Linear growth of instabilities on a liquid metal under normal electric field

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Linear growth of instabilities on a liquid metal under normal electric field

G. Néron de Surgy, J.-P. Chabrerie, O. Denoux, J.-E. Wesfreid

To cite this version:

G. Néron de Surgy, J.-P. Chabrerie, O. Denoux, J.-E. Wesfreid. Linear growth of instabilities on a

liquid metal under normal electric field. Journal de Physique II, EDP Sciences, 1993, 3 (8), pp.1201-

1225. �10.1051/jp2:1993192�. �jpa-00247897�

Classification Physics Abstracts

47.20 47.65 68. lo

Linear growth ^of instabilities

_{on a}

liquid metal under normal electric field

G. N4ron de

Surgy(~)

,

J.-P.

Chabrerie(~)

,

O.

Denoux(~)

dud J.-E.

Wesfreid(~)

(~) Laboratoire de Gdifie Electrique de Paris

(LGEP)(*)

,

Plateau du Moulon, 91192 Gif _sur

Yvette Cedex, France

(~) Laboratoire d'Hydrodynan~ique et M4canique Physique

(LHMP)

_(**), Ecole S~lp6rieure de

Physique et Chin~ie Ind~lstrielles de Paris

(ESPCI),

lo _rue Va~lq~lelin, 75231 Paris Cedex 05,

France

(Received

ii December1992, revised 27 April 1993, accepted 6

May1993)

Rdsum4. On salt q~l'~ln champ 41ectriq~le appliq~l6 normalement I la surface Ebre d'~ln fl~lide cond~lcte~lr _{a ~ln} elfet d6stabifisant. En restant dons le cadre d'~lne th60rie lin6aire, _no~ls

4t~ldions ici le d4veloppement des instabilit4s 41ectrocapillaires, dons le _cas trbs g6n6ral off la viscosit6 d~lfluide et _son 4paisse~lr sont q~lelconq~les. No~ls d4duisons di1f4rents comportements dons divers r6gimes et donnons des 6q~lations de dispersion analytiq~les dons le _cas des couches 4paisses et minces, inertielles et visq~le~lses. No~ls pr4sentons alors _~ln diagramme des di1f4rentes

simplifications possibles et _{comme no~ls} d4d~lisons _ces simplifications directement de l'4q~lation g4n4rale il est plus facile de pr4ciser les limites de validit4 des hypothbses. No~ls montrons a~lssi les similarit4s et les di1f4rences _avec le _cas d'un liquide tombant d'un support solide plan

(instabilit6s

de

Rayleigh-Taylor)

_en absence de champ 41ectrique ainsi qu'avec les instabilit4s dons les ferrofluides.

Abstract, It is well known that _an electric field that is applied normally to the free surface of

a conducting ^{fluid has} _a

destabilizing

effect. Here

we study ^{the linear}

growth

of electro-capillary

instabiEties in the _very

generil

case where the viscosity of the fluid and its thickness _are of any value. We derive the asymptotic behavio~lr in various regimes and give analytical dispersion equations in the _case of thin _or thick, inviscid _or viscous films and _compare with previous results.

Then _we present _a diagram of the corresponding simplifications of the dispersion relation and

as we derive them directly from the general equation, _{we are} able to derive their conditions of

validity explicitly. We also show the simflarities and the differences with the _case of _a fiq~lid falling from _a solid flat plane

(Rayleigh-Taylor instabilities)

without electric field and with

ferrofl~lid instabilities.

(* ^{URA CNRS} N°127

(** ^{URA CNRS N°857}

1. Introduction.

We shall consider

(see Fig. I)

_a

conducting

and

incompre8sible

film of

liquid metal,

_mercury for

instance, laying

_{on a} metallic flat electrode. The undisturbed free surface of the

liquid

is at _{z =} 0 and the electrode _at

z = -a. Another electrode is

disposed

at _{z =} b, ^with vacuum for 0 < z < b and _a

potent1al

V is

applied

between the electrodes. The geometry ^{will be}

supposed

to be infinite for both _~ and _g.

The linear

stability

of such _a film under _a normal electric field Eo _"

V/b

has

already

been studied

(Tonks

_[1], Frenkel [2],

Taylor

[3], ^Melcher

[4-5],

^Nekrovskii

[6]).

The

study

of the thick

(a

and b

infinite)

inviscid _case with normal modes of

perturbation

of the type

((r)

₌

((~,y)

=

Aexp(st ik.r)

(where

_s is the

growth

rate of

perturbation

and

k,

the modulus of

k,

is the horizontal _wavenum-

ber)

is easy ^to

develop

and well

known, leading

to the

following dispersion

relation:

ps~

=

-Sk~

₊

eoE(k~ pgk (1)

where

p is the

density

of the

liquid,

S the surface tension of the

liquid-vacuum interface,

_go the

permittivity

of _vacuum and _g the

gravitational

^field.

It _can be _seen with this relation that _a destabilization of the

liquid

surface

(Re(s)

>

0)

occurs

only

above _a critical field

Ec:

Ec

₌ ^~

ll

eo associated with _a

critica1wavelength lc

lc

₌ 2gr

~ Pg

Let _us consider _a twc-dimensional

(~, z) problem,

where

((~)

₌

Aexp(st ik~).

In order to introduce the thickness effects _we shall first consider _a horizontal

length

scale l~ =

1/2gr (with

I =

wavelength

₌

2gr/k),

and _a

vertica1length

scale noted lz. The value of lz will be taken _as the lowest value between the film thickness _a and

1/2gr.

For

2gra/1

» 1 the film will be called thick

(the

vertical scale is then the

wavelength),

^for

2gra/1

< 1 the film will be called thin

(the

vertical scale is then the fluid

depth).

VaCUUIII ^Z

z=o ,fij ^{ill /I.. ii}

~

._.illi@1:(~' _g v

: a

~

~

x

Fig,I.

Schematic view of the experimental device.

thin thick inertial inertial

i

thin _thick

viscous _viscous

2na/~

Fig.2.

_Diagram of the different kinds of behaviours.

For the

study

^of the effects of

viscosity,

_we shall then consider _a

Reynolds

number _: Re

=

inertia forces

/

viscous forces. If Re _» I _we may

neglect viscosity

and call the film inertial _or

inviscid,

dud if Re < I _we may

neglect

inertia and call the film viscous.

We shall denote

by

_q the

dynamic viscosity

of the

liquid

and _v

=

q/p

the kinematic

viscosity

of the

liquid.

The

Reynolds

number is:

~~

~~~ l'~

q

(I/lz)~~~~l/lz)~j

U

Therefore _we have in the extreme _cases:

-for _a thick

film,

where the vertical scale is

1/2gr:

j2 _~j

~~

2v(2gr)2

^~ 2vk2 -for _a thin

film,

where the vertical scale is _a:

Re=

~a~jsj

v

This

Reynolds

number _compares the viscous

penetration length (a

fundamental

length

in

hydrodynamics)

_lvp

=

/@

with the film thickness and with the

wavelength.

It _can be

written:

~~~~~ ~2

~~ _~

[(21r/1)~ $(l/lz)~j

On the

diagram

of

figure

2 _we have noted the different kinds of

(asymptotic)

behaviour.

We _{can see} four different

regions.

The inviscid

(thick

and

thin) regions

_can be related _to the inviscid

dispersion

relation. In this article _we shall

give analytical

results for the two other

regions,

and also for the _case of

high viscosity

and moderate thickness. We extend the asymptotic

analyses

of

Rayleigh-Taylor instability

of

Hynes

_[7] and Limat _[8] to the electrc-

capillary instability.

The

general

_case, which takes into account the

viscosity

_v of the

liquid metal,

its thickness

a and the _vacuum thickness b between the free flat surface and the _upper electrode is not

always

^well treated.

Indeed,

_{some papers} concerned with this

problem ([6,

9,

10])

contain _a

faulty dispersion

relation.

Nevertheless,

_a correct

expression

for the inviscid _case and for the thick-film _case

(the

thickness _{a - cc} has

already

been

proposed.

For this last _case, instead of

punctua1numerical

treatments of the thick

dispersion

relation [9], we propose to show the

asymptotic

behaviour of _very viscous

liquids (corresponding

to the thick viscous _case in

Fig.

2)

for which the dominant

wavelength

will be shown to be

independent

of the

applied

field. We shall _{use a} method similar to that used in the

study

of

gravity

waves

by

Leblond and Mainardi

ill].

In order to compare the

respective properties

of different metals _or

alloys,

_we ^shall _use

dimensionless values: _we shall _see that the results

depend

_{on a}

capillary length

_lc and _a viscous

length

iv

(characteristic

of the

fluid),

on an electric

length

_l~

(characteristic

of the fluid and of the

applied potent1al V)

and _on the

geometrical lengths (thickness

_a of the

liquid film,

distance b between the fluid and the _upper

electrode).

We shall check that the

stability

domain

depends only

_on the

capillary

and the electric

length

values

(and

is therefore

independent

of

viscosity

and

geometry),

while

dynamics (I.e.

^{the linear}

selection of the most unstable

wavelength)

involves all the

lengths.

We shall also

point

out the

analogy

with the _case of _a fluid film in _a

gravitationally

unstable situation

(Rayleigh-Taylor instability)

studied in the works of Chandrasekhar [12],

Hynes

_[7]

and Limat [8].

2. The characteristic scales of the

problem.

Having

^{identified different}

regimes (thin

and thick

layers,

inertial and

viscous),

_we find it useful to compare the thickness _a with different

typical

^distances.

The

liquid

meta1is submitted _to

capillary,

viscous and electric forces. For each force _{we can} build _a characteristic

length. By comparing

the effects of surface tension

(Sk~)

to those of

gravity (pgk)

in

(I)

_{we can} construct _a

capillary length:

i~ =

si/2p-1/2g-1/2

This

length

is

always

of the order of10~~cm for the usual

liquids

in the normal

gravitational

field.

In _a similar way, we can compare the effects of the electric field

(eoE]k~)

to those of

gravity

to build _an electric

length:

le =

(eo/2)E(p~~g~~

For the critica1field

(Eo

"

Ec)

this

length

is

equal

to the

capillary length

and is greater ^than

lc for

larger

fields.

Near the critical field the

wavelength

is of the order of lc in the inviscid _case

(in

fact

equal

to

2grlc).

^{It will be} shown that it remains true at finite

viscosity.

For greater ^{electric fields the} value of the

linearly

most unstable

wavelength

includes _a combination of lc and l~, it will be

shown to be

usually

of the order of

I(/le

for le »

lc.

In fact _we have

le/lc

=

El /E(

which is the control parameter ^of our

problem,

and that will be noted 4l. Then _we shall _see that

lc/4l

scales the

wavelength

and that

increasing

the field will shorten the

wavelength (but

not in the thick viscous

case).

We _can also build _a viscous

length.

For this _we consider _a dimensionless number built

(as

_a

Stokes number for

particles) by comparing

_a time of stabilization due to

viscosity (Tsta)

^with

a time of destabilization due to

gravity

for _a free

falling drop

_(Tdesta)1

Table I.

-Physical

^{vdues for different}

liquids.

(g/cm~)

S

(dyn/cm)

_v

(cm~ Is) (cm)

iv

(cm)

d

Li 0.5 400 lA _x 0.88 1.2 _x 5.5 _x

Ga 6.1 718 3.5 _x

10~~

_{0.34 2.3 x}

10~~

_5A

x

10~~

Sn 7.0 612 2.8 _x

10~~

_{0.30 2.0}

x

10~~

x

10~~

Al 2.4 914 3.I _x

10~~

_{0.63 2.I x}

10~~

x

10T~

Au 17.3 l140 2.7 _x

10~~

_{0.26 2.0 x}

10~~

x

10~~

Hg

13.5 470 1-1 _x

10~~

_{0.17 1.0}

x

10~~

x

10~~

Silicon oil 1.0 21 10 0.15 0.56 5.7

Water 1.0 70

10~~

_0.27

10~~

x

10~~

fluid 1.0 10 0.5 0.1 6.3 _x

10~~

x

10~~

Tsta " l~

IV

Tdesta "

(I/g)~/~

~~

~a ^{li~~/~ ~/~~~~/~}

^~~~

where _a viscous

length

_{[13] can} be defined:

iv _"

V~/~g~~/~

This

length

is

only

determined

by

the

physical properties

of the fluid

(contrary

to the above introduced viscous penetration

length).

It is of the order of 10~~cm _for

a

liquid

metal like mercury ^at room temperature.

We shall write St

=

d~~

when

= lc:

d

=

(iv/ic)~/~

which will

play

_an

important

role in _our

problem.

The other

lengths

in the

problem

_are the

geometrical lengths

a and b, and the

wavelength

1

(= 2gr/k).

We _{can use} the different

length

scales to build dimensionless

values;

_we shall _use the

capillary

scale _as the reference scale _as it will be _seen that this scale is the _most

important

in

defining

the

wavelength.

We therefore expect

a/lc

and d to be cruc1al parameters. In table I _we

give

_a list of values of

density,

surface tension and kinematic

viscosity,

the

corresponding capillary

and viscous

lengths

and the

corresponding

value of the parameter d

(we

took the _same elements

as [9] and added silicon oil _{to compare} with

Rayleigh-Taylor instability

and

magnetic

fluid _to compare with the similar situation of

magnetic

fluid under

magnetic field).

We _can

similarly

introduce characteristic times. The _more

usefil

will be the

capillary time, typical

of _a free

falling drop,

and defined before _{as Tdesta}

tc ₌

(lc/g)~/~

₌

S~/~p~~/~g~~/~

The other time in the

problem

is the

growth

time _or inverse of the

growth

rate of

perturbation:

~

j-1

3. General _case.

We shall _now solve the _more

general

_case of _a viscous

layer,

and _we shall leave _open the

possibility

to be in the _case of

Rayleigh-Taylor instability (we

then _suppose the

density

of the upper fluid _to be greater ^{than the}

density

of the

underlying

fluid: oil

falling

in air for

example).

With _g directed from the fluid _to the _{vacuum, g}

=

(0,

0,

g),

_{we are} in the _case of _a fluid located under _a solid flat

plate,

which is the

Rayleigh-Taylor

_case. We shall let the

possibility

to choose this in the

equations by using

_a constant x

equal

to I if g =

(0,

0,

-g)

and to -I if g =

(0,

0,

g).

In the

following

_we shall denote

by ((~, y)

the vertical

displacement

of the

interface, v(~,

_y,

z)

the

velocity

of the

fluid, e(~,

_y,

z)

the electric field in _vacuum and _n, the unit vector normal _to the

interface,

=

(8~(, 8y(, -1)

_ci

(8~(, dy(, -I)

at first order

/1

₊

_(8~()~

₊

_(8y()~

The system ^is

represented by

the

following equations (for

_an

incompressible fluid,

with q and S

constants):

div

^{v = 0}

^)for

^{z <}

^{((~,y) (in} the1iquid metal)

p[$v

+

(v grad)v]

=

-grad

_{p +}

nAv

_{+ pg}

curl _{e =} 0

for _{z >}

((~,y) (in

the

vacuum)

div _e

= o

which _mean bulk conservation, ^the Navier-Stokes

equation

dud Maxwell equations

(without magnetic field).

The

boundary

^conditions

(where [Xi

₌ value of X above the interface value of X under the

interface)

are:

1(

= uz

u~8~( uy8y(

at z =

( (free

surface

condition)

-~j]n;

+

[T,k

+

~(~]nk (S/R)n,

at _{z =}

( (stress

balance _at the

interface)

n x e = 0 at z =

( (liquid

metal

condition)

v = 0 at z = -a

(velocity equal

to zero at the

electrode)

e~ = ey = 0 at z = b

(electric

field normal _to the

electrode)

where _we defined the stress tensor:

the viscous rate-of-strain tensor:

~ik " 11

(~~k~t

+

~~k~k)

and R~~. _curvature of the interface

(positive

if directed towards the

fluid) R~

=

()

+

))

m

iai~

+

aj~)

We

shall,

from _{now on, use} dimensionless values with

capillary

scales _as references:

lc for

length,

tc for the time, the

capillary Laplace

_{pressure pc =}

S/lc

₌

/@

for _pressure,

Eo "

V/b

for electric field.

In this _{case, we} note:

~'

=

~/lc, y'

₌

y/lc,

^z' ₌

z/lc, I'

"

(/lc,

a'

=

a/lc,

^b' ₌

b/lc,

t'

#

t/tc>

P'

=

(P+ xPgz)/Pc,

e'

=

e/Eo

We recall that 4l

=

(1/2)eoE](pgs)~~/~

=

E( /E(

and d

=

(lv /lc)~/~

_{as was}

previously defined,

with _x

= I if _g is directed like

(o, 0, -1),

which is _our

problem,

and _x

= -I if _{g is} directed like

(0, 0, 1),

^which

corresponds

to _a film in _a

gravitationally

unstable

position (Rayleigh-Taylor).

To make the different

expressions

easier to read _we shall

drop

the

primes:

We obtain

j~~ ^v~ _grad)v

=

-grad

_{P +} dAV

l~~~

^{~ ~ ~~~~ ~~}

curl _e

= 0

for _z >

((~,

_Y)

div _e

= 0

with the

following boundary conditions,

where for the curvature we have restricted calcula- tion to the first order term:

81(

_{= Uz}

U~8~( Uy8y(

^at z =

(

-(-P

₊

x(

_81~

( 8(~ (

⁺

)8~(

+

~l(el e( el)8~(

+

2e~ey8y( 2e~ezl+

-2d(8~u~ )8~( d(8~uy

+

8yu~)8y(

+

d(8~uz

₊

8zu~)

₌ 0 at z =

( -(-p

+

x( _8]~ ( ₈₍₂ (

+

)8y(

+

4l[2e~ey8~(

+

(-e]

₊

e( e])8y( 2eyez]+

-2d(8yuy)8y( d(8~uy

+

8yu~ )8~(

+

d(8~uz

+

8zu~

= 0 at z =

(

-P ⁺

x( 8]2( ₈₍₂₍

₊

=

=

4l(-2e~ez8~( 2eyez8y(

+

(-e] e(

+

e()]+

-2d8zuz

₊

d(8~uz

₊

8zu~)8~(

+

d(8yuz

+

8zuy)8y(

= 0 at _{z =}

(

e~ +

d~(

= 0 at z =

(

ey

+dy(

₌ 0 at z =

(

uz = up = uz = 0 at z = -a

e~ = ey = 0 at z = b

P(o)

^{" -4l}

At zeroth order _we have: ejo) "

(0, 0,1)

~(o) " °

At first order the

problem

is solved in

Appendix

A and _we obtain _a system of six equations for six unknown constants.

This leads to annulate the

following

determinant:

sinh(ka)

0

sinh(qa)

0 0 -s

2k~

sinh(ka)

0

(k~

+ q~)

sinh(qa)

0 0 0

((

+

2dk) cosh(ka) f

+ 2dk

2dq cosh(qa) 2dq

-214l

cosh(kb)

_{x +} ^k~

0 0 0 0

sinh(kb)

ik

k k

cosh(ka)

_{q q}

cosh(qa)

0 0

0

sinh(ka)

0

sinh(qa)

0 0

with q~ = k~ _{+ s}

Id.

The formal wavenumber _q relates the horizontal wavenumber k of the

perturbation

and the vertical wavenumber

/fi, giving

_an idea of the penetration of the viscous

dissipation

inside the

layer (with

dimensions it reads q~ = k~ _+s

Iv).

This determinant has

already

been obtained in reference _[9] with _a

slight typographical

_error. It leads _to the

following

transcendental

equation as an

implicit dispersion

relation between the dimensionless

perturbation growth

rate s and the dimensionless wavenumber

k,

for different values of the external

applied

field Eo

written in terms of the ratio 4l:

4qk~[q

^k

coth(ka) coth(qa)] _(k~

+

q~)~[q coth(ka) coth(qa) k]+

~~~

_{~~ ~ ~}

= [k~

24lk~ coth(kb)

+

Xk][q coth(qa)

k

coth(ka)] (2) inh(~a)~inh~qa)

d~

The parameter x is I in a

gravitational

stable

layer

and -I in _a

gravitational

unstable

layer.

For 4l

= 0

(no applied

electric

field)

and _X

" -I _we retrieve the results of Chandrasekhar [12]

for the

Rayleigh-Taylor instability

4.

Asymptotic

behaviour of the

dispersion

relation.

We recall that _{we use} dimensionless values

(relatively

to

capillary

values and without

primes)

unless otherwise stated.

We _can first _see that if

(k, q)

^is a solution of

equation (2)

then

(k, q)

is also

solution,

q

being

the

conjugate

of _q. We shall limit _our

study

to solutions where

Im(q)

>

0,

with k real _as is usual in the

temporal theory

of linear

stability.

We _can also _see that _s

= 0

(which

_{means q}

=

k)

is

always

^{solution of}

equation (2).

Apart

from _this trivia1solution _we look for the unstable

regime,

therefore

Real(s)

> 0, ^which

implies Real(q~)

> k~ and

Real(q)

> k. We _can

numerically

check that the sc-called

'principle

of

exchange

of stabilities'

(Real(s)

= 0

implies Im(s)

=

0) applies

in _our

problem (Appendix B).

As _s

= 0 is

already solution,

_we shall first obtain the limit of equation

(2)

when _{s -} 0.

In this _case Sk

= q k

+~

)

can be factorised in

equation (2)

and after

simplification by

6k 2 k

we obtain _a

dispersion

relation of the type: Sk

+~

P(k),

valid _{near s}

= 0, ^{and for which} s = 0

(which

is then

equivalent

to Sk

=

0)

is not _a triv1al solution.

In fact _we then obtain the

dispersion

relation of the _very viscous _case

(as

_qa -

ka,

obtained if Sk _- 0 for _a

given,

which allows the

simplified equation (14)

of the viscous _{case, can} be

obtained

by doing

either _{s -} 0 _or d _-

cc). Performing

_{s =} 0 in this relation

gives

the

equation

of the _curve of

marginal stability:

P(k)

= k~ 24l

coth(kb)k

_{+ x}

= 0

which is therefore

independent

of the

viscosity

and of the

depth

a of the

layer.

In

Appendix

B _we also show that if

P(k)

< 0

(unstable region),

then

Im(s)

= 0 and there exists _a solution with

Real(s)

>

0,

^{while if}

P(k)

> 0

(stable region),

then

Real(s)

< 0.

In the

following

_we try to find the

asymptotic

behaviour of the

growth

rate _s in the different

regimes,

^{thick and}

thin,

viscous and

inviscid,

in the

spirit

of the studies of

Hynes

and Limat.

For the different

asymptotic dispersion

^relations _{s =}

s(k),

^the condition for

instability

is always that k should

belong

to the linear allowed band where

P(k)

< 0.

Outside this

band,

_{we are} not

always

^allowed to use the

simplified dispersion

relations: for

example,

in the well-known inuiscid _case the

dispersion

relation of type

(I)

^s~

=

-kP(k)

cannot

strictly

be

applied

for

P(k)

> 0: s~ would be

negative, giving

_a

purely imaginary

growth

rate _{s =} iw

but,

for _any finite

viscosity,

the

imaginary

part ^of the

dispersion

relation would

give Im(s)

= 0,

introducing

_a contradiction

(in

fact there is _a real part of _s that will

damp

the

wave).

We have nevertheless

plotted

_curves for 4l _< 1 where

P(k)

is _never

negative

and for the

regions

where

P(k)

< 0 for 4l _> 1 in order to show how the _curves evolve with the field.

One of the interests in

analyzing

the

asymptotic regimes

is to derive

explicit

relations for the fastest

growth

rate s*

= sm~x obtained for

ds/dk

= 0. Indeed if the external field reaches

a

supercritical

value

Eo

>

Ec sufficiently quickly

(T =

Eli

< I

/s*)

it is

possible

to

achieve,

t

at least

transiently,

_a wavenumber selection mechanism of linear type

following

the s* ₌ s*

(E)

curve. If the field is

applied

too

slowly

the transition will _occur before the "desired" field Eo

will be reached

(for

_{a very}

slowly applied

^field the transition _{occurs near} the critical

field)

and

once the instabilities have

begun

to grow, a full non-linear treatment would be needed in order to

predict

how the _structures evolve.

We know

experimentally (see _[14])

that _a

quicker

installation of the field

(if

^{the field} is greater than the critical

field) produces

_more

peaks

for _mercury under _a normal electric field.

In _numerous related

problems

_{we can} encounter similar

dispersion

equations: for instance in

magnetic liquids (see [15]), formally equivalent

when the

magnetic permeability

_p - cc, and in the _case of

ripplons

at the

charged

interface of

phase-separated

^{~He -~} He mixtures excited

by

_an electric field

(see

Leiderer

[16])

^the

dispersion

equation is

analog

to the inviscid _case.

Savignac

and Leiderer [17] ^{studied the} case of _an interface

crystalline-superfluid He; they

have

experimentally

shown that the

instability, again

similar to ours, grows with

ds/dk

= 0 and

gives

_many more

peaks

for

higher

fields than for onset.

This situation is

extremely important

for

technological applications

in order to obtain

Liquid

Metal Ion Sources

(LMIS)

with _a

typica1wavelength

smaller than lc.

4.I. THICK _FILM. We first

study

the _case of _a thick film of

liquid

metal. The thick film

hypothesis implies

that ka _» I and Real

(qa)

» I, so that

coth(ka)

_ci I and

coth(qa)

_ci I, the

dispersion

relation

(2)

leads _to:

4qk~ (k~

+

q~)~

=

j[k~ ^{24lk~ coth(kb)}

+

xk] (3)

It is

possible

to solve _{q =}

q(k)

and there _are four roots. It is also easy in this _case to

verify analytically

the

principle

of

exchange

stabilities. In order _to solve the

equation

it is _more

convenient to write it _as:

16

(k~

⁺

j)

k~

=

(2k~

⁺

j)~ j _[-k~

+

24lk~ coth(kb) xk] ~

(4)

It is _now

possible

to solve _s

=

s(k).

_{If q} is solution of

(3)

then

(4) just

adds its

opposite

since solution but since _{we are}

only

interested in _s

(which

varies like q~) ^{it is of} no

importance.

Equation (4)

has four solutions for _{s, not} all of which

correspond

to

growing perturbations

(with Real(s)

>

0).

We shall

study simplifications

of

(4), corresponding

to the different thick film

regimes.

4.I.I. Thick inertial

(inviscid)

fiIni. In the _case of _a

vanishing viscosity,

Re » I (j s

/(2dk~)

_» for _a thick

layer), equation (4)

_can be

simplified, leading

to the well-known

case of _an inviscid thick

liquid

film

(also

valid if d

= 0

exactly)

s(~_; =

-k~

₊

24lk~ coth(kb) xk (5)

Instabilities exist if the

growth

rate s is real and

positive.

When the distance b between the fluid and the _upper electrode is

large compared

with the

wavelength

_so that

coth(kb)

_m I this condition is fulfilled without onset for x = -I

(Rayleigh-Taylor instability)

and for 4l > 1 when _x

= I. The allowed wavenumber k

belongs

to the linear band

C(4l)

defined

by:

4l-

x<k<4l+

_x

The maximum

growth

rate

(ds/dk

=

0)

is

given by

the relation:

obtained for the wavenumber

~* ^{24~ +} 44~~ 3X

tk.I ~

The

validity

conditions for this

regime

^become

l~~a

^{» I thick}

^hypothesis

d~(2~~~

_{~ ~~~~~~~~ ~~~~~~~~~~}

Near onset (4l -

1)

s - 0 and k _- kc ₌ I, ^which is

only possible

for the limit of inviscid

film iv _- o

(d

_-

0),

_so for _non-zero d _{we cannot}

apply

^{the inviscid}

hypothesis

in the limit

4l _- 1. In other _terms the viscous

penetration length

_lvp - cc near onset and the viscous

dissipation

then exists _on all the finite

layers.

In

figure

3a _we

plot s~(k)

_(X

" I, ^electric

case), keeping

in mind that the above-mentioned

conditions of

validity,

_even if

they

^{hold for}

k*,

_are not

necessarily

verified far from the maximum.

4.1.2. Thick non-inertial film. We shall _now

study

the corrections introduced

by

_a finite

viscosity.

For this _{case we can} write

equation (4)

^{in the}

following

form:

16

(k~

⁺

II

^k~

=

(2k~

+

II

^~

_(~l" ^j

^~

(6)

Let _US ~~~~°~~~~'

~

2dk~

₁

~d

j

^~

8tk.I i~~i _~~~"

where Re; is a

Reynolds

for _a fluid where the

dispersion

relation is

supposed

to be the _{same as} in the inviscid _case and

(

is the ratio between the effective viscous

growth

rate and the

growth

rate for the fluid

supposed

to be inviscid and is

always

smaller than I

(as

will be _seen

later).

We have therefore Re

=

(Re;

₌

(IS:

this

Reynolds

should be much greater ^than I to allow the inertia1film

approximation.

If the

viscosity

^does not affect much the

dispersion

relation

we have

f

_ci I and Re _ci Re;.

We obtain the

following equivalent equation:

s2 /fl

^Thick s I Thick

~I~<I

~I~=1 4~<1

~l~=0 ^'°' _j~

o-o o-S I-o I-S 2.o 2.5 3.o o-o o-S I-o I-S 2.o 2.5 3.o

a b

~

fi T

o.2

(~=o 5)

^S

(a=Q,5

o-o ~ ,~

~~~

~~~~~~

.o.2 4$=1

~

~ ~

~J~<i ~l~<I

~l~=0 ~l~=0

o-o o-S I-o I-S 2-o 2-5 3.o o-o o-S I-o I-S 2.o 2.5 3.o

d

lll ^~~

~J~>1

%

~

~

«>i

fi

-o,os

'(

m=i

d=20)

_~ ~~ ~D<1 4~<1_

~ ~~

4~=o

k

^°"°

k

o-o o-S I-o I-S 2-o 2-S 3.o o-o o-S I-o 1-5 2.o 2-S 3-o

e f

Fig.3.

The different dispersion relations:

s~(k)

for thick inviscid, non-thick inviscid and thin in- viscid

s(k)

for thick-viscous, non-thin viscous and thin-viscous.

@~(@ +

2f)

= 1(@ +

f)~ ii

^~

(7)

which is _a well known

quartic equation

obtained in the

study

of

gravity

waves

[18, 11].

^The difference between

gravity

_waves ^and our

problem

is that _we consider solutions where _s is

real,

while for

gravity

waves s is

purely imaginary (in

this _case solutions _are

sought

for _a

purely

(

~~

ⁱ

~

o o

', 2 4 6 8 10

-1

-2

-3

(

_m -0.46 6

-4

Fig.4.

The real solutions of the q~lartic equation

(7).

imaginary 9).

Since _{we are} interested

only

in the unstable

regime,

_we

plot (

versus in

figure

4

only

for

>

0, showing

two real branches

(for

which

(

<

I).

The condition

ds/dk

₌ 0

(maximum growth perturbation) leads, together

with

(7)

to the condition:

l~

^%~~

@~(2@ +

3()

= [(@ ⁺

()~ l] 2(9

₊

()9

^8~

(8)

sik.;

4.1.2.I Thick

quasi-inviscid

film. If the

liquid

has _{a very} low

viscosity,

iv < lc

(d

<

I),

the

dispersion

relation is _not

strongly

modified and this is the _reason

why

_we call the

regime quasi

inviscid. As the _{value of} d is small for

liquid

metals this _case is

interesting

from _a

practical point

of view.

For this condition 9 _< 1, which _means that _{we are} in the _case where the thick inviscid

hypothesis

_can be

applied,

equation

(7)

leads to

(=1-9+O(9~)

when9-0

(9)

or:

sik.qi "

[-k~

+

24lk~ coth(kb) xk]

^~~~

^2dk~ (lo)

This

corresponds

_to the

region

9 _< 1 in

figure

4 where the two branches

collapse.

The term 9 in

equation (9)

describes the viscous attenuation of

growth

in the quasi-inviscid

regime,

it

shows how the dispersion equation is affected

by

_a small viscosity. It should nevertheless be

noted that this relation is not valid _{near s}

=

0,

and _cannot be

used,

in

particular,

to determine the linear band of

stability.

This correction

brings slight changes

to the

dispersion

relation which also

slightly

affect the value of

k(~,;.

In the _case of

gravity

_waves the viscous attenuation is real and does not

change

the

pulsation

of the _wave

(s

is

purely imaginary

in the inviscid

case):

the _{waves are} attenuated but the

pulsation

is the inviscid _one. Here it affects the

growth

rate because _s is _a rea1number.

4.1.2.2 Thick _very viscous film. If the

viscosity

is very

large,

_we expect a different behaviour.

If Re < I

(the

film is

viscous)

_we should have

f16

_< 1 which

implies (see Fig. 4)

9 _» 1,

so

Re;

< I.

Reciprocally,

^for

Re;

<

I,

_as

f

< I, _we have Re

=

f

Re; < I _so that Re < I is

equivalent

to Re; < I. In this viscous _case

equation (7)

^leads to:

~3

~ ~

~4

~

~~'

~~~~ ~

~ ~° ~~~~

the other branch of solution

being f

= -0.46 9 when 9 _{- cc,}

corresponding

to a stable

regime.

We _can write

equation (II)

to first order in the form

(which

is valid

only

for _x

=

I,

_as it

tends to _cc for k _- 0 if _x

=

-1):

sik.~ "

i-k

+ 24l

coth(kb) ~j ⁽¹²⁾

We _{can note} that _sik.v

"

Re;sik.;

and _as

Re;

< I, the value of _s is much smaller than in the

inviscid _case.

In

figure

3b _we

plot s(k)

for _x

= I.

When the distance between the fluid and the _upper electrode is

large compared

with the

wavelength (coth(kb)

_m

I), instability

exists if 4l > 1 for _x

= I. Allowed wavenumbers k

belong

to the linear band

C(4l).

The maximum

growth

rate

(ds/dk

₌

0)

is

given by

the relation:

S~k,v "

(~ l)

obtained for the critical wavenumber

~tk.v

_" ~c " 1

The existence of this

regime

is

governed by

the

following

conditions:

la ^~

^{» I thick}

^hypothesis

(4l

1)

< viscous

hypothesis

2d

The second condition holds for fields _near the critical _one (4l ^almost

equal

to

I),

but d values

are

usually

too small for it to be valid in strong ^fields.

Let _us stress an

important point:

for fields _not far above the critical field (4l m

I)

the

unstable

region

is around k

= I, as the

instability

domain is the _same for all kinds of

behaviour, corresponding

to k in the linear band

C(4l).

Then _{we cannot} expect to find _a dominant mode for which the

wavelength

will be of the

order of the thickness _{a or} the viscous

length

iv, at least _near the critical field. For

higher

fields _{we see} here that in the viscous

infinitely

thick _case the

capillary length

is still the

typical

scale.

Moreover,

_as d is

usually

_very small for

liquid

^metals

(10~~),

this _case

only applies

in

practice

_very close _to the threshold field.

In the

Rayleigh-Taylor

_case the situation is

completely

different:

for _x

= -I and 4l

= 0 there is _no

dispersion

relation

giving

_us the fastest

growth

rate: the

corresponding

maximum of _s

(we

should _{moreover go} further in the

k-expansion

_as k

= 0 is the

maximum of

(12))

does _not

satisfy

the

hypothesis

9 _» 1, ^{which leads} to a different behaviour.

Let _{us assume} that d _» I

(viscous length

»

capillary length)

for k _< I

(wavelength

»

capillary length),

_sik.;

"

k~/~,

therefore:

(a similar simple relation cannotbe btained

in

the

203(20

+

3t)

=

j(o

+

t)2

11

j4(o

+

t)o

11

(13)

Relations

(7)

and

(13) give:

Sri 0.66

so that

k _ci

0.49d~~/~

or, in dimensional form k _ci ~'~~

iv

s ci

0.46d~/~

_{or, in} dimensional form _{s ci} 0.46 ~ iv

as obtained in [12]. ^{This is the}

typica1situation

when surface tension _can be

neglected

in the interface of miscible

layers,

_{as occurs} in many

geophysical applications.

One checks that k _< I for d » I. The thick

hypothesis

reads

0.49a/lv

_» I _so that the thickness effects _{are more}

important

than those of

viscosity.

Here _{we see} that _{we are} in _{a case} where the

wavelength

is of the order of the viscous

length,

an

important

^{difference with the electric} _case. ^We _can ^also note that Re _< I but _we have not

exactly

Re < 1.

4.2. VERY vIscous FILM. If the

depth

_a has _a finite

value,

it is still

possible

to

simplify

the

problem

when the influence of

viscosity

is

large enough.

If the condition a~ _s

Id

< I is

satisfied,

I-e- if the viscous

penetration length

_lvp

=

@

_is

_larger

_{than the film thickness}

a, them the bottom friction _can reach the interface and _a

Taylor

expansion in powers of

(qa ka)

leads to:

1 ~ ~~ ~ ~~

xj sinh(ka)cosh(ka)-ka

~~'~ ~

2d ^{~ ~°~} k

cosh~(ka)

+

(ka)~

^~~

If k satisfies ka < I, a~ _s

Id

< I is

equivalent

to Re < I.

When ka » I,

equation (14)

reduces to

equation (12),

which _can be used for x = I. That is, the _very viscous thick _case is

equivalent

to the

infinitely deep

viscous _one. As _we have _seen that

Rayleigh-Taylor

instabilities exhibit _a different behaviour in the

infinitely deep

viscous case, we expect _a ^{difference with the electric} _case ^for

large

film thicknesses in the

very-viscous

regime.

4.2.1. Thin viscous fiIni In the _case of _very thin films

(ka

<

1)

the viscous

dispersion

relation

simplifies

and leads to:

sin_v =

[-k~

+

24lk~ coth(kb) _xk~] (15)

which _can be

directely

obtained from lubrication

theory.

Indeed this is the _case when the

dissipation

exists in all the

layer depth.

The lubrication

approximation

of

creeping

flow

implies

that the deformation of the interface induces _a

tangent1al gradient

of _pressure, with _a half- Poiseuille flow for the

boundary

conditions of _our

problem.

One

important

feature of this limit is the cubic

dependence

of the

growth

rate _on thickness. This result is very useful in the

problem

of

charged

thin membranes

(see

also [19] ^{where thickness} a is very small and _van der Waals forces need to be included. We call this the _very thin

layer case).

When the distance between the fluid and the upper electrode is

large compared

to the

wavelength (coth(kb)

_m

I)

this condition is fulfilled without _onset for _x

= -1

(Rayleigh-Taylor instability)

and if 4l > 1 for _X

" I. The allowed wavenumber k

belongs

_as usual _to the linear

band

C(4l).

The maximum

growth

rate

(ds/dk

=

0)

^is

given by

the relation:

s(~,v

₌

$

(34l

⁺

8x)~ ^8x 4l)

obtained for the wavenumber

~~ ^{34~ +}

-

~"'~ 4

The

validity

conditions become

l~~~

< I thin

hypothesis

d~ (34l

⁺

@)~ _(h

4l)

^{< 1 viscous}

hypothesis

In the _case of

liquid metals,

since the value of d is

small,

the value of _a should be _very small for the second condition to hold if _{we are not near onset.} We

plot s(k)

in

figure

3f for the

electric _case.

4.2.2.

Very

viscous "non thin" film.

Figure

3d shows the

plot

^of

s(k) (for coth(kb)

₌

I,

x = I: electric

case)

in _{a case} where the film is neither thick _nor thin. We shall _{now see} the difference with the

gravitationally

unstable _situation.

Let _us

study

^the

Rayleigh-Taylor

^behaviour

(x

= -I and 4l

=

0).

If _{we suppose} that k < I

(let

_us

point again

^{that this}

hypothesis

is not

justified

in the electric _case, since the unstable domain is around k

=

I)

_{we can} write

equation (14)

_as

1

sinh(ka) cosh(ka)

ka

s =

~

~

(16)

dk cosh

(ka)

+

(ka)~

which has _a maximum for

k* _m

2.121a

s* _m

0.16/d

where the conditions of

validity

_{are now a} » 2.12 and

0.16a~

_<

d~,

_so that

viscosity

effects

are more

important

than those of thickness.

Here _we find _{a case} where the

wavelength

is of the order of the thickness _{a, so} that in the

Rayleigh-Taylor

_{case we can} have _a

wavelength

of the order of

lc, iv,

_{or a}

depending

_on