

^condition

The effective viscous torque

Fy

^{reads :}

In absence of

secondary

^flow

(1Ja

⁼

nb)

^this torque would be :

so that the contribution of

D Vx

^not

only

^reduces

Ty eff

^{but also can make its}

^sign change; leading

^{to a}

stabilizing

^{mechanism in some}

regions

^{of the}

sample.

This is

particularly

^{the case near the}

boundary plates

of the cell

(Z ~ 1)

^{where a}

high

shear leads one to

expect ^a strong

destabilizing

mechanism. Since

r zl Nz ~ -

^{Z the}

instability

^condition

(*) simply

reads :

The

region

^{where it is} fulfilled shrinks

when ’1b decreases,

^{so that we}

qualitatively

understand the increase of

E’

^with

1 - 17 b

_llaⁱⁿ

^figure

^4.

For all

physical

situations

with ’1b

na the dis- tortion which _appears

corresponds

^{to the lowest}

critical Ericksen

number,

^{that is to} say : ^a distortion with _average twist and transverse flow as observed

experimentally.

^Our

instability

^with average

splay

should not be confused with the metastable

regime

^A

described in ref.

[1].

^{As a matter of}

^fact, regime

^A

displays

^{an average twist but no} transverse flow at all

(Vx(Z)

⁼

0), keeping

^{the trace of a} feature which appears in the non-linear

domain,

^{since it is obtained}

by starting

^{from a}

large

distortion.

3 . Z MAGNETIC FIELD EFFECTS

(CASE

OF AN IDEAL NEMATIC WITH

K1

⁼

K2 = K; F

⁼

F’).

^-

Equations governing

^{U and V now read :}

with

where

is a Fredericks field

[6] (3).

For the

T-solution, boundary

conditions

again

lead to eq.

(3.14)

but with the

changes :

Eq. (3.18) corresponding

^{to an} S-solution is also

slightly

modified but we

again

expect ^a distortion with average twist at threshold and concentrate on this

case.

Eq. (3.15)

^{now reads :}

The

applied magnetic

^field

H, parallel

^{to the} unper- turbed

director,

^{has a}

stabilizing

^{effect and we} expect

Erc

^{to be an}

increasing

^{function of F.}

Then, Xl

grows with F and

consequently :

we may

neglect

the I.h.s. of _eq.

(3.20)

^which

simply

reads :

A;iY - Yci - n

or

Recalling

^that

and since

increases

with F,

^the

approximation

which leads to eq.

(3.21)

^{is more and more accurate. No such} appro- ximation exists for the other

type

^of

instability

^and

the

complete

calculation has to be worked out.

1121

Numerical results are

given

^on

figure

^{6. Once}

again

we expect ^the distortion with average twist which

corresponds

^{to the lowest} Ericksen number for all F.

Fluctuation

profiles corresponding

^{to this case are}

FIG. 6. ^- Critical

Ericksen

^{number as a function of the magnetic}

field measured the distortion with by ^{F =} mean

Xa H 2 (d)2 K 2

twist to ^for develop ^{the S and T} whatever the field is.^{case. One expects}

FIG. 7, ^{8. -} Fluctuation profile ^for the T-distortion at a given F=AO. Fig. ^{7.-F= 10 or} H ~ 2 Hc. Fig. ^{8.-F= 150 or}

H ~ 8 Hc.

given

^on

figures

^{7 and 8 for H N 2}

Hc(F ~ 10)

^and

H - 8

Hc(F ~ 150) respectively.

^The

aspect

^{of the} distortion at threshold _may be understood

qualitati- vely

^{as follows :}

- Opposite

^{to the case of the}

simple

shear flow

[4], here,

^{the shear rate s =}

aZvY

^is not constant across

the

sample.

^The

corresponding instability

^mechanism

is the

strongest

^{close to the}

plates,

then decreases and vanishes

midway

between them.

- On the other

hand,

^the

applied

field tends ^to

align

the director back

along

^Ox.

Across the thickness of the

sample

^we may then

-

distinguish

^{the central} part, ^{where the}

stabilizing

effect of the field

dominates,

^{from the}

neighbourhood

of the

plates,

^{where the}

instability

grows

[6].

This feature is obvious on

figure

^{8 for H - 8}

H,

but the

tendency

^was

already

^{indicated on}

figure

⁷

for H - 2

Hc.

^Note however that _eq.

(3.21)

^which

gives Ec(f)

^cannot be derived from a

simplified

model in terms of a constant shear across the unstable

layers.

3. 3 GENRatL CASE

(F # F’).

^{- Since no obvious}

variable or function

change

^{allows to come back to}

a system of the form

(3.19),

^we

develop

^an

approxi-

mate method in terms of Fourier

expansions [3].

The method which will be outlined below is _very

powerful

^{due to the fact that it takes into account}

the

parity

property ^{and the}

boundary

^conditions

simultaneously.

^{Let us} consider the case of a distortion with average twist

(Ny

^even function of

Z). Eqs. (3.7)

and

(3. 2)

^{now read :}

N,

is odd and must vanish at Z ⁼ ± ^{1 so that we}

assume :

Since _eq.

(3.23)

is linear we may consider

Ny

^{as a}

superposition

where

Ny(n)

^{are the solutions of}

which fulfil the

boundary

^conditions

... T IMB/r-7 I w -

The

general

solution of the reduced

equation :

hich is an even function of Z reads :

and a

particular

solution of the

complete equation

(3.26)

^{is :}

so that the solution we are

looking

^{for reads :}

Now

Ny

^and

^NZ

^must

^verify

^eq.

^(3.22).

^This

^{gives :}

The I.h.s.

of eq. (3.27)

^{reads as} the Fourier

expansion

of the function defined

by

the series at the r.h.s. In order to

identify

the coefficients we

multiply

^{both sides}

by

^sin

(pnZ)

^and

integrate

^{from zero to}

1 ;

^{then we} get :

This system may be written as follows :

with À

= Er 2, ^Then,

the coefficients

are understood as the components ^{of an}

eigenvector

^A

of the matrix A,

corresponding

^{to the}

eigenvalue À :

and the

highest

^real

positive eigenvalue Àc

^{will corres-}

pond

^{to the} critical Ericksen number

given by :

The exact solution is

given by

the infinite series

(3.24)

^and

(3.25);

^an

approximate

^{one will be}

given by

the truncated series :

- To first order v ⁼ 1 we have

- To second

order, Ac(v

⁼

2)

^{is the}

highest

^root

of the characteristic determinant :

- and so on.

Including

^{more and more harmonics leads to the}

exact value. When F ⁼ F’ ⁼

0,

the convergence is rather

quick

^{as can be}

appreciated

^{on table II.}

The reason for this

rapid

convergence is

already

apparent ^{from the} aspect ^of

Nz

ⁱⁿ

figure

^{3 which}

is _very close to a sine wave. For

F,

^F’

large,

^{when the}

fluctuation

profiles

look like those of

figure

^{8 one}

expects ^{a much slower} convergence but the method remains

quite

^efficient.

TABLE II

Average

^twist

F and F’ defined

by

^eq.

(3.5)

^{measure the}

strength

of the external

stabilizing

^{fields. In order to} compare the ideal case F ⁼ F’ to the actual one F -# ^F’ we

shall take the

geometrical

^mean

JFF’

^{as a}

^global

measure of the

stabilizing

^{forces. In} absence of elec- tric

field,

^{this choice is}

equivalent

^{to the}

change

^from

Ki, K2

^to

^K

⁼

J K1 K2,

^an

approximation

^which

leads to

good

^results every time the

anisotropy

^{of the}

elastic constant does not

play a

crucial role. Numerical results

plotted

^on

figure

^{9 are in}

good

agreement with the

experimental

^ones.

1123

FIG. 9. ^- Critical Ericksen number as a function of external fields measured by

for different values of F’/F : 1,1.5,2. All calculated points ^{fall close}

to the curve given ^{as S on} figure ^{6 which} corresponds ^{to F = F’.}

With

while the

experimental

^ratio is in the range 16-19.

It _may be noticed that the calculated

points

^for

Ki

^{= 1.5}

K2

^and

Kl = 2 K2

^are

practically

^{on the}

same curve as those for

K1 = K2 = k

^{which were}

obtained

^{much more}

simply

^{from eq.}

(3 . 21). However,

this fact does not reduce its

efficiency

^{to the method}

which will be

employed

^{in cases when no exact solu-}

tion can be found

(roll

instabilities described in ref.

[1])

or when the exact solution is too

complicated.

^{As an}

example

^{let us come back to the case of a distortion}

with _average

splay.

^Then

assuming

we may

deduce Yx

^{from eq.}

(3.6)

and from the

boundary

^{condition J}

and

Nzn)

^{is the} solution of _eq.

(3 . 7)

which fulfils the

boundary

^condition

N,(Z

^{= +}

1)

^{= 0.}

x

Replacing Ny

^and

^Nz = Y ^Nzn)

^{in eq.}

^(3.2)

^and

n=1

following

^{the same} steps ^{as before one obtains a} system ^{similar to}

(3.28).

^The

only

modifications are :

- the

interchange

^{of F and F’}

(with f

⁼

,FF),

- the addition of the term :

(which corresponds

^to the contribution of the inte-

gration

^constant

K)

^{to the} r.h.s. of

(3.28).

^{In the}

limit F ⁼ F’ ⁼

0,

^this

supplementary

^term

simply

reads :

Corresponding results,

^{when one} varies the order of

truncation,

^are

given

^{in table III. Once}

again

^the

convergence is rather

good,

which could be

expected

from the aspect ^of

Ny(Z)

ⁱⁿ

^figure

⁵ which looks like a

sine wave. However one should observe that the

limiting

^{value is reached}

from below, opposite

^{to the}

case of ^a distortion with average twist

(table II) (4).

TABLE III

Average splay (*)

(*) MBBA : nb/na, ^{= 0.572.}

4. Conclusion. ^- In this _paper, ^we have

given

^a

theoretical

description

^{of a} Poiseuille flow

instability corresponding

^{to a} uniform distortion of the director.

As for the

simple

shear flow

instability,

^{the Poiseuille}

case is well described

by

^{means of a} linearized

theory.

Locally,

^the mechanisms are similar in both _cases, which is revealed in the

expression

^of torques ^and forces

(Eqs. (2.2)-(2.4)). ^However,

^{the presence of a}

non-constant shear modifies the symmetry

properties

of the unstable modes. In the

simple

^{shear case two}

(4) ^{This seems to remove} part of the interest for a variational principle governing this kind of problem since such a principle

would not have a definite extremum property.

types of solutions

(ny, nz, ^vx)

^and

^{(n y, nz, vx)}

^could

exist. The

instability

^{threshold was} obtained for the first one which was associated with a lower elastic energy. In the present ^{case the}

parities

^of ny

and

are

opposite

^{in a}

given

mode. Solutions are of the form

(ny, ^{n’, vD}

^and

(ny, nz, ^v’)

^{and the elastic energy no}

longer plays

^a

prominent

part.

Actually,

^{it is the}

contribution of the transverse flow to the viscous torque ^which separates ^{the two solutions as discussed}

in §

^{3.1.2. When no} transverse flow is present

(1Ja

⁼

nb)

the

instability

thresholds are

equal

for solutions with average twist

(nye)

^or

splay (nz).

^{For the more realistic}

case nb na’ due ^to the _presence of transverse flow

one expects ^an average twist distortion associated with

+d/2 a net transverse flow

(f+d/2 dz Vx =1= ^{0 .}

Exact solu-

- d/2

tions _may be derived when

Ki

⁼

K2

^{and no electric}

field is

applied.

^{For the}

general

^{case we have}

develop-

ed a

powerful approximate

^{method which is}

being

extended to the case of the roll

instability

^{in alternat-}

ing

flows. Threshold values and the

predicted

symme- try of the unstable modes are in

good

agreement ^with

experimental

^results

presently

^available.

Acknowledgments.

^{- The} authors would like to thank E.

Guyon

^{and P. Pieranski at}

Orsay,

^{N. Boc-}

cara, and J. P. Carton at

Saclay

^for very

stimulating

discussions.

References

[1] Preceding paper.

[2] GUYON, E., PIERANSKI, P., ^J. Physique colloq. ³⁶ (1975) ^C1-203.

[3] CHANDRASEKHAR, S., Hydrodynamic ^and Hydromagnetic ^Sta- bility (Clarendon Press, Oxford) ^1961.

[4] MANNEVILLE, P., DUBOIS-VIOLETTE, E., ^J. Physique ³⁷ (1976)

285.

[5] PIERANSKI, P., ^Thesis Orsay (1976).

[6] ^{For a review see : DE} GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) ^1974.

[7] GÄHWILLER, Ch., ^Mol. Cryst. Liq. Cryst. 10 (1973) ^301.

[8] PIERANSKI, P., GUYON, E., Phys. ^{Rev. A 9} (1974) ^404.

[9] PIERANSKI, P., GUYON, E., Phys. ^{Lett. 49A} (1974) ^237.

[10] ^{Handbook of} Mathematical Functions, Abramowitz M., Stegun ^I. A., ed. (National ^{Bureau of} Standards) ^1972.