Liquid layer deformation under horizontal thermal gradient

HAL Id: jpa-00246308

https://hal.archives-ouvertes.fr/jpa-00246308

Submitted on 1 Jan 1991

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Liquid layer deformation under horizontal thermal gradient

M. Papoular, D. Camel, J. Favier

To cite this version:

M. Papoular, D. Camel, J. Favier. Liquid layer deformation under horizontal thermal gradient. Journal

de Physique I, EDP Sciences, 1991, 1 (2), pp.143-151. �10.1051/jp1:1991120�. �jpa-00246308�

Classification

PhysicsAbstnucts

68.10 44.25

Shom Communication

Liquid layer deformation under horizontal thermal gradient

M.

Papoular(~),

D.

Camel(~)

and

J.J.Favier(~)

(~)

^Centre de Recherches surles Tr6s

Basses'Ibmp£ratures, C-N-R-s-,

166

X,

38042 Grenoble

Cedex,

France

(~) D£partement

d'Etudes des Mat£riaux, C-E-N-G-, 85X, 38041 Grenoble

Cedex,

France

(Received

17 October199f

accepted

in

final fern

5

December1990)

R4sum6. Nous £tudions _une couche

liquide

h surface libre _en convection sous un

gradient

ther-

mique appliqu6

horizontalement.

Capillarit£

et

gravit6

5e

conjuguent

pour ^donner

une16gdre

indi- naison de la surface, sous un

angle qui depend

de

1'£paisseur

de la couche. Ce r£sultat peut

presenter

un certain int6rdt _en solidification

Marangoni. surtout,il

d6bcuche surle

principe

d'une _mesure

pr6-

cise du coefficient de

temperature

de la tension de surface.

Abstract. We consider a free-surface

liquid layer convecting

under an

applied

horizontal temper-

ature

gradient.

The surface gets

slightly

^tilted

by

_an

angle

which

depends

on

layer

thickness

through competing capillary

and

gravity

effects. This result

might

^{be of} some interest in

Marangoni

solidifi-

cation set ups. ^{It also} opens up a new way ^of

accurately measuring

the surface-tension temperature coefficient.

I. Free-surface

tilting.

A free-surface

liquid layer, subjected

to a uniform horizontal

temperature gradient

G

=

dT/dx,

sets into convection. The motion is controlled

by

two dimensionless

quantities,

the Grashof and

Reynolds-Marangoni

numbers _:

Gr =

~

^v ~~~ ReM

₌

-~ ^v' ~/ _(l)

associated, respectively,

with bulk thermal convection and

surface, thermo-capillary

convection.

The notations _{are :}

h, layer

thickness

(along

^vertical direction

z)

_g,

gravity fl

=

-dp/p dT,

thermal

expansion

_p,

specific

mass v =

q/p,

kinematic

viscosity

_; and a'

= da

/dT, temperature

coefficient of surface tension

(usually negative).

^{We shall} assume that the

temperature

varia- tion of

fl

and a'

themselves,

and of _q, can be

neglected.

From the outset, we cxclude from the

144 JOURNAL DE PHYSIQUE I N°2

present

^discussion very

large

values of Gr and

ReM

for which structures with

vorticity

and _non-

negligible

advection forces

~w (jV) £

,

are known to

appear ill.

In

particular

,

we shall

keep

_:

Gr «

(Gr)c

_m

^10~.

In

practice,

for

liquids

with small vhcosities like

liquid metals,

this will restrict thickness h to the millimetric range ^unless one works in _a low-

gravity

environment, h is _a sensitive

parameter

as it enters Gr with the fourth power. We do not consider very ^small thicknesses either

(h _~

0.2 mm,

say)

in order to avoid film-adhesion

problems.

The

problem

15

considerably simplified,

of course, for _a thin

layer (I.e. large aspect

^{ratios in} the _z and _y

directions)

_: this is the _{case we} shall be

mainly

interested in here. In that case, a

one-dimensional

solution, ignoring

_any deformation of the

surface,

has been

given by

Birikh [2].

The

purely Marangoni velocity profile v~(z),

^for

instance,

is

parabolic.

This is an excellent ap-

proxirnation

to the

velocity,

but does not

completely

describe the pressure ^{field. The free surface} must somehow relax the stresses it is

subjected

to,

by deforming slightly.

The

corresponding

con-

tribution to the vertical

pressure gradient

is absent from Birikh's solution.

This correction is

straightforward

ⁱⁿ the _case of bulk thermal convection where

gravity

auto-

matically provides

for the vertical pressure ^field : p = po +

pg(h z),

_po

being

the pressure ^{at free} surface. The result

(see

Ref.

[3])

^is a small thickness

gradient, parallel

to

grad

^T

h~(z)

₌

hi _(Po/P)~~~

_{+ 3}

_{(« «o) /Pg (2)}

where slow variations in _p and _a have been

assumed,

which 15

hardly

restrictive _a5

long

a5 the

applied temperature grad

ient G does not exceed _a few tens of

degree

cm- ^I The

subscript

^denotes

values at _a reference

point

z = zo.

In the

gravity-dominated regime

described

by equation (2), h(z)

is _seen to increase in the _same direction _{a5 a} for small

thicknesses,

^and in the _same direction _as

p-I

for

larger

thicknesses. The

origin

of the

driving

force is

mainly capillary

_: a'

(mainly

barometric _:

pgh~fl)

in the first

(sec.

ond)

case.

Usually,

both _a and p increase towards the cold end.

So,

both

types

^{of convection}

(thermocapillary

and

bulk-thermal)

drive the upper

part

^{of the}

liquid

from the hot to the cold site.

Our aim in this work is to

provide

a detailed discussion of free-surface deformation in both

regimes

where the

restoring

force is

governed by gravity

_g

(as

in Ref. [3] ^and

Eq. (2)),

or surface tension _a. A well-defined criterion allows _{us to}

distinguish

between the two. The latter

regime corresponds

to

low-gravity

conditions.

Throughout,

_we

neglect

the

weak, induced,

transverse

temperature gradient

which decreases _a5 h times Pr [2]

(the

Prandtl number Pr 15 much smaller than _one in

liquid metals).

In the next

section,

_we describe _a

general formalism,

based _on non-dimensional

analysh

and

covering

the various

regimes

for

driving

and

restoring forces,

and various cell

aspect-ratios.

In section

3, coming

back to the

gravity-dominated regime,

_we stress that surface deformation basi-

cally

reduces to a

small,

uniform

tilting

of _the free surface which

opens

up the

principle

of _a novel method for

measuring

the surface-tension

temperature

coefficient a'

(Sect. 5).

^{In section}

4,

we show that the

capillarity-dominated regime also,

is describable in

simple physical

terms, ^both

for narrow cells

(width

I much smaller than

length L)

and wide _ones

(I

»

L).

In the latter case,

~2

the

tilt-angle,

in the middle of the

cell,

_goes as

ja'.

2. Dimensionless-number formalism.

Free-surface

shapes,

irt the weak-deformation

limit,

have been studied in the literature in terms of non-dimensional

analysis (see

for _ex. Ref. [4]

),

_{e.g. in} the

zone-melting

context. In thin

section,

_we

Z

,/Y

~

j---

° hix) ^° ho

°

&J

~i~~~ii~_-~-

q-

---2

Fig-

1. Geometrical characterization of free-surface deformation temperature

gradient

^is directed

along

Oz. Cell vidth,

along Oy,

is £.

adapt

the

corresponding

^formalism to our

liquid-layer problem.

In sections 3 and

4,

we show that the

physically interesting limiting

cases where

gravity

or surface

tension, respectively,

dominate the

restoring force,

admit of

simple

direct solutions.

In terms of the

geometry

defined in

figure I,

we introduce dimensionless

lengths

and velocities _:

h

ho

_{~ y z}

~

ho

^{' ~} L' ^~ £' ^~

ho j3)

Vx ₌

I,

Vz ₌

I

U0~ U0z

where the

subscript

"zero" refers to the

middle-point

O.

Defining

three dimensionless numbers which refer

velocity, gravity

and surface

tension, respectively,

to

viscosity

_:

Re=~,G=~),S=~

(4)

v v pv

(Re,

for

example,

h the

Reynolds number),

and mo

aspect

^ratios :

A ₌

~°,

_B

=

° (5)

we

get

^the dimensionless

pressure

^field as

~

~(~

~~~'~'~~

_~° ~ ~~~~

~e

~ ~ ~

~ _le ^~~ _~~

~

~~ ~/

~~~

p(~,

_y,

z) obeying

Navier-Stokes

equation

of course, with the associated

boundary

^conditions

(see,

e-g-,

Eqs. (19)

to

(22),

in Sect.

4)

_po ^is the gas pressure ^at free surface.

Expanding,

for a thin

layer (A,

B «

I),

the pressure ^and

velocity

fields to first order in defor-

mation,

we

recognize

the

viscosity

term A

~j

_{to be of order b the local}

_slope

_{of the deformed}

surface.

Thus,

A

~j/

^{is much smaller than the} pressure ^{P. To order} zero, the latter reduces to Birikh's pressure.

So,

_we find the local thickness H

IX

_{as a} solution of _:

Po(X)

= ~

/~ _((Or ^emj

_X

=

~

H ~

(A~~/

^{+ B}

~/j ₍₇₎

e e e

146 JOURNALDEPHYSIQUEI N°2

where

Po(X)

is Birikh's solution [2] for the pressure ^{field at the surface. Let} us, for

example,

consider very ^{wide cells} : I »

L,

I-e- B < A. The last

equation gives

:

II ^G)~~~j

3

R~~

or

/4

~~

2A S _x

g

^1/2

~~~~

2

GA

ji ^g~'/2j

~~

A

~S~

^{~ ~~~}

sh

~ ~

When

gravity

dominates

(G

>

SA~)

,

tills

equation

reduces to :

H(x)

=

~

~~M °~/~ ~ ~-(Q/s)1/2 _~-i

(1 ~) ^{~-(g/sl'/2 ~-i} ₍₁

~

~) _~j

2

GA

2 2 2

(9)

The surface is

uniformly

tilted _over most of cell

length,

and

edge

effects _occur

only

over a char- s

~2

^1/2

acterhtic reduced distance

,

proportional

to the

capillary length la /pg)1/~

H reaches its maximum value _near the end wall _:

~ 3

ReM Gr/4

A'ax ~

i _GA (lo)

When

capillarity

dominates

(SA~

»

G)

,

the surface

profile

obtains _{as an}

expansion

of the

hyper-

bolic-line functions in

equation (8)

_:

H(X)

₌

(4SA~)

^~'

(ReM Gr/4) X~ ( _ill)

(with

our

notations,

X

= ~ at the cell

ends).

The deformation is _{now a}

cubic,

instead of

linear,

function of distance X and the deformation

amplttude

ts given

by

:

Hmax

₌

(4SA~)

^~~

(Or /4 ReM) (12V5) ⁽¹²⁾

and _occurs at X

=

(2v5)

3. Free-surface

tilting

under

gravific restoring

forc~

From the above

analysh,

we recover the limit where

gravity provides

for the

restoring

^force.

Equa-

tion

(9)

is

equivalent

to an

expansion

of

equation (2)

to first order in deformation _:

~ 2~h ^{~~ ~ ~i} ih ^"

~

~~~~~~

~~~~

where

((z)

is local vertical coordinate of free surface

(h(~)

in Sect.

2).

The surface deformation

essentially

reduces to a

small,

uniform tilt with

slope

b _: the curvature is

negligible-

The above _so-

lution,

of _course, is valid

provided

the cell dimensions

L,

and

£,

are much

larger

than the

capillary

length (a /pgl'/~

_{a few} millimeters

usually.

The

sign

^{of tilt}

angle depends (for

a

given sign

of G

=

VT)

on

average

^{thickness h}

(denoted ho

in the

previous section).

^{In the usual} case where thermal

expansion fl

is

positive,

while surface-

tension

temperature

coefficient a' is

negative,

_{we can} define a characterhtic thickness

hc given by

_:

h)

=

~"

= 4

~~~~~

Pgfl

Gr

(14)

With

ordinary

values of

parameters

^for

liquid metals, hc

is about half _a centimeter

typically.

The surface

slope

b is

positive (negative)

for h smaller

(larger)

than

h~. h~

is the thickness for which bulk and surface convections

conspire

to restore

horizontality

of free surface.

(Of

_course, for h~

to be

properly

defined

by equation (14),

our initial criterion _: Gr <

(Gr)~

must be

fulfilled,

that h _: with

fl

₌

10~~

_K~ I and G

= lo

Kcm~',

_kinematic

_viscosity

v = q

/p,

in

cm~s-I,

_{must be}

_larger

than _ce

~~))~ ^).

This

suggests

an

original

_way of

measuring a',

_as we shall _see in section 6.

On the other

hand,

the flow

velocity

on the free surface itself may ^{be shown} [2] to vanish for

equal

and

opposite

Grashof and

Reynolds-Marangoni

numbers. For _a

given imposed

thermal

gradient,

the overall

intensity

of flow is then

considerably

reduced which may ^have

interesting applications,

in directional solidification for instance.

(It

should be

stressed, though,

^that the

velocity profile v~(z)

now

displays

an inflection

point

in the middle

plane

of the

layer,

which in the limit of low

vhcosity

_may result in _non

stationary

instabilities

(see

e-g- Ref.

[5]).)

^The

corresponding

characterhtic thickness

h~

is

given by

[2] :

A short

development

is in order here. The _case

fl

_> 0 and a' _> 0 is _more

interesting

in

practice

than the _one considered in reference [2]

~both fl

and a'

negative).

It could be arrived at

by doping

with _a surface-active

agent

^under

appropriate

conditions of

temperature

^and chemical

potentials (e.g.

bhmuth in

tin,

see Refs.

[6, 7j).

Then a' _e da

/dT

has _a maximum _{as a} function of

temperature.

^The

point

is that surface tension is

density

of surface free energy :

a =

f,

_{= e,}

Ts, (16)

The

high-temperature

side is dominated

by

increase in

entropy,

so that a' is

negative (as

in _a pure

liquid)

while the

low-temperature

side is dominated

by

^increase in energy e,

(since

more

and _more Bi atoms leave the surface _as T

/).

With

ordinary

values of

parameters,

h~ is of order I _cm.

4. Free-surface tilt under

Marangoni-induced

_pressure field.

Thh _case

corresponds

to

low-gravity

conditions. The "barometric" pressure ^h now

negligible compared

to flow-induced pressure. ^The

corresponding

criterion writes

simply

_:

pga «

a/r (17)

(or

<

SA~,

_see Sect.

2),

where _a is deformation

amplitude

and I

/r

is maximum characteristic surface curvature. The latter is taken to be modest in the _sense that _r »

L,I,

the cell

length

L and width £ themselves

being

_on the scale of centimeters for

example. Only

in _a

low-gravity

environment

(g ~ 10~~go, typically),

will condition

(17)

be fulfilled.

148 JOURNAL DE PHYSIQUE I N°2

Let _us consider _a shallow container which is wetted

by

the

liquid,

with _{an amount} of

liquid filling exactly

the container the free surface b

pinned

at the wan

edges

and tends to

stay parallel

to the bottom wall. The

temperature gradient

is directed

along

Ox

(e.g.

T

decreasing

from left to

right, Fig. 2).

~

o

f,90 _{____~}

[---b--- ^I

_(~

~Gii~/~S~~~~~~~)

J ^'

L L~-.---

Fig.

Z schematics of free-surface deformation in a wide cell _: £ > L, when

capillarity provides

for the

restoring

force axis of symmetry :

perpendicular

to

plane

of

drawing, through point

O.

The Birikh solution [2]

provides

a valid

approximation

to the

velocity profile v~(z)

for both thermal

(bulk)

and

thermccapillary

convections. This is _a one-dirnensional solution however 16 =

0) and,

_as

such,

it fails to describe

properly

the "internal"

prissure

field associated with the flow.

lb do so, the full 2d

problem

16

#

0 and _vz

# 0)

must be solved

self-consistently,

that

is,

with the constraint _{: vz}

=

bv~

at free surface. This _was not necessary, ^{in section}

I,

with the "external"

barometric pressure ^field pgz.

The Navier-Stokes

equations

now read _:

tip fi~U~

^fi~U~

$

^~

_fiz2

^~

fi~2 tip fi~Uz

_~ ^fi~Uz

~~~~

$

^~

_{fiz2 fi~2}

the

velocity components being

linked

by

the condition of

incompressibility

_:

~~

⁺

~~

^{= 0.}

~ z

The

boundary

conditions _{are :}

(I)

no

slip

at

wall,

₍₁₁₎

vanishing

normal and

tangential

stress

components

at free

surface, (iii) geometrical

constraint _on

velocity

at free surface _:

z = 0 _{ux = uz =} 0

(19)

~ ~ ~°

~~~)

~

'

~)

~

~~~

^~

~

~~~~

z =

(

_uz =

bux (21)

(((~)

_z coordinate of free surface at abscissa _{~ z =} 0 at bottom

wall).

The

(small)

tilt

angle

b

=

)

results from

subjecting

the

velocity

field _vx, at

every

^abscissa ~,

~

to the return-flow condition _:

f _vx(z)dz

= 0

(22)

o

The

general

situation is

relatively complex

because the solution must involve _an

optimized trade-off,

at every

point

on the free

surface,

between the _nvo radii of curvature rL and _ri respec-

tively along length L( II Ox)

and width

fill Oy)

of cell.

Let _us first _assume the cell _to be

verywide

_: £ _» L. Then the curvature is

essentially longitudinal fi2j

(I.e. along L)

_ri » rL At every

point

_x,

^r-I

_e

rjl

= p is

given by Laplace

^formula

(the

fix small

vhcosity

term

2q ^~~~ being neglected)

:

fiz

l~~ ~

^{~ ~ ~}

^j~)

~

~

jr

₌ 0 at

mid-point O, Fig. 2).

The "Poheuille"

overpressure gradient being given by

_{[2] :}

Direct

integration gives

_:

(((~

₌

0)

₌

( lx

⁼

))

=

h),

_so that _:

~

~2,,

bo ₌ @(~ =

0)

₌

(26)

in

agreement

^with

equation Ii)

of section 2.

Now, point

M of

figure 2,

with maximum elevation _a and abscissa _xM

=

j _b,

_{h determined}

bY

)

"

°,

th~tls

M

~ ~~~~

~ ~'~ ~~~~

Then,

_a

slightly

overstimated evaluation of deformation _a is _:

p Note

case,

a

get _{: a} ~ 8 pm. But a increases apidly

deformation _{a ceases}

to

be _much

than

_{thickness .}

Consider now the opposite^ase

of

a long cell _:

L

i.e. a depression

of

_freesurface in the left-hand

side of

the cell,

a

welling_in

(i12)~

~(~) ~30)

"

2rijx)

150 JOURNAL DE PHYSIQUE I N°2

With I ₌ I cm, L = 10 cm, and the same values _as above for

a'la,

G and

h,

the maximum

deformation _{a x ~} ~ is about 30 pm.

This case is

perhaps

less

interesting, experimentally,

than the first one,

although

^a' shows up ⁱⁿ formula

(30)

as it does in

equations (26)

^and

(28).

A mixed cross-over

(r

^I = r

j

+

r/

_~, rL m ri is

expected

for cells with _{more or} less square

shape.

The treatment in this paper ^{could be extended} to a

system

^of mo fluid

layers separated by

an

interface. One

might

even consider _a

phase-separated _system approaching

its critical

point T~.

The

equivalent

of

equation (26)

^would then lead to an evaluation of the

quantity

_(T~

T)

~,

= p,

~

a

that

is,

the critical

exponent

^{of surface tension} a = ao

(T~ T) (The experiment

would be delicate

though,

due to the

necessity

of

working

with the smallest

possible

thermal

gradient,

and of

maintaining

the interface

pinned against

the

walls.)

5. Conclusions. A method for

measuring

^a'.

We have

discussed,

as a function of

layer thickness,

the free-surface deformation under _an

applied

horizontal thermal

gradient.

This was done with _an

analysh

ⁱⁿ terms of dimensionless

quantities

and with

emphasis

on both cases where

gravity,

or

capillarity, provide

for the

restoring

force.

The latter case is relevant _to

low-gravity

conditions. It would be

interesting,

under such condi-

tions,

to

try

^{and check}

(e.g. photographically)

the square

L-dependence

of

angle

_@o

(formula (26)),

or the cubic

dependence

of

amplitude

_a

(Eq. (28)),

and the overall

shape

of the deformed surface

as sketched in

figure

2.

In the former case, under

ordinary gravity,

and

tempera

ture

gradients

G _a few tens

degrees _per

cm, the characteristic deviation is of order _a fraction of I fb

(and

the characteristic thickness

h~, Eq. (14),

is _a fraction of _a centimeter for _a

liquid metal).

We stress

again

that _we have

neglected throughout

_any

temperature

variation of a' and

fl

them-

selves,

as well as of

viscosity.

Should this

assumption

break down

(under larger

thermal

grad

ients for

instance),

the various dimensionless numbers _we have introduced in sections I and 2 would vary

along VT,

and the treatment would be invalidated. The Poiseuille

gradient

in

equation (24)

for

example,

would _now have to be included in any

integration along

O~.

These considerations

yield

the

principle

of _a new,

simple

but _accurate method for

mcasuring

the surface-tension

temperature

coefficient a'

=

$. _Working

with h _<

h~,

_an

optical

measurement of the

tilt-angle

leads _us

through equation (13)

to a,

,

if thermal

expansion fl

is known

(avoiding,

at the _same

time,

rotational-flow

problems

if Gr

IA

₌

h~) happens

to be _>

(Gr)~).

Of _course,

expression (26)

for the «-restored

(I.e. low-gravity) situation,

also enables _us to mcasure a'. The relative

precision

of the _mo choices

(normal

or low

gravity) depends,

_among other

factors,

_upon

L,

the cell

length.

This method would be

ideally

suited for

liquid

metals where the

temperature dcpendcnce

of

a'

itself

(and

of

fl

and

~)

^is

relatively weak,

and

vaporization

effects _can be

ignored.

References

ill

see for

example

: LAURE P and ROUX B. C.R~4.S. MS

(1987)

1137.

[2] BIRIKH

R-V,

f

AppL

Mech. rich

Phys.

3

(1966)

43.

[3] LANDAU ^{L. and LifsHnz}

E.,

Course Theor.

Phys.,

Fluid

Dynamics (Pergamon Press) 1959,

_p. 237.

[4] LAI ^{Ct. and} CHAT

A.T,

in Proc.

36'~

_{Int. Astr.}

Congress,

^stockholm

(Oct. 1985).

[5j ^{BEN HADID}

H.,

PhD

Tl1esis,

Marseille

(1989).

[6j ^{BRAGIMOV K.I.} et

al,

Sov

Phys.

^DokL 9

(1964)

227.

[7j CAMEL D., TISON P and FAVIER

JJ.,

EN.R.s. _Conference "Ccnvection, surface Tension and Micro-

Gravity" (Mons, Belgium)

1986.

[8] ^LANDAU L. and LifsHnz E., Fluid

Dynamics (Pergamon Press) 1959,

_p. la.

Cet article a 6t6

imprim6

avec le Macro

Package

"Editions de

Physique

Avril 1990".