Orientational instability of nematics under oscillatory flow

(1)

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Orientational instability of nematics under oscillatory flow

A. Krekhov, L. Kramer

To cite this version:

A. Krekhov, L. Kramer. Orientational instability of nematics under oscillatory flow. Journal de

Physique II, EDP Sciences, 1994, 4 (4), pp.677-688. �10.1051/jp2:1994155�. �jpa-00247991�

(2)

Classification

Physics Abstracts 61.30G 47.20

Orientational instability of nematics under oscillatory flow

A-P-

Krekhov(~)

and L.

Kramer(2)

(~) Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany

(~) Physics Department, Bashkirian Research Center, Russian Academy of Sciences, 450025 Ufa, Russia

(Received

27 October 1993, revised 5 January1994, accepted 6 January1994)

Abstract. Generalizing _a recently introduced approximation scheme valid when the director relaxation time is large compared to the inverse frequency of _an oscillatory flow, we derive equations for the time-averaged torques _on the director for _a prescribed plane shear flow. Whereas for _a linear flow field

(simple

Couette flow) the _average torques vanish, _one has for Poiseuille flow

(and

_more general flow

fields)

torques that tend to orient the director essentially

perpendicular

to the flow plane. For flow-aligning materials the orientation parallel to the flow is also

(weakly)

stable. Including the effect of homeotropic surface alignment _we estimate the threshold of the oscillation amplitude for the out-of-plane transition. The results _are essentially recovered

land improved)

by direct numerical simulations.

1. Introduction.

Liquid crystals

exhibit

interesting

^flow

phenomena

due to the

coupling

between the director and

velocity

field. The flow

properties

of _a nematic

liquid crystal

_are characterized

by

Leslie viscosity coefficients _al,

, a6, of which two, _a2 ^and 03, describe the

coupling

between flow and director orientation [1-3]. ^For a

steady velocity

field

viz) along

the z-axis

(plane

shear

flow)

the director will tend to

align

in the shear

plane ix

_y

plane)

at _an

angle

bfl ₌ tan~l

(a3 /02)~/~

with the z-axis if _a3

la2

_> 0, ^which is

usually

the _case [4, 5].

However,

in _some materials _one has

a31a2

< 0

(especially

_near the transition to a smectic

phase)

and then instead of flow

alignment

_one has _a

tumbling

motion [6]. ^In ^the ^usual ^situation ^where the director is

aligned

at the surfaces of _a

liquid crystal layer

such

hydrodynamic

_torques _can lead _to orientational instabilities. A _recent review of these effects is

given

in [7].

When the

velocity

field oscillates

periodically

^and

symmetrically

around _zero the situation becomes _more

complicated.

In _a

given

an account of much of the

existing

work _on this

subject,

which _we will not repeat here [8]. ^We ^have ^also ^introduced

a method based _on _a two-time scale

analysis

to extract the slow

averaged dynamics

of the director under _a

prescribed oscillatory

^shear ^flow ^field. The director motion _was restricted

(3)

to the shear

plane.

In this paper we

generalize

the treatment to include

out-of-plane

director motion. The

general averaging

method for the director

equations

_are

presented

in section 2 and

we

apply

it in section 3 to the

simplest small-amplitude plane

Poiseuille flow.

Surpflzingly

the

problem

_can

again

be solved

analytically.

For the _case of

homeotropic

director

alignment

the critical flow

amplitude

for the

out-of-plane

transition is calculated. The numerical simulations

presented

in section 4 confirm the

analysis

and extend them _to

arbitrary

flow

amplitude.

In

section 5 _we comment _our results with _a view _on

possible experiments

and

give

_an outlook for future work. Some detailed

analytic

formulas _are

presented

in the

Appendix

A.

2. Basic

equations.

We consider _a nematic

layer

of thickness d confined between two infinite

parallel plates.

The z-axis of _a Cartesian coordinate system ^is ^chosen normal to the

bounding plates,

the

oscillating

flow in the z-direction and the

origin

in the center of the

layer.

With this choice _one _can write for the director i1 and

velocity

Q

n~ = cos b _cos

#,

_n~ ₌ sin

#,

_n

z = sin b _cos

#,

U~ "

U(t, Z),

_Uy "

U(t,

Z), _Uz _" 0,

ill

where

b, #,

_u, _u _are functions of t and _z. In

general,

a full solution of the four

coupled_

hydrodynamic equations

for

b, #,

u and _u [1, 2] with

time-periodic boundary

conditions for

u(t, z)

at z =

+d/2 (Couette flow)

_or

time-periodic

_pressure

gradient 8P/8z (Poiseuille flow), u(t,

_z

=

+d/2)

_m o and appropriate

boundary

conditions for

b, #

is

quite

_a formidable task.

For

simplicity

_we will here treat

u(t, z)

_as _a

prescribed 2x/w-periodic

function in t with

u(t

₊

x/w,z)

=

-u(t, z),

_so that the

time-average

<

u(t,z)

>= 0. In the _same

spirit

_we set _u

=

0. This should be _a

good

approximation ^for small oscillation

amplitudes A/d

< I

(A

is the maximum

displacement

in the

oscillatory flow)

because the director distortion is small.

Moreover,

for

larger amplitudes

the

approximation

should still be reasonable for low

frequencies

such that the viscous

penetration depth fi,

where _o is _a

typical viscosity,

is

larger

^than

the thickness d. Then the flow field is distorted

by

the fact that the effective shear

viscosity

is

z-dependent

_as _a result of the director distortion. The distortion is

(presumably)

not very

significant,

except

maybe

for Couette

flow,

where the undistorted flow field

(constant velocity gradient)

does not lead to any effect

(see below).

With the dimensionless variables

I

= wt, I

= z

Id,

fi

=

u/dw, ₍₂₎

the

equations

^for

b(t, z), #(t, z)

_can _now be written _as [1-3]

~,t

~(~)U

z ~~[~ll(bi

~)~,zz

+

a2(bi

_~)~~z ~ ~l3(~i

~)~,zz

₊

a4(~i _~)~~z

_~

a5(~i~)~,z~,zji (~)

#,t K'(b)M(#)u,z

₌

e~[bi(b, #)b,zz

+

b2(b, #)9)~

+

b3(~, ~)~,zz

+

b4(~,

_@~~z +

b5(~, @~,z~,zj, (~)

where the tiides have been omitted and

K(b)

=

(I

cos~ _b sin~

9) Ill Ii, M(#)

=

~

sin

#

_cos

#,

2 1

=

o31a2,

e~

=

I/(Tdw),

_Td

='fid~ _/Kii, ₍₅₎

(4)

the notation h, ₊

8h/8i, h'( f)

₊

8h/8 f

has been used

throughout

and

a,(b, ii, b;(b, #)

_are

al16, ii

₌ cos~b ₊ k2

sin~

_b ₊ _(k3 _k2

sin~

_bcos~

#, a216, #)

₌ _[k2 + (k3

k2)

^cos~

ii

sin b _cos

b, a316,

_WI

= k2

ii

sin b _cos b tan

#,

a41b,

_WI ₌

(2k2

k3

iii

sin b

cosb,

a516,

_WI ₌

-2[cos~

b ₊ k2

sin~

b ₊

2(k3

k2

sin~

_bcos~

#]

tan

#, bi16,

_WI = (k2

ii

sin b _cos b sin

#

_cos

#,

b216,_WI = [sin~^b + k2 cos~b ₊

2(k3 k2) sin~

bcos~

#]

sin

#

_cos

#,

b316,

ii

= sin~b ₊ k2 cos~b ₊ (k3

ii ^sin~

bcos~

#,

b416, 11 =

-(k3 ii ^sin~

b sin

#

_cos

#,

b516,11

₌ ^-2 ^sin ^b cos

b(k2

k3 cos~

# sin~

_WI,

(6)

and

k,

₌

K,,/Kii.

Since the

frequency

of

oscillatory

flow _w is

usually

much

larger

than the inverse director relaxation time I

/Td (I /Td

~

10~~ s~~ for d

= 100

pm)

_we _may _assume e~ < I.

Neglecting

in _a first step the terms on the

right-hand

side of

equations (3, 4),

the

remaining

first-order

ordinary

differential equations can be solved and the solution bo is obtained

implicitly

from

/°° (

=

glt,z), glt, z)

=

/~

^dtu

^z(t, ^z), ⁽⁷⁾

so that bo _"

bol~, glt, z))

is _a

periodic

function in t. Note that for the

flow-alignment

_case,

1 > 0,

K(b)

vanishes at b

=

+bfl

and

(7)

cannot be used for _~

= +bfl

(then

_bo ₊

+bfl).

In

principle,

the

b-integral

in

(7)

can be solved

analytically

and

bo(~, g)

_can be

expressed

in terms of

elementary

functions _[9] but for _our _purpose this is not needed. Then the solution

lo

_can be obtained from

~° /l~

-

~ dtK~(b°)Uz

= in

ill°1

⁽⁸¹

and

#o(x,~,g(t,z))

is also _a

periodic

function in t. We choose the

origin

of t such that

g(t

₌

o)

₌ 0 _so that _one has

bo(t

=

0)

₌ _j

#o(t

=

o)

₌ _x.

Thus,

from

(7), (8)

bo ^oscillates around _~ and

lo

around _x,

but,

in

general,

< bo

>#

_~ and <

lo >#

_x. We conclude from

(7), (8)

that

neglecting

the elastic

coupling

leads to _a continuous two-parameter

family

of

periodic

oscillations of b and

#

that _can be

parametrized by

the

"phases"

_~ and _x. The

only

fixed points

are #o _" o, bo ₌

+bfl

and

#o

=

x/2.

One

easily

_sees from

equation (8)

that the

trajectories

in the

#,

b

plane

_are

given by

~°

~~

~~~/

^~~~

In

figures

la and 16 _we have

plotted

the

trajectories

for two _cases with I > o and I < o,

respectively.

The three

trajectories correspond

to ~ =

x/2

and _x

=

x/18, x/6

and

x/4, but,

as seen from

equation (9),

other combinations of _~ and x lead to the _same _curves.

Depending

on the

amplitude

_a

only

part of the _curves _are

actually

traced out

by

the director. For

large

a and I _> o the oscillations saturate at the

flow-alignment angles

whereas for I _< o the director motion is in

principle

unbounded

(extend Fig.

lb

periodically).

A similar behavior

was obtained

previously

_[9].

In order to

investigate

the influence of the elastic

coupling

in equations

(3, 4)

_we _use the method of

multiple-scale analysis

_[10].

By introducing

_a '~slow" time T

=

e~t,

that modulates

JOURNAL DE PHY~)QUE II -T 4 N'4 APRIL 19~4 ~

(5)

r/4 y=n/4

_,,...,__

=~~~8

_."" '".._ ~)

n/6 :/

,---,, "_

.~ ," ',

./ ,' ',

:" l' ',

n/12

,' ',

/ ,

l~o

0

6tj

__... ....___

:.." "". b)

:. ...

:." "...

:.' "..

n/6 _.." "...

,

...

:" ,' ,, "..

:.' ,, _, _..

, , ...

, ^,

, ,

n/12 ,' ',

,,

o

0 n/2 R

6~

Fig. I. Trajectories in #o,_Ho plane obtained from equation (9) for 1 > 0

(a)

and 1 < 0

(b).

the

periodic

behavior _on the "fast" time

scale,

_so that b

=

b(t,z,T), #

=

#(t,z,T)

and

at

_- at +

e~8T,

_one _can formulate _a

systematic perturbation expansion

of the form

b

= bo +

ebi

₊ e~b2 +

,

#

=

#o

+

e#i

+ e~#2 +

,

(lo)

where all functions

b,,

#, are

periodic

in t. At order e° _one has the solutions

(7, 8)

where the

"phases"

_~ and x are now allowed to

depend

_on _z and slow time T and _are undetermined at this order. At first order in _e _one has

~

~~~

~' ~

-K'~bo

)~j) _at _K'(bo ^M'(~o _)u zj'

~~~~

where L is the linear operator of the

perturbational equations

and _we _can choose

(bi

" o,

ii

"

o). Finally,

at order e~ _we find

~

b2 ~bo,T

+

albo,zz

+

a2b(,z

~

a3~0,zz

+

a4~(,z

~

5b0,z~0,z)

~2

~~0,T

+

blbo,zz

+

b2b(,z

~

b3~0,zz

+

b4~(,z

~

bsbo,z~0,z

~~~j

where _a, ₌

a,(bo,40),

b,

=

b,(90,40).

Since L has the two null

eigenvectors 8~(90,#o)

and

8~(90, 40)

_one has two

solvability

conditions for the

inhomogeneous

linear

equations (12)

which take _on the form

<

V+[F

_>=

0,

<

W+[F

_>=

0, (13)

where F is the

right-hand

side of

equation (12)

and _we define the scalar

product

<

V[U

_>=

) ^~~~~ _dt(ViUi

+

V2U2). (14)

K t

(6)

Here

V+,

W+ _are _two

linearly independent

null

eigenvectors

of the

adjoint

operator L+

L+(j+,j+)=0, L+(W/,W/)=0,

L+

-

(K,,ilii~llll°1"

^~

a~

ri~g~~~,~l~~u l

(IS)

One

easily

finds

V~

=

(),°)

,

W~

=

~-g~~((

,

)) ₍₁₆₎

,n ,n ,x o,x

Using

the fact that

(90, lo) depends

_on _z and T

only through

_~, _x and _g, equations

(13)

_can

be _cast into the form of evolution

equations

for the

phases

_~ and _x

~J,T = Bi ₊ Ci~J,z +

Dix,z

+

Ei~J,zz

+

Fix,~~

+ GiJ~)~ ⁺

Hix)z

+

IiJ~,zx,z (17)

x,T =

82

+ C2~,= +

D2x,z

+

E~~,z~

+

F~x,~~

+ G~~)~ ⁺

H~x)z +12~,zx,z (18)

where

B,, C,,..

,

I~ are scalar

products involving

the functions _~, _x,

90, lo,

_g,z and _g,zz. The

expressions

_are

given

in the

Appendix. Equations (17, 18)

describe the slow evolution of the

phases

_~ and x. Bi, 82 _are in

general

_nonzero _{if g is}

spat1ally varying,

I.e. if _u

zz

#

0

(see Appendix).

Nonzero Bi and 82 _means that there _are net bulk torques

acting

_on _~ and _x. If

spatia1variations

of _~ and _x

play

_no role the stationary solutions of

(17, 18)

_are

simply given

by

the _zeros of

Bi,

82.

3.

Small-amplitude oscillatory

Poiseuille flow.

Let _us consider _more

explicitly

the _case of small

g(t, z),

which

essent1ally

amounts to A

Id

< I, where A is the maximum

displacement

in the

oscillatory

flow. Then _one has from equations

(7, 8)

for 90 and

#o

_up to the second order _{in g}

90 = ~J +

K(~J)g

+

K~(~J)K(~J)g~

+

,

40

₌ _{x +}

K'(~J)M(x)g

₊

)lK"(~J)K(~J)

⁺

K/~(~J)M'(x)lM(x)g~

₊

₍₁₉₎

The

simplest

situation is obtained for

plane oscillatory

Poiseuille flow:

u =

a(4z~ 1)

cost, _g = 8az sin t.

(20)

Then Bi, 82 have _no

explicit

_z

dependence

and for

spat1ally

uniform _~, x equations

(17, 18)

with

(19, 20)

lead _to

~,T "

a~B1(~, X),

_Xi

"

a~B2(~, X), (21)

where the functions

fli(~/, x), fl2(~/, x)

_are obtained from the

general expressions (Appendix) ti _{(~, x)}

=

3211ai (~, x)

+

a5(~, x)M(x)iK'(~)K(~)

₊

a2(~, x)K~(~)

+a3(~, x)M(x)iK"(~/)K(~/)

₊

K'~(~/)M'(x)1

₊

a4(~/, x)iK~(~/)M(x)i~i,

£2(~, x)

₌

3211bi (~, x)

+ b5

(~, x)M(x)iK'(~)K(~)

₊

_b2(~, x)K~(~)

+b3(~, x)M(x) ill" (~)K(~)

+

K'~ (~)M'(x)1

₊

b4(~, x) iK'(~)M(x)i~i ₍₂₂₎

The _a,, b, _are defined in

(6).

(7)

For

in-plane

motion, I-e- _x

= o one has

82(x

₌

0)

= 0 and

fli _lx

"

0)

=

16[(cos~

_{~ +} k3

sin~ ~)K~ (~)l'. (23)

In the

flow-alignment

_case

(I

>

o) fli(x

=

o)

has three _zeros

corresponding

to two stable

equilibria

at ~ = 0 and _~

=

x/2,

and _an unstable

equilibrium

at the

flow-alignment angle

~ = 9fl ₌

tan~~(a31a2)~/~

_[8]. For

non-flow-aligning

nematics

(1

_<

0)

there remains an

unstable

equilibrium

at ~ = 0 and _a stable _one _at _~

=

x/2.

By linearizing 82

around _x

= 0

((x(

«

ii

_one _can

study

^the

stability

with respect to out-

of-plane

motion. For the

typica1relations

between the elastic constants

K22

_<

Kii

<

K33

and

(I(

< I _one finds that in the _case 1 > 0 the

equilibrium point (~

₌ 0, _x ₌

0)

is stable with respect to

out-of-plane

motion and

(~

= 9fl, _x ₌

0), (~

₌

x/2,X

"

0)

_are unstable. For the

case 1 _< 0 the

in-plane

motion is

always

unstable.

Out-of-plane equilibrium points

of

equations (21)

_can be

easily

found for _~

=

x/2.

Then

Bi(~

_"

x/2)

= 0 and

fl2(~

"

x/2)

=

~~

sin x cos

x[(k3

k2 + 1 k2

lk3)

cos~ _x _I

sin~ x] (24)

(1- 1)2

has _zeros at x =

0, x/2

and for 1 > 0 also at

xo "

tan~~(~/(k3

k2 + k2

lk3)/1). (25)

Finaliy

another

equilibrium

point

(~i,

_xi

),

with ~i " 9fl, _xi _*

x/4

_appears in the _case 1 _> 0.

Complete phase diagrams

which show the

trajectories

of the system

(21)

_are

plotted

schemat-

ically

for the _cases 1 _> 0 and 1 _< 0 in

figure

2a and

figure 2b, respectively.

For 1 _> 0 _one has

two attractors, ^but since for

typical

nematics 9fl is

small,

_one expects ^the ^solution

(~

₌ 0,_X

"

0)

to be _very

weakly

stable

(small

^domain of

attraction) compared

to

(~

₌

x/2,

_x

=

xo).

For

1 _< 0

only

the fixed

point

_x

=

x/2

is stable. In both _cases

(~

₌

x/2,

_x ₌

0)

is unstable with respect to out-of-

plane

motion.

The above

analysis

shows the

possibility

of _a

(spatially homogeneous)

transition to out-of-

plane motion,

but it cannot

give

_a threshold because

(stabilizing) boundary

conditions _on the

director _are not included. We _now take into account

homeotropic boundary alignment

in _an

approximate

_manner within _our scheme.

Returning

to the evolution

equations (17, 18)

and

linearizing

with respect to x around

(~

₌

x/2,

_x

=

0)

_one obtains with

(19, 20)

x,T "

a~bx a~izx,z

₊

/x,zz, ₍₂₆₎

where

"

~i

~jj2

(~3 ~2 + ~2 ~~3 Ii

?

"

11

~>)2

l~~~~ + ~~

~~l'

/

= k3

(27)

The

boundary

conditions for _x at z = +

1/2

_are

x(z

= +

1/2)

= 0. The modal solutions of equation

(26)

_are of the form

x =

exP(aT)U(z) (28)

and

u(z)

satisfies the differential equation u"

fl2zu'+

_au

= 0,

u(z

=

+1/2)

₌ 0,

(29)

(8)

n/2 _n/2

I x

z z

0 0 O O

0 6jj n/2 0 n/2

aj ~j

Fig. 2. Phase diagrams for the spatially uniform solutions of evolution equations for 1 > 0

(al

and 1 _< 0

(b).

where

o =

a~il / _al /, fl

=

a~il /. ₍₃₀₎

2 With the transformation _u

=

exp(flz~ /2)w(z)

this equation _can be reduced to

w" ₊

[a

+

fl fl~z~]w

₌ o,

w(z

=

+1/2)

= o.

(31)

It is well-known that for this

problem

there is _a minimum

eigenvalue

_o +

fl

that is real and

positive.

The solution of

(31)

can be

expressed

in terms of _a confluent

hypergeometric

function

ill]

~° ~~~~ ~~~

~~'~ ~' '~~~~'

_~~~~

From the

boundary

conditions for _w _one has the

following

equation for

o(fl)

The

dependence

of

a(fl)

obtained from the numerical solution of

equation (33)

is

plotted

in

figure

3. From

equation (30)

^follows a =

cfl

a where _c

=

2ili,

=

al /.

_Thus _for

a >

cfl

one has a _< o and

perturbations

_{of x}

decay

in

time,

while for _a _<

cfl

the

growth

rate is positive. The critical value fl~ and the

corresponding

threshold

amplitude

_a~

=

fill

_of

the

oscillatory

^flow _can be found from the intersection of the universal _curve

a(fl) (Fig. 3)

and the

straight

line _a

=

cfl

whose

slope depends

_on

liquid crystal

material parameters. For

example,

with the MBBA

liquid crystal

_parameters Kii _" 6.66 _x

lo~~~, K22

_" 4.2 _x

lo~~~,

K33

" 8.61 _x

10~~~N,

₀₂

" -lloA _x

lo~~,

a3 " -1.I x

lo~~Ns/m~ _[12,

_13]

one has _a~

= o.4.

Thus for _a _< _a~ the

growth

rate is

negative

and

in-plane

motion is stable

becoming

unstable for _a > a~. From equation

(27)

_one _sees that I has _a small influence _on the threshold _as

long

as

iii

< 1.

(9)

1o o

80

o= cJ

60

~

40

20

~~0.0 _5.0 _{lo 0} _15.0 _{20 0}

fi

Fig. 3. Dependence of

a(fl)

corresponding to the minimum eigenvalue equation

(31).

4. Numerical simulations.

In order to test the

analytical

results direct finite-difference simulations of the basic

equations (3, 4)

for the Poiseuille

velocity

field

(20)

_were

performed. Homeotropic boundary

conditions

for the director 9

=

x/2,

# ₌ 0 at z =

+1/2

and MBBA parameters were used. We indeed found the

instability

of the

in-plane

motion. The

out-of-plane

distortion of the director _can be described

by

the

angle

#m

corresponding

to the

midplane

_z ₌ o

(there

the

angle

is time

independent

since for Poiseuille flow

u,z(z

=

o)

₌

0).

In

figure

4 _we show

ii

_as _a function of the flow

amplitude

_a for different

frequencies

of the

oscillatory

flow. Near threshold _one has

#m

_+~

(a a~)~/~

_so the bifurcation is

supercritical leading

to a continuous

change

of #m with _a.

The critical

amplitude

_a~ decreases

slightly

with

increasing

flow

frequency,

but remains

higher

than the value _a~

= oA obtained from the

analytical approximation.

The

discrepancy

^is not

alarming given

that in the

analytic approach

_we assumed a~ _< _I. _Moreover the

homeotropic boundary

conditions for 9 _are

only

^satisfied in the average in the

analysis.

Numerical simulations of

equations (3, 4)

with

homeotropic boundary

conditions for prescribed

oscillatory

Couette flow

u =

a(z

₊ _cost

(34)

2

confirms that there is _no

out-of-plane instability

_up to

high

flow

amplitudes.

5. Discussion.

Our

investigation

shows that for

plane oscillatory

Poiseuille flow in _a

homeotropically

oriented

sample

_one has _a transition to

out-of-plane

motion

through

_a

supercritical

bifurcation. Our

treatment is valid in the

frequency

_range

I/Td

< w <

a/pd~

where _o is the effective shear

(10)

06

- f=10 Hz

- f=50 H2

H- f=100 Hz

-f=500 Hz

0.4

cv E

£+

0.2

~'~

55 60 0.65 0.70 0 75

a

Fig. 4. Dependencies of

#(z

= 0) _on the amplitude of oscillatory Poiseuille flow for different

frequencies.

viscosity, and then the threshold

amplitude

is

proportional

to the

layer

thickness d and

indepen-

dent of _w with corrections of the order

(Tdw)~~/~

due to the

boundary layers.

For

flow-aligning

nematics the

homogeneous

orientation

parallel

to the

velocity

^field is

linearly stable,

but

large

fluctuations in the flow

plane beyond

the

flow-alignment angle might

destabilize the state

(see Fig. 2a).

For such materials

(homogeneous)

orientation

perpendicular

to the flow

plane

_can in

principle

also become

unstable,

but the

hydrodynamic

torques _are small in this situation and the attractor _near

by.

For

non-flow-aligning

nematics the orientation

parallel

to the

velocity

field _can become

unstable,

but

again

the

hydrodynamic

torques are small. For such materials the orientation

perpendicular

to the flow

plane

is

absolutely

stable.

Clearly

these results differ

considerably

from what is known for

steady

_{flow [7].}

We have not

investigated stability

with respect to

spatially periodic

fluctuations. From the results for

elliptic

shear flow [14] we expect that

out-of-plane

motion _can

undergo

^such _an

instability leading

to a

periodic

roll pattern.

For

simple oscillatory

Couette flow _no

instability

is found within _our framework. This _ap- pears ^to remain true when

spatially periodic

fluctuations _are considered [14] ^unless ^inertial

terms _are

included,

which

surprisingly

_appear to be

responsible

for _an

instability

at low fre-

quencies (down

to below loo

Hz) [15-17].

Actually

inclusion of inertial terms into _our treatment is

straightforward

_as

long

_as the oscillation

amplitudes

_are small. Then the restriction _w _«

a/pd~

is _overcome and in the

high- frequency

_range the

small-amplitude approximation

is in fact

expected

to be valid. Then also

oscillatory

Couette flow should lead to average torques ^and

possibly

to _a

spatially homogeneous

reorientation. Work in this direction is in progress. Moreover _we

hope

to be able to

generalize

our method to include corrections to the

velocity

field

resulting

from director distortions in _a self-consistent _manner.

Clearly

it would be

interesting

to test _our

predictions experimentauy,

and _we

suggest doing

(11)

this with

homeotropic

director

alignment.

When

oscillatory

Poiseuille flow is induced in _a thin slab in the usual way

by application

of _a pressure difference _one must be _aware of the fact that _as _a result of

compressibility

the _pressure

decays along

the flow direction _z _on _a

length

b

+~

d/pc~/(3wa))

where _c is the sound

velocity

and _o _a

typical

shear

viscosity

_[18]. Since b _» d for all reasonable

frequencies, experiments

^should nevertheless be

possible.

The

decay

is avoided when Poiseuille flow is induced

by oscillating

in

parallel

both

plates

of _an

open-ended

slab.

Finally

^let us

point

out that effects related _to the _one treated

here,

where

systematic

forces result from

oscillatory

and

spatially varying

excitation, _are known in Plasma

Physics

under the _name of

"ponderomotive

forces"

(see

_e.g.

_[19]).

Acknowledgments.

We wish to thank A. Buka and H. Schamel for

helpful

discussions. Financial support ^from Deutsche

Forschungsgemeinschaft (SFB213

and

Graduiertenkolleg

"Nichtlineare

Spektroskopie

und

Dynamik", Bayreuth)

_are

gratefully acknowledged.

One of _us

(A.K.)

wishes _to thank the

University

^of

Bayreuth

for its

hospitality.

Appendix.

The

solvability

conditions

(13)

with definition ofthe scalar

product (14)

and

linerly independent

null

eigenvectors

^of the

adjoint

operator

(16)

take _on the form

~

)~0,T

>"<

)[a190,zz

⁺ ^a29~,z ^~ ^~l3~0,zz ⁺

^a4~~,z

^~ ~l590,z~0,z) >1

(~~)

,n ,n

~

)§~0,T

^> ^<

^~~~

^90,T ^>-<

)[(bl

jl'~al)~0,zz

⁺ ^(b2

l'~a2)~~,z

~

,x _, ,x ,x ,n ,n

(b~

jjj a~)io,zz

+ (b~

jj~ ^a~)ii,z

⁺ ^(b~

jj~ a~)90,zio,zi

>,

(36)

, ,n ,n

where _a,

=

a~(90, lo),

b, ₌ b~(bo,

lo)

_are defined in

(6)

and < F _>=

) f/~~~dtf.

From equations

(7, 8j

_one has bo ₌

bo(~,gl,

40 ₌

40(x,~J,gl,

where _J~

=

J~(T,zj,

_x

=

x(T,zj,

g =

g(t, z)

and

b0,T

"

b0,q~,T, ~0,T

_"

~0,q~,T

+ §$0,XX,Ti

bo,z = bo,~~J,z ⁺

bo,gg,z, 40,z

= 40,~~J,z ⁺

40,xx,z

+

40,gg,z,

bo,zz ₌ _bo,~~J,zz + bo,~~~J)z ⁺ 2bo,~g~J,zg,z +

bo,gg,zz

+ bo,ggg)z,

40,zz

₌ _40,~~J,zz + 40,~~~J)z ⁺

40,xx,zz

+

40,xxx)z

+

2jo,~x~J,zx,z

+

2jo,~~J~,~g,~ +

2jo,~~x,zg,z

+

40,gg,zz

+

40,ggg)z. (37j

Then for the coefficients in evolution

equations (17, 18)

_one has

i~l

_<

)ial(~0,g§,zz

⁺ ^b0,gg§~z) ^~ ~l2b~,g§~z ^~

~l3(~0,g§,zz

⁺ _~0,gg9~z) +

,n

a4~~,g9~z ^~ a5b0,g~0,g9~z) ~i

1~2 " <

)I(~l ^)~~ ~ll)(b0,g9,zz

~ b0,gg9~z) ^~ (~2

l'~a2)b~,g9~z

~

, , ,

(12)

(~3

~'~a3)(~0,g9,zz

+ ~0,gg9~z) ⁺ (~4

~'~a4)~~,g9~z

~

o,~ o,~

(b5

°'~a5)90,g40,ggizl

>, o,~

Ci = < 12ai bo,~g +

2a2bo,~bo,g

+

2a340,~g

+

2a440,~40,g

+

a5(bo,~40,g

+

bo,g40,~)lg,z

>, o,~

C2 - <

H12(bi Iii ^ai)bo,na

+ 2(b2

Iii

a2)bo>nbo,g + 2(b3

ta3)40,ng

⁺

2(b4

fia4140,~40,g

⁺ ^(b5

(°'~ asl(bo,~40,g

+

bo,g40,~llg,z

>,

o,~ o,~

Di = < 12a340,xg +

2a440,x40,g

+

a590,g40,xlg,z

>, o,~

~2

" <

)[~(~3 _jl'~ ^a3)~0,Xg

^~ ^~(~4

_jl'~ a4)~0,X~0,g

+ (b5

jl'~ a5)~0,g~0,X)9,z

~'

,x ,n ,n ,n

El = <

)ia190n

⁺

^a3'0ni

^>,

~~ " <

fi~~ IIl ^l~°~

+ ~~~

IIl ^~~~'°~l

>'

~1 1

" <

fa3~0,X

~'

o,~

' ~ ~~

~~~

~~ ~~

Gi

= <

£la190,w

+ a~91,~ ⁺

a340,w

+

a441,~

⁺

asbo,~40,4

_>,

o,~

G2 # <

fii~bi _Illl ^al190nn

⁺ ~b2

Illl ^a2191n

⁺ ^~b3

IIl ^a31'°nn

+

(b4

~°'~

a4)#(

₊ _(b5

^~°'~ a5)90

n

lo

_n) _>,

90~

ⁿ

90~

Hi

o,~

H2 = <

ox on o,~

Ii

o,~

12 #

<

_fi12~b3

~ Illl

a3)'0nx + 2~b4

-

Illl a4)'°n'°x + ~b5

~

Illl 5)90n'0xi

~~

/

^~

^/(9)~

_~'

~~

/

^~

j/(b)l~

^~' ^~~~~

which

gives

~°'~

~~~~'

^~°'~

^~~~°~'

^~~~~

(13)

Analogously,

an ~° /l~j

=

an In l~l~~l ^ax

£~° /l~j

- °,

(41)

[j'° ^d# ^(bo)j

~~

x

M(41

^~~ ^~~

K(~)

40,n

=

M(40)~'(jj)~~, ^40,x

⁼

(~)j~, ^40,g

⁼

M(40)K'(bo). (42) j~~(~i~/~~~

~~~~~~~~~~~

~°'~~'~°'~~' ~°'"'~°'~~' ~°'~~'

_~°~~x,

40,gg, 40,no, 40,xg

are

simply

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23.

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