Journal de Mécanique Théorique et Appliquée, Journal of Theoretical and Applied Mechanics, Numéro spécial, 1986, p. 225 à 247
Modélisation of fluid-fluid interfaces with material properties by
R. GATIGNOL and P. SEPPECHER Laboratoire de Mécanique Théorique, Associé au C.N.R.S.-U.A. n° 229, Université P.-et-M.-Curie, 4, place Jussieu, 75252 Paris Cedex 05 ABSTRACT The interfaces are seen as thin three-dimensional layers. The fluid inside these layers is a mixture the internal energy of which depends on the density gradient of each constituent. The internal strengths are described by a second gradient theory. The corresponding equation set is then integrated through the layer. So we obtain the evolution equations of the interfacial physical quantities in which there enter the parameters of the fluids on both sides of the interface. Then by an asymptotic proccess the interfacial layer may be considered as a carrier surface of material quantities. As a special case the balance laws are derived for an interface without mass but carrying a surfactant. At last using the linear thermodynamic of the irreversible proccesses we give the interfacial transport coefficients.
RESUME
Les interfaces sont assimiìées à des couches tridimensionneUes de faibìe épaisseur. Le fìuide contenu dans ces couches est un mé Lanqe dont l: 'énergie interne dépend des gradients des densités de chaque constituant. Les efforts intérieurs y sont décrits par une théorie du second gradient. Le système d'équations décrivant ì'évoìution du miìieu est intégré sur ì'épaisseur de ìa couche. On obtient ainsi des équations ìiant ì'évoìution des paramètres interfaciaux aux paramètres décrivant Lee deux fìuides de part et d'autre de l:' interface. Par un processus asymptotique ì'interface peut al.one être considérée COmme une surface porteuse de propriétés matérieììes. Les ìois de conservation sont en particuìier, expìicitées dans ìe cas d'une interface sans masse transportant un surfactant. En utiìisant ìa thermodynamique ìinéaire des processus irréversibìes on donne ìes coefficients de transport interfaciaux.
with it
l.
INTRODUCTION
The
interfacial
an is
internal
energy
important
properties increase
into
to
account.
In this paper,
variations
the (l).
connected
give
to
the
Such
two fluids to
the
interface
is
the
case
is
often
phenomenon a more with
regarded
of
surface
complex the
as
a discontinuity
tension.
structure
introduction
and
of
But
in
some problems
to take more
surface
surface
material
viscosities
which
the dissipation.
of interfacial
use the
region between
of
in order
balances, some
parameters
laws of continuum
definition
of
to give an understanding
we regard
the
Limiting oneself
the interface
are
much
mechanics;
internal to a layer
larger
as
is
Journal de Mécanique Théorique et Appliquéel Journal 0750-7240/1986/22523/$ 4.30/© Gauthier-Villars
Inside
power method
given by the
with a thickness
that
of
is
of the relevant
equations
layer
the space
transition
than outside.
by the virtual
strengths
to the derivation a thin
the
interfacial
for a class
so-called small
Theoretical
of virtual
"second gradient
compared to
and
in which
the
Applied
region
we
motions, theory"
radius of
Mechanics,
the
R. GATIGNOL
226 mean curvature of
the
interface, it
is
P.
'"
SEPPECHER
possible by asymptotic approximation, to
replace the
interfacial layer by a discontinuity surface carrying material properties. The surface quantities are defined by integration over the thickness of
the
layer. Then we can wri te down the laws
of interfacial balances as well as an expression for the interfacial entropy production. First we note that from a physical point of view the properties of the medium in the interfacial region are appreciably different from these of the two adjacent fluids. It is difficult to precise the thickness of the layer. But the experimental work of Palmer, as mentionned in
(2),
the book of Hirschfelder, Curtiss and Bird
allows to say that the thickness for a liquid-
vapor interface is larger and larger, when the critical point is approached. hundreds
of molecular
diameters.
More,
the volumic mass is observed
It may be several
to be a continuous
function
of position in the direction normal to the interface. Under conditions well away from the critical point, however, the a
few molecular diameters (~,
interface
is
an expression
explicitly
(4).
recognized
for the surface tension
layer thickness is only
Nevertheless the three-dimensional in
statistical
mechanical
character of the phase
calculations
in
order
to
obtain
(4). Let
us
consider
a thin
transition
region
with
a finite
thickness, located inside a layer limited by two surfaces " by
"+'
and
and dividing two continuous media denoted
the indices
may
occur
through media
or
generally
more
On
equilibrium the
+
the
interfacial
two
in
l).
are
stress
the
mass
A
transfer
layer. We
suppose
two Newtonian
mixtures
their
contrary
(Fig.
+
and
that the
the
and
of
such
tensors
transition
fluids,
fluids;
are
layer
at
spherical. we
have
for
the medium a preferential direction, so that the stress tensor is not spherical at the equilibrium. The
parameters
mass,
in
the
+
and
two media
(volumic
...)
coefficients,
viscosity
pressure,
are
different but they
smoothly connect one another through
the
specific,
Figure l layer.
surface S by
be
we
introduce
( the normal unit vector to to
four
fluid regions
give a across We
To
an
(Fig.
+
l).
pointing from fluid
S
The space is
R'
R_,
+
divided
On
and
possible profile for a
quantity
can
IjJ
emphasize
the
two
following
interfacial
"true"
integration
(1)
2,
into we
when we go
point. of
to the definition
those of
the
introduced
by
layer of
the
quantities.
interfacial along
view :
of the "true"
interfacial quantities and the second to "excess"
Figure 2
Fig.
the layer in the normal direction.
the first one is related
A
imaginary
inside the interfacial region and we denote
the
quantity thickness
is of
the
real physical quantity describing the medium
MODELISATION
OF FLUID-FLUID
227
INTERFACES
An "excess" interfacial quantity is defined by the following integral s
(2)
(I/!
I/!ex
where
I/!
I/!+)
(or
represents the value
the assumption that this medium the boundary
S
With the filling
were
in
excess,
and that I/!
(or
~
f
- I/! )d~ +
of the quantity
occupies all the I/! +)
- I/!+) d~
+ (I/! a
in
!JJ
region
,
the
introduction of the excess values, the media
the s tJ; ex'
whole is
R_ U
regions
concentrated
R'
and
R' +
the
geometric
surface
on
of
I/!
and
+
+) under limited by
in R_ (or R+). are seen as
respectively, S.
(R+ U R~)
(or
coincides with the true value
(or
medium
U R'
R
while
That is
the
if
they
quantity
corroborated by the
equality
where
X
and
X+
are two values taken by the variable
+.
or
in the media
The thickness of the interfacial layer is not well defined, but we can expect given expression for
I/!:x
does not depend on it.
depends on the precise localisation
of the
surface
S.
(5).
in excess has been first given by Gibbs (1928)
that the
On the other hand, this expression strongly This description with the quantities
A suitable
choice for the surface
S
is
the one for which the interfacial mass in excess, ps , is zero (then S is called equimolecular ex localisation surface). Lastly we can say that with the excess values, the equations on S between the two media
and +
are exactly
jump relations. With
the
introduction
of
the
"true"
interfécial
quanti ties we consider a different approach. of
magnitude
densities interface
the
ratio
of
component
the
gradients
inside
and
of
the
outside
the
by a small adimensional
is characterized
E.
microscopic
variation
is
the
each
region
parameter In
of
of
The order
of
a
described by a
figure
description
quantity
(it
is
I/!
(the
graph similar to possible
to
€l
order
along the normal
the
direction
that given on
have
for
a
the more
complicated function).
o
In
the
macroscopic
interfacial
S the variation of
Thereby two-dimensional
of
through the layer,
I/! on the surface we
can medium.
S (Fig.
consider Such
a
that
description, is
thickness. The value
Figure 3
the value
layer
seen
as
(the a
order
surface
of the quantity
I/!
S
l),
the
without
on the surface
is undefined a priori, but keeping the memory of we may affect the material
surfacic quantity
as
3). we
have
shematisation
is
two
three-dimensional
easy
to
if
understand
media we have
separated no
mass
by
a
transfer
between the interfacial medium and the adjacent media. In
the case where
same, we must belonging
distinguish
to the
the material particles of the interfacial between
two velocities
layer and the velocity
WS
the material
medium are not always the
velocity
of the geometric surface
VS S.
of the particles A discussion about
228
R.
this
difference
has been
The order
of
The value and
of
exact
magnitude
true
is
of
layer
Ishii
(12)
account
(9),
and
(Deemer
and
on the definition
Dumais
the
only
on which
the
Chung
Goodrich
and the calculus
choice
layer
of
the
and
Shaposhnikova
of the surface
et
tension
al.(13),
alone
is
(8),
Albano, and
for
those
the
rarely
except
(for
1:
of an interfacial
(13)
layer
by
values,
consequence.
"excess" quantities,
interfacial
Gogosov
defined.
true
Deemer and Slattery
Za Bin
(l'l),
the
is of slight
of
the
well limited
for
the existence
of
for
is
region
and,
strinked using
alone
the
given,
is
equations
description
(8),
interfacial layer
on the
and Goodrich (11)
Naletova,
Slattery
the
We only mention those
(la)
detailed
SEPPECHER
asymptotically
interfacial balance
Gogosov,
However
of
depends
is
S
P.
(7).
thickness
of many works.
and Vlieger
(6),
quantity
the
the
the concern
quantities.
Gouin
the
position of the surface
Bedeaux of
of
interfacial
The derivation of layer,
recently given,
The thickness
1:+.
the
a
GATIGNOL &
"true"
taken
into
in the many works
instance
(14) ,
Brenner
(15). In
many
presented
analogy
works
as
a
with
relative
to
two-dimensional
the
power,
us mention
and those
surface local
with
enclosing
material are
in
the
material
the
the
(16),
material
Delhaye
(20) to
lead
very
of
the
language
to
a
the
interfacial
by
introduced
are
then postulated (8).
the theorem of the virtual
interface
for
are
Deemer and Slattery
distributions.
nicely
the
(18),
interfaces
are
laws
who apply
where
these
quantities
(17),
Dirac
distribution
symmetry
structure,
The interfacial balance
connected
and they
and
which
and Mazur
quantities
surface
property
a
of Daher and Maugin (19)
Albano
written
a singular
on
continuum.
approach
Bedeaux,
laws
transversality
continuum
with
and Prud 'homme
the
of
the
balance
interfaces
three-dimensional
in an integral form (Barrère Lastly let
the
is
seen
as
In
this
last
a singular work
three-dimensional
interfacial
stress
balance
tensor
are
the
continuum
laws.
then
very
The easy
to obtain. In using
this
the
paper,
components.
S
in
the
second gradient The
without
section
III
thickness.
The balance IV and
Again
in
stress entropy give
its
are
for
of
the
of
true
the
mass,
and,
In
by
the
last
using
the
the
section
of
quantities
linear
interface spheric
at
The
the
thermodynamic
of
the
with a
n
surface
general
form
•
are
surface
detailed into
the
for
the
irreversible
in
account.
interfacial irreversible
processes,
we
coefficients.
interfacial
zone
In
this
is
considered
section
describe
can
equilibrium
appear and
as
we present
is well known that
cannot
presence
E
o
and entropy on
by
layer
of order
expression
THREE-DIMENSIONAL DESCRIPTION OF THE INTERFACIAL ZONE
It
a mixture
adopted.
the
2.
components.
interfacial region
of
symmetry property of
derive
transport
evolution.
is
a surfactant
and the we
the
case
interfacial
momentum, energy
appearance
transversality condition
discussed.
of
in the
when we only keep the terms
interfacial take
the
description of
theory
schematisation
view
given
how to
the
the
detailed this
the
fluid mixtures
is
IV,
production
The
more
laws
we give
we present
concerns
law is
we explain
section
tensor
;
The point
of an interfacial balance
section
section II, theory
the
the
scope
three-dimensional
and
the equations
phenomena as in
a
discuss
characteristic
set
of
of Newtonian fluids
capillarity. of
the
fluid
Indeed
Newtonian anisotropy
continuum
equations
of
which
or even these
no specific
fluids.
with
Moreover interfaces
does
or
describe
of Newtonian
energy the
one
due
stress not
to
the
tensor appear.
MODELISATION
So these
equations
are not
OF FLUID-FLUID
suitable for
the
229
INTERFACES
description
of
the
evolution
inside
the interfacial
zones. Then we are a
fluid
thermodynamic
think
the
to
in
be
1901,
intersticial flux
the
flux
way than
of
stress the
in
of
suggested
the
us
given
to
consider
and
the
such
classic
fluid
that
is
general
are
:
the
The interfacial
large.
of these
a function
fluid
end we apply
theory.
a consequence
a function
an important
energy
flux
for
difficulty second
the
is
It
large
of
natural
gradients.
its
added
law;
then
internal
: this
principle
have
balance
by
internal these
entropy,
(Gurtin
they
energy.
expression
an energy
for
(23).
of
precised
However
the energy In
flux term
have
balance
using
the
the case of
strengths
order
called what
the
of
to
"flux
possible
powers
(l).
one
description introduces
specified value.
virtual
(Germain
the medium has
classic
naturally
It
law with a well
principle theory
where
than
difficulties.
energy
by
second gradient
in
case
the remove
in the
the
(21) the
for to
obtained
V
an
This
in
the
method
component
only.
an This
class
has
been
Here
these
(25).
a mixture
isotropic
ve loci ty its
barycentric
laws for
medium which
E.
{l, ... .n },
velocity.
We
each component leads
í
s a mixture
We denote
n
with
First
let
us recall
the
virtual
the
virtual
v
class
continuous strengths
af
the power
power
is
an objective it
ai,
and
Bi
to
the
power
the
d
form on
:
motions
is
V
in a Galilean of
forces
for
the space
(second gradient (an objective
no chemical
of
volumic
of the mixture
reaction.
The mass
__1_+VV 3t .
quantities
and external
components volumic
o
dt
principle
acceleration
internal
derivatives
of
theory).
quantity
frame
and for
V
a system each
fields
is
considered of
n
The virtual
an absolute equal
to
the
virtual motion. vectors
Vi
power of the
is a quantity which
with
internal
does not
depend
is expressed). pint
(4)
respect
of
cons ide red virtual
second partial
on the frame where
of
virtual
n mass
equations
V. Ji
and
chronology
the
we have
following
i +p i V.V+
with
by
assume that
to the
dP d"t
(3)
where
the
To this
Application of the virtual power principle
Let
with
the
thermodynamic
allows
is
and Gouin
extended
2.1
The
is
a more
(21).
density gradients
energy
description tensor
mechanic
motions
are
sum of
describing
and Gouin
on the interfaces
(2~
with
working"
for
virtual
by
equations
Dunn and Serrin 1984 (24),
appropriated
used by Casal
and
limitative
internal
Korteweg
appropriated description
balance
of
(Casal
could take.
a unique
results
set
by the presence
that
difficulty
A more
of
less
of energy
incompatible
this
form this
extra
a
of such materials encountered
overcome
with
the
capillarity"
mass of each component and of the gradients of these volumic mass.
Already the study
in
excess
we assume
the volumic
of
construct
characterized
that
Therefore
seems
to
principles
regions are to
going
"endowed with internal
Ci first
- Iv are
tensors
of
two indices,
order so
the
l,
2 and 3. way to
The tensors
write
the
V V Vi
expression
(4)
are is
symmetrical not
unique.
230
It
R. GATIGNOL
becomes unique if we lay down conditions
with respect to the first two indices
(25).
P.
&.
SEPPECHER
to the tensors
The objectivity
Ci
that is to say the
property of
pint
symmetry
has the following
consequences n ¡;
(5)
a
i
a
i=l For simplicity,
is a symmetric
the virtual power of the external pext
(6)
text.V dv +
JV n
where
V
z
(l/p)
forces is supposed of the form
Fext.V
Lav
with
z
p
i=l The external
forces
ds
n
i Vi
p
tensor.
p
i
i=l
are of simple
form and
they
act in a similar
wayan
each constituent
of the mixture. The virtual
power of the
acceleration
quantities
pace
where
with
is the acceleration
vector of the
pace
(7)
fv
p
r.v
It amounts to the same to suppose components playa
are
is a priori
very
near
each
other
i
dv
r
v
~ {l,
order
i that
We assume
--ª-at
with
ri ~ r, in
component.
...
V
,n}.
we could
+
V. V V.
The velocities of the different assume
that
the
inertial
forces
weak part in the diffusion phenomena (25).
From
the virtual
power principle,
for each system
V
and
for each virtual
motion of
iì,
we have
fav
(8)
By integrating by part the terms by taking virtual obtain
velocities
Ili
last
once and the terms a neighbour
of the
Ci
twice,
boundary
aV
of
and
V
we
:
v.ië +
(9)
where
: V Vi
which are zero on
Fext.V ds
¡¡i
and
indices
:
?
are the tensors deduced
from
lBi
"2aßy We obtain the following
local
(la)
p
r
equations
with
a
V V
and
Ci
by permuting the first and the
MODELISATION
(11)
a
OF FLUID-FLUID
INTERFACES
i
i
q;i+~V."C.
V.Bi + V V
O
p
The equation (lO) is the equation (11)
231
i
l, ...
obtained by taking all the virtual
,n
.
velocities
as
equal,
and
by taking all the velocities
equal to zero except one of them. The equation (lO) is the classical balance law for the momentum that we can also write d
(12)
dt g + g V.V
the tensor ~
2.2
- V.~
with
of order 2 is symmetrical and is called
Application of the first
g
p V
the stress tensor ..
and second principles of the thermodynamic
At every time the material derivative of the energy of a system is the sum of the power
Q.
of the external forces and of the rate of the received heat
Denoting by
U
the internal
V and using the kinetic energy theorem (obtained from the virtual power
energy of the system
principle by taking the real velocities), this first principle can be written
(dU/dt) =
Q_
pint.
In a local form it becomes:
du + V V dt u .
( 13)
Here
u
denotes
the
internal
unit of volume and
q
energy
per
unit
of
volume,
r
the
rate
of
the heat current. By using the equation set (11),
energy
source
per
the equation (13)
becomes
(14)
~+ dt
n
(15)
r - V .. q + il:
u V.V
E
(V
1:
i=l
V V + V.E
ci +~ l (iSi - V pl
Vi
.ci)
.Ji)
We emphasize that in the energy equation (14), there appears an extra energy current denoted by
E. To distinguish this extra energy current from the heat current
q
is absolutely
necessary
before defining the entropy current and before using the second principle of the thermodynamic. Indeed we adopt for the entropy current the expression only; here, in a classic
as usual, form :
T
(16)
~~ + s V. V + V. (~) -
where
is the massic entropy and
cr
pi
( 17)
and their gradients
du
1:
ij The
coefficients
~
i
are
¡.j i
u
q
f
cr i: O
cr
is an objective function depending on the entropy
Vpi.
n T ds +
i=l with
related to the current
the entropy production per unit of volume.
Now we aSSume that the internal energy s, the densities
q/T
is the absolute temperature of the medium. We use the entropy balance
u
i
dp
Then we have i
n +
n
1:
i=l V i,j
€o
z
¡.ij Vp j. d (Vp i)
j=l {l, ...
,n}
called chemical potentials. The coefficients
/j
might be
232
R.
called
s
GATIGNOL
"cocapillarity coefficients" ; in the
coefficient (21).
The
coefficients
P. SEPPECHER
case of one component only
are determined in
À ij
n
at most.
n
For
2.3
larger than 3,
only
the quantities
Àij
jgl
À 11
is the capillarity
a unique manner vjp
only
are uniquely
if
n
is
3
determined.
Expression of the entropy production
Let us remark that
the equations (3) imply
n ¿
(l8)
Àij Vp
i
n
~i a
with
Ai V vi + C
Ai
~i _j_(V i) = a .Ji dt p
j=l
+ Il
V V vi
i Àij vpj. V Vp
¿
j=l n
Ai Il
z
n
Ai
¿
C
[2
where
+ Vp i. Vp j [2)
p
Àij
i
Vp j
®
[2
j=l
denotes the unit tensor of order 2.
We assume that equation (17) dissipation
vpj
Àij(Vpi®
j=l
and
the
equation
(l7)
the equations (3),
is
always
(13)
valid
and (16)
(Axiom of the
we obtain
_ .9T
V T + (Ts - u + n
- pint -
n
z
n
By using (18)
.9T
V T + (Ts - u + i
n
+ igl ( a n
+
-
~i a
the
¿
d dt
+
V V
Bi)
V V
i )J
i
êi)
assumptions.
not depend on the
in
theory VJi
of the and
Ji.
V.Ji
We restrict
fluid mixtures.
V V Vi.
study
to a simple
of the
The first of these assumptions
Therefore
Since the quantities
are related by the relation
our
second gradient
in the case of one component only
O
The quantities
Ji
V V Vi
sed
in
.
)
V Ji
[2
1
nor on the first gradient of the mass currents
occur from the terms
i
Vp
P
stresses do
paper of Casal and Gouin
i=l
(p i)2
we make some simplifying
viscous
Bi)
(Ili
¿
/)V.V +
-
(Ili
Vi
usual
- Bj)
(Vp i)
n
i
p
(--.-+
¿
i=l n
In order to go on,
n i=l
i p Il i _ Bi
+ ih(C
tion is
(Ilj
j=l
we arrive to
To
in which
¿
)Jj)V. v +
Àij Vp j
¿
¿
n
pj
j=l
i=l j=l
the
state). From the expression for the
Ta: To
model
local
the following
E {l,
we assume V V Vi
that
; the
E {l,
o ; ...
no dissipation
so we have
,n}
is propo-
second a s sump-
are independent,
... ,n}
i~l V.Ji
v
(21)
velocities
we have
may
233
MODELISATION OF FLUID-FLUID INTERFACES
where
OC
is
a tensor
which does not
depend
on the
index
i.
n jh
V. (Àij
V'p j)
u +
n 1: i=l
Let us denote ,i
i
W
W
P
Ts
~N
- P
D
The dissipation expressions
and the
the
the
limits
considered
terso
of
expression
intricated
to
A complete
the
a mixture. term
stress In the
set
entropy
the
to
the
obtained
classic
current
is
the
-
Vp i ® Vp j
in the
equation
of
the
the to
a::D i
(26),
study
V.V)Vpj
irreversible specify
objectivity
processes
a set
of
principle V'p
i
results
.E.
l,
among the
gradients
i P
P
different
lj(v.Ji
]
term
dissipation
small
is
(20)
term
energy
without
introduced
from
of
laws
for
Curie)
is
thermostatic
the
+pi V.V)Vjp
parame-
n 1: i=l
Ts - u +
E
n 1: i=l
equation energy
set
current
(24). Moreover
spherical.
Wi pi
Wi Ji
for
n
following heat
conventional
current,
only we have
P
Ts - u+pV.(Àll
eN
- p 12
E
Àll
_ À 11
Vp)
Vp ® Vp
p(V.V)Vp
(21)
the
we have
[2
and Gouin
classic in
by Dunn and Serrin
phenomenon is no longer
a mixture with the l,i Ji is called = q + ih " (l/TXq' - 1: wi Ji).
of Casal
the
constitutive
(principle
i
the
follo-
V V'
+ p
parameters
(l6),
the
-
Àij(V.Ji
leads
In the case of a continuum with one constituent
We find
(14) have the
(25).
extra
W
Ji +
n 1: j=l
vector
(l4),
equations
q' quantity
n 1: i=l
of
the
we have
of a medium with
quantity
Ji
in Ref.
r io i ,
hand
~N
case
given
(3),
corresponds
of
-eN
the
Àij
involved
V T - V(w,i)
The application
is
On one
tensor
We find
current
dissipation
\..l,i
usually
n 1: j=l
i
~N
thermodynamic
the
appearance
study
u,i
linear
for
mixture.
due
E ; this
the
(20)
The equation
the
n 1: i=l
E
In
for
..9. T
Ta
(21)
the
energy
p
:
(20)
of
extra
)..l,i
n 1: i=l
-
[2
-z:
-z:
wing
-
and Dunn and Serrin
(24).
and
difference
consequent
ly
: the
234
R.
Let of
us assume that
the linear
the viscous
the
thermic
thermodynamic
stress
of
and
the
intrinsic
o With five transport
must verify
dissipations
.!.(V
us
(21).
this
Lastly
define
a
last
an
of
interfacial
to be straight
be
expression the
is more complete
inequalities
the
gradients
the
~o.
to
is
well
to
WS
than
these
the
five
expression
coefficients
the
the
has
is
small
where
of in
such a
a small
of
we
some
¡;
have
a
O
region,
thickness
with
in
with
are
of
these
components.
TI
very
addition
The lines
1;.
in this
have
mixture
components
the directions
direction
we then
the space
much
large the
larger
than
gradients.
property
ç
of the field
We
that are
all
assumed
layer. direction. In the case
a contact
layer.
layer
small
and
¿
surfaces
But
for
S.
l).
Let
normal
surfaces
such as
line
or
a
contact
where
the
large
point
the
study
Let
presentation
layer us
of
is
be
I;
the the
are
I;
to
parallel
the distance
rate
of
volumic
masses
common direction surfaces
of
between
convenience
a motion
¿
and
¿
and
S,
that
surfaces
equal
of
the
densi ty
only which
of
will S
WS
by
the normal
(Fig.
+
of
l)
order
the normal
= O,
and
F,
the displacement
component
appear
on itself.
¿+
is
by F,
We denote
and ¿
the layer. We denote
in motion.
remark
which defines
LO, of
¿.
located inside
equations
interfacial
some
component
the
and
by two extreme
surface
.
«
(E
surfaces
as parallel
interfacial are
The
Öo
scales,
respectively
the
problem the
a tangential
region
masses
the normals to the interfacial
moving surface
defined.
- (ß + v)2
LAYER
being
be an isodensity
In,a dynamic of
as
and appear
gradients S
region
layer
inside
the
Let
co-ordinate
velocity
that
O
¡;
+ ~ - 2ß)
À(y
volumic
directions
ratio
each constituent,
=
for
(25).
inside and outside
ÖO
layer
the
We delimit
O
have a preferential
different
in Ref.
Let
on which
this
and to define
have
of which. is
the
layer
The interfacial gradients
¡;
three-dimensional
gradients
the considered gradients
for
+ Y + ~ + 2v
We assume that
density
theory
results
TVPT
we give
OF THE INTERFACIAL
consider
the
it
outside.
Then the
following
:
MODELISATION
Let
the
Vp
+ Vv)
V
2
coefficients
and Gouin
À
Inside
uncoupled.
to
® I; + I; ® 1;.0)
À
3.
are leads
tensor
where
by Casal
SEPPECHER
irreversible proccesses
ß(O.I;
chosen
P.
GATIGNOL &.
later
W':_ on,
of we
WS give
MODELISATIONOF FLUID-FLUID
To each
physical
quantities
quantity
and
f defined
in the interfacial layer,
~+
f~
We define
the
particles
interface
ps
per by
p
down the
(25)
ò
J
is
the unit
current of
fV
t
of
W dv
In
and
law for
a quantity
form for
a volume
W + W V.W + v. (W(V -
t;
area
W V
with
equation
the
field
f
V
is
W
which
which
is
Let
A
Ws,
and
be a piece limi ted
denote
by
v
normal
to
C, the
T
= I; A v
generated
of
equal
V
(w(V
- W) + J).I;
ds +
fA
the
on
S
the the
S
outward
and
pointed vector
the velocity
of
introduce to
the the
on
lateral
and as
surface
C,
and
the
the
S,
A,
surfaces the
surfaces
A~.
surface
for
C we to
such
C
leaning on
(24)
+ W.V).
a/at
Along
to
cut ted
the
asso-
tangent
normals
and
space Annex).
uni t normal
outside
thin
We
volume
A~, A
by the boundaries
J).n
C.
vector
We also
by
=
(IS/1St
curve
unit
the
whole (cf.
moving with
the
tangent
source
velocity
time derivative
W
and
- W) +
(w(V
aV :
of
the
in
the
equation
rate
with
WS
n
densities
limited
the
to
of
by
A the
(w(V
~
firstly
surface
defined
by
(Fig. 4).
A~
4
of volume,
¢l
with the velocity
and
fA
is
equal
o/1St
write
+
and
by
pieces
wdv
W,
denoted
T
unit
a volume moving
and by
by
IV
of
VS
dv
V
av,
ciated
1St
per
We have to
(26)
velocity
limited by the
quantity
(24),
and
Figure
material
+ J)
W)
+ f aV (w(V - W) + J) .n ds
the
the
density associated
volume.
unit
classical conservation
ò
density per
s
and secondly in an integral
(24)
where
two interfacial
( f)
d~
(Ishii (12)
(23)
We write
f
interfacial mass
on the
in a local form,
we associate
(f)
(22)
the
235
INTERFACES
ds
- W) + J).I;
fv
ds
dv
+ In
order
the
to
introduce
integral
denote
by
expressions H
form
(cf.
mean
curvature
the
Annex),
the
interfacial quantities
appearing
in
three-dimensional and
of order
H
the LO
this
last
tensor
of
mean curvature. (or of order
de f í.ned in equation. order
(22), each
2 associated
We shall
larger
In
than
limit LO
our
we are point
of
with
the
study to
in order
going the
to
transform
surface
second
interfaces
to consider
S
we
fundamental
plane
with
a
inter-
236
R. GATIGNOL &
faces).
Let
us
remark
On these conditions
that
~/L
we have
P. SEPPECHER
O(E)
at
each
point
inside the
interfacial
layer.
we have
fv
f dv
(27)
(fA
fA
f
(Cf
C
fCf.(1l2
d~
~
i
(28)
~+
f
ds
f.n
+ O(E»
- .~).
and in
We
is
P.
&
source
entropy
at lj;
one
densities.
and
a
and
represents
This
priori
the
the
same
the
inside
mass,
assumption
partial
time
total
is
volumic
the
and
momen-
released masses
if of
ljJ
each
constituent.
4.1.
Balance
From tuent
the
law of masses
equation
(3)
and
by
using
the
formalism
of
section
III
we obtain
(36)
p
By adding
i-consti-
the
is
(36)
equations
and
by using
the
definition
W//:
of
»//
_ Ws
by
the
O,
S
divides
normal (II
is
to
S
nega-
and positive
of
expression
in
the
are
the
the curvature
orthonormal
tensor associated
frame defined
by the
with the second fundamental
principal
directions
of
S,
is
following
where
and
tensor
as being
H
two principal
the tensor
with the
curvature
radii.
O
O
O
O
the
orthonormal
frame
We construct a three-dimensional
components
O
in
F
equation The
case).
In each point
the
>
F
region
when the surface
in the opposite
form
E
in
two regions
defined by the
O
O
two principal
directions
of
S
and the normal vector
f;.
Surface
e. of
divergence theorem
Let
B
Along
C
A,
be
a vector
we denote
and by
the
T
field by
one uses
{V Il (f; Il B)l.f;,
is
unit to
E. Let
vector C
A
be a part
tangent
such that
T
to
= f;
S,
of
normal
Il v (Fig.
S to 4).
limited C
and
by the
curve
pointing
We have:
JeB.hllf;)di
the classical it
defined in the
tangent vector
Je B.v di
where
v
easy
theorem to verify
of
Stokes.
that:
If
L
represents the operator
such
that
L(B)
out
MODELISATION
L(B) where
~2
OF FLUID-FLUID
INTERFACES
245
(12 - ( () : V B - (V.()(B.()
is the unit tensor of order two. Let us remark that the operator
X
denoted by
in Barrère and Prud'homme
L
is the operator
(16).
We introduce the following notation P = ~2
-
A//
(
(
projection operator on S (p = ~ = ~2)
A.~
=
s
B// = ~.B
A
is a tensor
where
~
V .B s V
with
f
V B IP.V
= B.P
BJo =(B.(l(,
or order 2 or more,
surface divergence
f
,
surface
of
gradient of
B f
So we have L( BJ.)
'1.(
O
L(B)
With the introduced notations the surface divergence is written
J Surface transport
S
We define the velocity the point Let book of
P us
W
and
M
recall
the
(29)
transport
theorem
us denote by
WS
E
of
is the orthogonal projection of
Germain
M
JA
B. ( dA =
öB JA 16T
t
as is
it
is
given,
used
to
mention
+ B V. W-B. V
wl. (
dA
it is easy to obtain
ò
t
J A fdA
or ò
t
JA
J A l if' +f öt f dA
V.W
Ws. ( =-(aF/at)/ l'IF I. -+
-+
OM = OP + PM ( where
S.
WS
t
its displacement veloci-
by using the relation
on surface
s s /ö
WS
satisfies the relation
on a surface
and where the notation
ö
f (
in motion and let
F(x, t) = O
in each point
ó
B
S
is now
is moving with the velocity
With
B. v di
theorem
Let us suppose the surface ty. The equation of
C
- f (.('1 W).() dA
for that
instance, the
in
the
surface
S
R. GATIGNOL
246
P.
&
SEPPECHER
REFERENCES
(l)
P. GERMAIN La méthode des puissances 12, p , 235-274 (1973).
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J. Méca., vol.
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INC.
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(4) Y. ROCARD Thermodynamique. Masson, (5)
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J.W. GIBBS The Scientific Papers of J. p. 219-274 (1961).
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Gibbs.
(6)
R. GATIGNOL Conditions de saut à travers une interface de Mécanique. p. 318-319 (1985).
(7)
R. GATIGNOL Liquid-Vapor interface
(8)
conditions
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Actes
A.R. DEEMER and J.C. SLATTERY Balance equations and structural vo l . 4, p , 171-192 (1978)
models for
phase interfaces.
ALBANO, D. BEDEAUX and J. VLIEGER On the description of interfacial properties using singular a dividing surface. Physica, vol. 99A, p. 293-304 (1979).
(ID)
J.F. DUMAIS Two and three-dimensional
interfacial
(11)
F.C. GOODRICH The theory of capiUary (1981).
excess
(13)
(14)
M. ISHII Thermo-fluid
du 7ème
vol. l,
Congrès
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(9) A.M.
(12)
INC. New York,
Int.
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V.V. GOGOSOV, V.A. NALETOVA, CHUNG ZA BIN and G.A. SHAPOSHNIKOVA Conservation laws for the mass, momentum, and energy on a phase interface excess surface parameters. Fluid Dynamics. Vol. 6, p. 923-930 (1983). H. BRENNER Micromechanical derivation of the differential Interface Sci. 68, nO 3 p. 422-439 (1979).
at
equation
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for
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true and
J.
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H. GOUIN Tension superficielle dynamique et effet Marangoni pour les interfaces liquide vapeur en théorie de La capiUarité interne. C. R. Acad. Sc. Paris, t. 303, série Il, n? l, p , 5-8 (1986).
(16)
M. BARRERE et R. PRUD'HOMME Equations fondamentales de l'aérothermochimie.
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(17)
J.M. DELHAYE Jump Conditions and Entropy Sources l: p. 395-409 (1974).
(18)
J.M. DELHAYE Local Instantaneous Equations. In : Thermohudraul-ice of two-phase systems for industrial design and nuclear engineering. Edited by J.M. Delhaye, M. Giot and M.L. Riethmuller, Hemisphère Publishing Corperatio~ Washington, p. 95-115 (1981).
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two-phase
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Flow, vol.
MODELISATION
OF FLUID-FLUID
INTERFACES
247
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N. DAHER and G.A. MAUGIN The Method of Virtual Power in Continuum Mechanics. Application to Media Presenting Singular Surfaces on Interfaces. Acta Mechanica, vol. 60, p. 217-240 (1986).
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D. BEDEAUX, A.M. ALBANO and P. MAZUR Boundary conditions and non-equilibrium (1976).
(21)
thermodynamics.
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P. CASAL et H. GOUIN Relation entre l 'équation de l 'énergie et l 'équation de mouuement: en théorie de Korteweg de la capillarité. C. R. Acad. Sc. Paris, t. 300, série II, nO 7, p. 231-234 (1985).
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Tome 1,
Théorie Générale.
Masson et Cie, Paris
