A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

A Stochastic Model of Crystallization in an Emulsion
G. Vallet - D. Trujillo
Laboratoire de Mathématiques Appliquées, ERS-CNRS 2055
Université de Pau et des Pays de lAdour, 64000 Pau, France
guy.vallet@univ-pau.fr, david.trujillo@univ-pau.fr
Abstract :
The main feature of phase changing in dispersed medium is its random behavior. The aim of
this work is to study a stochastic model for the crystallization of emulsions dropplets. We adapt
here some technics of stochastic partial differential equation to prove existence and uniqueness of the
solution. We then give some numerical simulations and compare our results with the experimental
observations.
Keywords: Stochastic partial differential equations, non linear systems.
AMS Classication: 34F05 - 35R60 - 60H15.
1 Introduction
The purpose of this paper is to use nonlinear stochastic partial differential equations techniques to
describe the phase changing in a dispersed medium (here an emulsion). A common way of dealing
with the microscopic properties of an emulsion has been to replace the thermodynamics unkowns
by some kind of averages (see Dumas et al [6] and Vallet [16]).
Exepted in some particular cases (see water emulsions presented at the end of the numerical
simulations chapter), we obtain a reasonable model by using a deterministic one. Nevertheless, the
study of the effects of small uctuations of the deterministic model by some stochastic perturbations
is interesting :
rstly, for physical modelling reasons (see for example Grecksch et al [8] or Holden et al [9] for the
Stratanovich integration),
secondly, for mathematical reasons, because of the lack of compactness for that kind of problem
(see for example Bensoussan et al [3] , Lions et al [13] for the Stratanovich integration or Holden
et al[10] for stochastic conservation laws).
Therefore, the theory of nonlinear stochastic equations has been develloped in many papers, using
multi-valued functions, time discretization, Galerkin approximation, splitting methods, ..., mainly
in one dimension space.
For some stochastic calculus tools we refer to Friedman [7], Grecksch et al [8], Holden et al [9]
and Oksendal et al[14];
for stochastic partial differential equations we refer, moreover all of the above citations, to Bensoussan
[2], Breckner et al [4], Capińsky et al [5], Krylov [11], Kuo et al[12], without forgetting
Pardoux [15] and Walsh [17].
Modelling :
One of the better known energy storing method is the latent heat one.
It mainly consist in shifting the temperature of a given material to a given value using a certain
amount of energy, provided it is cheap enough. This brings about a phase change in the structures
of the material. When the material reverts to its initial temperature, the inverse phase change
yields a part of the energy stored during the rst phase change.
Experimentation, on an industrial scale, has shown that, for efficiency reasons, the phase changing
material (PCM) has to be scattered in droplets into an other material, which does not change
1

phase during the experimentation. We thus consider an emulsion containing the phase-change
material. However, repeating the melting-freezing cycles, destroys the dispersed structure of the
emulsion. To avoid that, a surface-active agent (surfactant) can be used and so the behavior of
the emulsion is closed to the behavior of solid matter. Therefore, the effect of free-convection can
be ignored.
The object of this paper is to talk about the crystallization of an emulsion.
This process is based on two phenomena (cf. Dumas et al [6]) :
1) the undercooling, dened as the difference between the melting temperature TF
and the
crystallization temperature, increases as the sample size of PCM decreases. For example, for water,
3
3
the undercooling is 14 K for a few cm macrosamples and is 38 K for a few m microsamples.
Moreover, any droplet which temperature is less than TF
may crystallize as soon as any perturbation
or shock occurs. Where from the second phenomenon :
2) the main feature of emulsions crystallization in an instable dispersed medium (here an
emulsion with temperature less than TF
) is its stochastic behavior. Samples of PCM which
are apparently identical, will not transform at the same temperature during the cooling process.
This leads to the notion of nucleation rate. The Nucleation Theorie gives us a function J,
the
deterministic part of the crystallization speed, per unit of time. In the sequel, we consider the
nucleation rate as a stochastic perturbation of this speed J.
We assume that J is a positive lipschitz function of the temperature with, J(0) = 0 and J( x)
=
0 , ∀x ∈ [ T , + ∞[
.
∞
F
One mays nd a presentation of the experimental context in the last section.
Let us denote by u the temperature of the emulsion and ϕ( t, x)
the proportion of crystallized
droplets, in the neighborhood of a point x and a time t .
Then, the mathematical modelling proposed by Dumas et al [6] is based on :
the nonlinear heat equation with a heat source term proportional to the speed of crystallization,
i.e.
∂u
∂ϕ
( u)
= ,
∂t
∂t
with some Fouriers type boundary conditions
in
(1)
∂ u
∂
N
( ) ∞
= ( u u ) , on = ∂
where u is a given temperature on the boundary ,
the following stochastic differential equation for the speed of crystallization
∂ϕ
∂t
= (1 ϕ) J( u) + b( u, ϕ)
∂w
, in
∂t
where 1 ϕ is the proportion of uncrystallized droplets, J( u) the nucleation rate, w the standard
real Wiener process and b the random part of the nucleation phenomenon.
A rigorous formulation of our system will be given below.
2 Notations and hypothesis
From now on, we consider the following notations :
is a bounded domain of R with a smooth boundary and an outward unit normal ,
for T > 0 , Q is the cylinder ]0 , T [ and its lateral boundary ]0 , T [ ,
2
(2)
(3)

( , Ϝ, P ) is a complete probability space, E the expectation and ( Ϝt)
t ∈[0
,T ] a right continuous
ltration such that Ϝ contains all Ϝ-null
sets,
denotes the real valued standard Ϝ -Wiener process,
t ∈[0
,T ]
t
H = L () , V = H () , H = L ( , H) = L ( , dP dx)
,
2 2 2
V = L ( , V ) , H = L (0 , T, H) = L ( Q, dP dxdt)
and
2 2
V = L (]0 , T [ , V) = L (]0 , T [
, V ) ;
|| . || and || . || denote respectively the H and V norms,
V
∞ ∞ ∞
0 0
∞
∞
0 0 0
F F
x
. the function , dene by ( x) = ( s) ds; which satises for all x and y,
x
′
. the function G, dene by G( x) = ( s) ds,
s>t s
0
2 1 2 2
is a lipschitz continuous increasing function with (0) = 0 and 0 < m M
(therefore, this model does not lead to a degenerated parabolic equation),
> 0 ; u ∈ L () , u 0 ; u ∈ L () denotes the cooling temperature imposed on the
boundary, with u positive, non increasing with respect to t and regular,
ϕ ∈ L () , 1 ϕ 0 ( ϕ = 0 in the experimentation),
J is a lipschitz continuous, positive function with, J( x ) = 0,
for any x of R and any x of
[ T , ∞[
, where T is the fusion temperature,
b is a lipschitz continuous, positive function on R such that b( u, ϕ)
= 0 if :
. u 0
(as the temperatures are in Kelvin, nothing may happen for non positive temperatures),
. u TF
;
(there is no crystallizations for a temperature above the fusion temperature),
. ϕ 0;
(some more crystallizations may occur only if some crystallizations already exist in the neighborhood,
i.e. ϕ > 0),
. ϕ 1;
(there is nothing else to crystallize if ϕ 1),
( w( t))
Moreover, for technical reasons, we need :
3 Recalls on Stochastic Integration
0
( x)( y x) ( y) ( x) ( y)( y x ) ,
. and the function , dene by ( x ) = max(0, min( x, 1)) ,
C is any constant that we do not need to precise.
0
We recall here some results concerning the Itô integration, which can be found in Grecksch et al
[8].
( , Ϝ, P, ( Ϝt) t0)
is a standard ltered probability space i.e. ( , Ϝ,
P ) is a probability space,
( Ϝt) 0
is an increasing family of
algebras of Ϝ, Ϝ0 contains all Ϝ-null sets and Ϝt
=
∩ Ϝ .
3
2
′

( w( t)) t ∈[0
,T ] is a continuous process with values in R , and w(0)
= 0.
For every s < t, w( t) w( s)
is a Gaussian real valued random variable with mean 0 and
variance t s.
Moreover, ∀s < t, ∀f
Ϝ -measurable
n
L 2(0 ,T ; L 2(
,H))
E[( w( t) w( s)) f] = 0 and E[( w( t) w( s)) ] = t s.
About the Itô integration :
2 2
t
∀f ∈ L (0 , T ; L ( , H)) , ( f( s) dw( s))
is a continuous Ϝ -measurable process and
If f ⇀ f then
and
s
t0 t
t t
2
2
E[ f( s) dw( s)] = E[ f( s) ds ] .
T
T
f ( s) dw( s) ⇀ f( s) dw( s)
n
L 2
0 ( , ,P,H)
0
. .
f ( s) dw( s) ⇀ f( s) dw( s ) .
0
L 2(
, ,P, C([0
,T ]))
0
The Fubini Theorem :
2 2
Let f ∈ L (0 , T ; L ( , H)) and h ∈ H with | f| h then
t t
f( s, ., x) dw( s) dx = f( s, ., x) dx dw( s) P a.s.
The Itô formula :
2 2
Let f, g, M ∈ L (0 , T ; L ( , H)) such that ∀t ∈ [0 , T ] ,
t t
M( t) = M + g( s, ., . ) ds + f( s, ., . ) dw( s)
then, for any enough regular ϕ : R → R :
n
0
0
0 0
t
ϕ( t, M( t)) = ϕ (0 , M ) + ∂ ϕ( s, M( s)) g( s, ., . ) ds+
t t
∂ ϕ( s, M( s)) g( s, ., . ) ds + ∂ ϕ( s, M( s)) f( s, ., . ) dw( s)+
0
2
t
1 2
∂2, 2ϕ(
s, M( s)) f( s, ., . ) ds.
2 0
So E ϕ( t, M( t)) dx = E ϕ (0 , M ) dx +
0
0
t
E ∂ ϕ( s, M( s)) g( s, ., . ) ds dx +
0
t
E ∂ ϕ( s, M( s)) g( s, ., . ) ds dx +
0
t
1
2
E ∂2, 2ϕ(
s, M( s)) f( s, ., . ) ds dx .
2
0
0
0 0
1
2
2
0
4
0
0
1
2
2

The Stochastic energy equality :
2 2 2 2
Let f, M and M in L (0 , T ; L ( , H )) , L (0 , T ; L ( , V )) and
2 2 ′
L (0 , T ; L ( , V )) respectively, Ϝt
-measurable, such that for t in [0 , T ] ,
t t
M( t) = M + M ( s) ds + f( s, ., . ) dw( s)
then M can be consider as a continuous adapted process and, P a.s., for any t,
2
|| M( t) || = || M ||
t
+ 2 < M ( s ) , M( s ) > ds+
4 Denition of a Solution
The couple ( u, ϕ)
∈ V H is a solution of the system (1) - (2) and (3) if
n
∞ ∞
n
V ′ ,V
t t
2
2 M ( s) f( s) dx dw( s) + f ( s, ., . ) ds.
∀v ∈ V, ∀t ∈ [0 , T ];
t
( u( t) u ) v dx +
t
∇( u) ∇v dx ds +
∞
( u u ) v d ds =
5 Existence of a Solution
5.1 Discretization of the system
0
t t
ϕ( t) ϕ = (1 ϕ) J( u) ds + b( u, ϕ) dw( s ) .
0
u0 and ϕ0 being given, let us inductively construct the sequence ( u n, ϕn)
n in the following way :
( u n+1 , ϕn+1) ∈ V H is the solution of the system : P a.s. and any v in V ;
( u u ) v dx + t ( u ) v dx + t ( u u ) v d =
∞
n+1 n ∇ n+1 ∇ n+1 n+1
0
0 ϕ 1,
and
( ϕ( t) ϕ ) v dx;
In order to discretize the system (4) - (5), let us consider, for N 1 and n ∈ { 0, 1, 2 , .., N}
:
n
0 2
0 0
0 0
0 0
0 0
In order to prove the existence of a solution to our system, we propose to adapt the demonstration
given in Breckner et al [4] or in Grecksch et al [8] for a stochastic partial differential equation. This
method (the Rothe method) is based on a time discretization, implicit for the partial differential
equation and explicit for the stochastic differential equation (for the Itô integration).
( ϕ ϕ ) v dx;
n+1 n
ϕ ϕ = tJ( u ) (1 ϕ ) + b( u , ϕ )( w w ) .
n+1 n n n n n n+1 n
0
0
t =
T
N
=
,
t n T,
w = w( n t ) ,
u = u ( n t ) .
5
(4)
(5)
(6)
(7)

Remark 1 For technical reasons (lost of the maximum principle for ( ϕn)
), we need to use
J( un) (1 ϕn) instead of J( un) (1 ϕ n)
. We then have to prove that, at the limit when N goes
to innity, the solution ϕ obtained satises 0 ϕ 1.
Proposition 1 There exists a sequence ( u n, ϕn) n∈ , ( u n, ϕn)
Ϝnt
-measurable, with values in
V H,
satisfying (6) and (7).
Proof. The result leads from a classical xed point argument (see Grecksch et al [8] for example
in the context of stochastic calculus).
5.2 A priori estimates
Lemma 2
N
2
2
∀n ∈ { 0, 1, .., N } , E|| ϕ || M so t
E|| ϕ || M,
N
1
Proof. Since [( ϕ ) ( ϕ ) + ( ϕ ϕ ) ] = ( ϕ ϕ ) ϕ ,
tJ( un) (1 ϕn) ϕn+1 + ϕn+1 b( u n, ϕn)( wn+1 w n)
.
As, ϕ b( u , ϕ ) is Ϝ -measurable, E ϕ b( u , ϕ )( w w ) =
E ( ϕ ϕ ) b( u , ϕ )( w w ) + E { ϕ b( u , ϕ )( w w ) } =
E ( ϕ ϕ ) b( u , ϕ )( w w )
1
2
E( ϕn+1 ϕn) + Ct. 4
Since, E tJ( u ) (1 ϕ ) ϕ C t E( ϕ ) + 1 ,
2 2
2
E( ϕ ) E( ϕ ) + E( ϕ ϕ ) C t E( ϕ ) + 1 .
n n
2
E( ϕ ) + E( ϕ ϕ ) C t E( ϕ ) + C
n
2
C t E( ϕ ) + C + C tE( ϕ ) .
n
2
Whereas t 0,
∃M > 0, ∀n ∈ { 0, 1, .., N 1}
,
n
and E|| ϕ ϕ || M.
n=0
1 2 2
2
[( ϕn+1) ( ϕn) + ( ϕn+1 ϕn)
] =
2
∀ n, E|| ϕ || Ce Ce
E|| ϕ ϕ || M t.
n=0
1 2 2
2
2 n+1 n n+1 n n+1 n n+1
n n n n t n+1 n n n+1 n
Whereafter,
n+1 n n n n+1 n n n n n+1 n
n+1 n n n n+1 n
n n n+1 n+1
2
n+1 n n+1 n n+1
2
n+1
2
n+1
2
k=0
k=0
Gronwall lemma (cf. Bainov [1]) since :
k+1 k
k=0
2
n+1 n
k=0
k n+1
2
k
2 Cnt CT
n+1
2
n+1 n
6
n
k+1
2

Proof. According to equation (7 ),
and so,
Lemma 4
Thus, the lemma leads from the lemma 2.
E|| ϕ ϕ || C t + Ct.
N
2
2
∀n ∈ { 0, 1, .., N } , E|| u || M so t
E|| u || M,
N
1
∞
n+1 n+1 n+1 n+1 n n+1
′
n+1 n+1 n+1
N
2
and t
E|| u || M.
Proof. Taking v = un+1
as a test function in (6), one has :
( u u ) u dx + t ∇( u ) ∇u
dx+
t ( u u ) u d ( ϕ ϕ ) u dx.
Since, un is Fnt measurable, E ( ϕn n) n =
+1 ϕ u +1 dx
E ( ϕ ϕ ) ( u u ) dx + E ( ϕ ϕ ) u dx
Lemma 5
2 2 2
n+1 n n n n n n+1 n
|| ϕ ϕ || 2 t || J( u ) (1 ϕ ) || + 2 || b( u , ϕ )( w w ) || ,
∃M
> 0,
n=0
n n + n n n n n
1 2
2
E|| ϕ +1 ϕ || E|| u +1 u || + t E J( u ) (1 ϕ ) u dx.
4
Moreover, as ∇( u ) = ( u ) ∇u
,
n n
1
n + k k
k V
2
1
+1 +
2
2
2
+1
+1 2
E|| u || E|| u u || m t E|| u ||
n n
1
∞
+ k L2
k k
2
0 + +
2
2
+1 2 t
2
E|| u || E|| u || () E|| ϕ +1 ϕ || C.
N n
2
V
N
1
n=1
n=0
Proof. Since is a lipschitz function, this result is obvious.
n
n=1
n V
n=0
n+1 n n+1 n+1 n+1
n+1 n n+1 n n+1 n n
1
n n + n n n V
2
∞
n n n
1
2 2
2
+1 +1 + +1
4
2
( ) + +
2
+1 2
E|| u || E|| u || E|| u u || m tE|| u ||
t
E
2
u d E|| ϕ +1 ϕ || C t.
Then, summing up from 0 to n,
we get :
∃M
> 0,
2 2
n+1 n
k=0
k=0
2
n+1 n
E|| u u || M,
k=0
k=0
t E|| ( u ) || M and E|| ( u ) ( u ) || M.
7
n
n+1 n
2
2

Lemma 6
Assuming that u ∈ V ; ∃M > 0, ∀n ∈ { 0, 1,
.., N } ,
Proof. Taking = ( n+1) ( n)
as a test function in (6); for any positive :
2 2
( n+1 n)( ( n+1 ) ( n)) + ||∇ n+1 || ||∇ n ||
2
||∇ n+1 n || n+1 n
( ) ( )
2
+
v u u a
t
u u u u dx u u
t
[ ( u ) ( u )] + t
2
( u ) ( u ) d
As t
is small enough,
2 1
|| ′ || ∞
2
n
2
E|| G( u ) G( u ) || M
k=0
k+1 k
n
2
and t || ( u ) ( u ) || M.
k=0
k+1 k V
∞
n+1 L2 n n V
2
2
|| || () + || +1 ||
2
t
a t
u
( u ) ( u ) +
a
2
( ϕ ϕ )( ( u ) ( u )) dx.
n+1 n n+1 n
2
n+1 n n+1 n n+1 n
and so :
1
( n+1 n)( ( n+1) ( n)) + ( n+1) ( n)
+
2
||∇ ( ) || ||∇ ( ) || + || ( ) ( ) || || ||
2
2
2
+ || n+1 n || V n+1 n n+1 n
Moreover, one has :
u u u u dx t u u d
t
t
u u u u
a
a t
( u ) ( u ) + ( ϕ ϕ )( ( u ) ( u )) dx.
2
E ( ϕ ϕ )( ( u ) ( u )) dx
2 2
2
∞
n+1 n n+1 n V n+1 L2
2 u ()
n+1 n n+1 n
2
2
|| n+1 n|| + || n+1 n ||
2
|| ||
1
b
E ϕ ϕ E ( u ) ( u )
2b
b
E ϕ
2
ϕ
E ( u u )( ( u ) ( u )) dx .
So, for b = and a = , we get :
′
2 || || ∞
n+1 n + n+1 n n+1 n
2b
1
2 2
( n+1 n)( ( n+1 ) ( n)) + ||∇ n+1 || ||∇ n ||
4
2
|| n+1 n || V n+1 n
( ) ( )
2
+
t
E u u u u dx E u E u
t
E ( u ) ( u ) + t E ( u ) ( u ) d
4
Hence, after summing up from 0 to n,
we get
k=0
0
t( ( u ) ( u )) ( u u )( ( u ) ( u ))
2 2 ∞
n+1 n n+1 L2
2 +
()
C E|| ϕ ϕ || t E|| u || .
n n
1
2
2
|| ( k+1) ( k)
|| + || k+1 k || V
4
t
E G u G u
E ( u ) ( u ) +
4
k=0
t
2
E||∇ ( un+1) || + t E ( un+1) d t E ( u0) d+
2
8

5.3 Existence
N N
N∈
N∈
t
E u t E u C E ϕ ϕ .
2
||∇
2
0 || ||
∞
+1|| 2
|| ||
=0
2 n
n
( ) + k L () +
k+1 2
k V
k
k=0
So, leading from the above lemmas, one has
n n∈
n
t
2
E|| ( uk+1) ( uk) || V C.
2
For any sequences ( x ) of H (resp. of V),
let us set :
N
N
N
x ( t) = x I
( t) if t > 0 and x (0) = x
N
N xk xk1
x ( t)
=
( t ( k 1) t) + xk1 I[(
k1)t,k t]
( t ) .
t
( x ) and ( x
) are some sequences in H (resp. V)
and one has :
|| x || t E|| x || ,
N
N 2 =
k
2
H
N
k=1
L∞
( H)
=
k =1,..,N
k
2
H
N N
N
1
2 t
= 3
k+1 k
2
H
N
∂t
k=0
N
1
2
L 2(]0
,T [
,V ′ ) = t
E
k=0
xk+1 xk
2 || V ′
t
One has the same kind of results, replacing H, H, H respectively by V, V,
V.
Lemma 7
N
N∈
- ( ϕ ) is a bounded sequence in L ( H)
and H as well,
N
N
N∈
- ( ϕ ϕ
) converges toward 0 in H,
N
N
N N
N∈
k ]( k1)t,k t]
0
- ( u ) ∈ is a bounded sequence in H,
L ( H) , L (0 , T ; L ( )) and L ( ) as well,
- ( u u
) converges toward 0 in H.
N N
As ( u ) = ( u)
,
N ∞
- ( ( u )) N∈
is a bounded sequence in L ( V)
and V as well,
N
N - ( ( u ) ( u)
) converges toward 0 in H.
N∈
n
1
2
2
|| ( k+1) ( k) || + ||∇ n+1
||
4
t
E G u G u E ( u ) +
4
k=0
k=1
k=1
k=0
|| x || Max E|| x || ,
|| x x || E|| x x || ,
∂x
|| || ||
The a priori estimates lead to :
∞
∞ ∞
5.3.1 Study of the stochastic differential equation
2 2
According to the lemma 7, it is possible to extract a subsequence of ( ϕ ) ∈ , still noted ( ϕ ) ∈ ,
N
∞
such that ϕ converges weakly towards ϕ in H (and L ( H) weak as well).
Step 1 : energy equality
9
.
N N
N N
(8)

Considering (7), one has, for any integer n,
and so,
where ε( t) goes to 0 in H with t
.
Moreover, ( J( u ) (1 ϕ )) ∈ is a bounded sequence in H,
as well as ( b( u , ϕ )) ∈ ; so,
there exists a subsequence of ( ϕ ) , still noted ( ϕ ) , and, J and B in H,
such that :
from H to H,
one has
n
1 n
1
k k
k=0
k=0
ϕ ϕ = t J( u ) (1 ϕ ) + b( u , ϕ )( w w )
n
0
( n 1)t
( n 1)t
N N
ϕ ( n t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s)+
t J( u ) (1 ϕ ) + b( u , ϕ )( w w ) .
Thus, for any t in [0 , T ] and n such that ( n 1)t < t nt, 0
0 0 0 0 1 0
t t
N N
ϕ ( t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s)+
Since and b are bounded functions,
0 0
t J( u ) (1 ϕ ) + b( u , ϕ )( w w )
0 0 0 0 1 0
t t
n n
J( u ) (1 ϕ ) ds +
n n
b( u , ϕ ) dw( s ) .
( n1) t
( n1)t t t
N N
ϕ ( t) ϕ (0) =
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) + ε( t)
0 0
N N
N
N N
N∈
N∈
N N
J( u ) (1 ϕ ) ⇀ J ,
N N
b( u , ϕ ) ⇀ B.
t t
Remarking that the applications u ↦→ u ds and u ↦→ u dw( s)
are continuous linear functions
0 0
t t t t
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) ⇀ J ds + B dw( s ) .
t t
ϕ( t) ϕ(0) = J ds + B dw( s) in H;
for any t (since the Itô integral implies that ϕ can be chosen continuous).
2at
2
Then, for any a > 0 , by applying the Itô formula with f( t, x) = e || x||
, we get :
2at
2
e E|| ϕ( t) || E|| ϕ || =
k k k+1 k
0 0 0 0
So, as N goes to innity, we get,
0 0
10
0 2
0
N N N
(9)
(10)

t
t
2as 2
2as
2
2 a e E|| ϕ|| ds + e E 2 J ϕ dx + || B|| ds.
Step 2 : discrete energy inequality
Let us look for a similar formula to (11), in the discrete case.
Let us multiply (7) by ϕ . Then,
1 2 2
2
[( ϕn+1) ( ϕn) + ( ϕn+1 ϕn) ] = tJ( un) (1 ϕn) ϕn+1+
2
( ϕ ϕ ) b( u , ϕ )( w w ) + ϕ b( u , ϕ )( w w ) .
2
a( n+1)t 1 a( n+1)t 2 ant 2 2
a( n+1)t 2
[( e ϕn+1) ( e ϕn) + e ( ϕn+1 ϕn)
] =
2
1 2 2 a( n+1)t 2ant 2
a( n+1)t ( ϕn) ( e e ) + e tJ( un) (1 ϕn) ϕn+1+
2
2
a( n+1)t e ( ϕ ϕ ) b( u , ϕ )( w w )+
2
a( n+1)t n n n n+1 n
e ϕ b( u , ϕ )( w w ) .
|| || || || || ||
|| ||
1 2 a( n+1)t 2 2ant 2 2 a( n+1)t 2
[ e E ϕn+1 e E ϕn + e E ϕn+1 ϕn
]
2
2at 2ant 2 1 e
2 a( n+1)t ae E ϕn
+ e tE J( un) (1 ϕn) ϕn dx+
2a
2 a( n+1)t e tE J( u ) (1 ϕ )( ϕ ϕ ) dx+
2 a( n+1)t 2
2 e
2
E b( u n, ϕn) ( wn+1 wn) dx +
E|| ϕn+1 ϕ n||
.
2
2 a( n+1)t
2
n+1
n+1 n n n n+1 n n n n n+1 n
So, multiplying again by e , we get :
Thus,
n+1 n n n n+1 n
Since (7) implies that
E J( u ) (1 ϕ )( ϕ ϕ ) dx = tE [ J( u ) (1 ϕ
2
)] dx,
one has,
e
0
2 a( n+1)t 2 2ant 2 2ant
2
e E|| ϕn+1|| e
E|| ϕn|| 2a t e E|| ϕn||
+
2
a( n+1)t 2e tE J( u ) (1 ϕ ) ϕ
2
dx + C( t)
+
2
a( n+1)t t e E b( u , ϕ
2
) dx.
Therefore, after summing up from 0 to N 1,
we get,
n n n+1 n
n n n+1 n n n
n n n
n n
N
1
2aNt
N 2
2akt
2
e E|| ϕ || E|| ϕ || 2a t e E|| ϕ || + C t+
k=0
N
1
2
a( k+1)t 2 t e E J( u ) (1 ϕ ) ϕ dx+
k=0
k k k
N
1
2
a( k+1)t 2
t e E b( u , ϕ ) dx,
k=0
0 2
0
11
k k
k
(11)

and,
e E ϕ ( T ) E ϕ 2 a e E ϕ dt+
2 2
0
2
T
aT
||
N
|| || ||
0
2at ||
N 2
||
N
1
k=0
2akt 2
a( k+1)t T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) ϕ dx dt + e E b( u , ϕ ) dx dt.
0
0
Step 3 : Lower semi-continuity
T T
Since applications u ↦→ u ds and u ↦→ u dw( s)
are linear continuous functions from H to
H,
0 0
T T T T
N N
J( u ) (1 ϕ ) ds +
N N
b( u , ϕ ) dw( s) ⇀ J ds + B dw( s)
H
0 0 0 0
and thanks to (10), ϕ ( T ) ⇀ ϕ( T ) in H.
Let us set
Remark that 0 and that (11) implies,
2aT N 2 2aT
2
T
T
+ E|| ϕ || 2 a
2as 2
e E|| ϕ|| ds +
2as
e E 2
2
J ϕ dx + || B|| ds =
0 2
0
2aT 2 2aT
N 2
+ e E|| ϕ( T ) || = e lim E|| ϕ ( T ) || L
where, we note L = E|| ϕ ||
+ lim { 2 a e E|| ϕ || dt+
0 2
N∈
2 N 2
T
0
2at
N 2
T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) ϕ dx dt + e E b( u , ϕ ) dx dt } .
Since the sequences
0
0
T
T
2at
N 2
2at
N N N
( e E|| ϕ || dt ) , ( e E J( u ) (1 ϕ ) ϕ dx dt ) ,
0
0
T
T
2at
N N 2
2at
N 2
( e E b( u , ϕ ) dx dt) and e E|| ϕ ϕ || dt
0
N
2C t ( e e ) + C t+
Thus,
N∈
are bounded in R,
a subsequence can be found, still indexed by N,
such that all of the above
sequences converge.
consequently,
T
2at
N 2
L = E|| ϕ || 2 a lim e E|| ϕ || dt+
0 2
T
2 lim
2t
e E
N N N
J( u ) (1 ϕ ) ϕ dx dt+
0
T
lim
2at
e E
N N 2
b( u , ϕ ) dx dt.
0
E|| ϕ( T ) || lim E|| ϕ ( T ) || .
= e lim E|| ϕ ( T ) || e E|| ϕ( T ) || .
12
0
0
0
N∈
(12)

Let us set,
Remark that L
and that,
T
L = 2 a
2at
N 2
e E|| ϕ ϕ || dt+
T
2
2at
e E
N N N
[ J( u ) (1 ϕ ) J( u) (1 ϕ)]( ϕ ϕ) dxdt
T T
+
2at e E
N N
2
( b( u , ϕ ) b( u, ϕ)) dxdt +
2at
N 2
e E|| ϕ ϕ || dt.
0
0
0
T T
(1 2 a + c( J, , b)) 2at N 2
e E|| ϕ ϕ || dt + c( J, , b) 2at
N 2
e E|| u u || dt,
0
T T
L = 2 a
2at N 2
e E|| ϕ || dt + 2
2at
e E
N N N
J( u ) (1 ϕ ) ϕ dxdt+
0
0
T T
2at N N 2
2at
N
e E ( b( u , ϕ )) dxdt + 4 a e E[ ϕϕ dx] dt
0
T T T
2at 2
2at N 2
2at
2
2 a e E|| ϕ|| dt + e E|| ϕ ϕ || dt + e E ( b( u, ϕ)) dx dt
0
0
T T
2
2at e E
N N
b( u , ϕ ) b( u, ϕ) dxdt 2
2at
e E
N N
J( u ) (1 ϕ ) ϕ dxdt
0
T
2 2at
e E
N
J( u) (1 ϕ)( ϕ ϕ) dx dt.
Thus, at the limit as N goes to innity, one has :
Since,
N
N
0
T
lim L = L E|| ϕ || + 2 a
2at
2
e E|| ϕ|| dt+
0 2
T T
lim
2at N 2
e E|| ϕ ϕ || dt +
2at
e E
2
( b( u, ϕ)) dx dt
0
T T
2at 2at
2 e E B b( u, ϕ) dx dt 2 e E J ϕ dx dt.
0
T T
2at 2
2at
e E ( b( u, ϕ)) dx dt 2 e E B b( u, ϕ) dx dt
=
0
N
N
13
0
0
0
0
0
0
0
0
0
(13)

we get,
T T
2at 2
2at
2
e E ( B b( u, ϕ)) dx dt e E B dx dt,
T T
2at 2
2at
2
lim L = L E|| ϕ || + 2a
e E|| ϕ|| dt e E B dx dt
N
0
T T
2
2at e E J ϕ dx dt + lim
2at
N 2
e E|| ϕ ϕ || dt+
5.3.2 Study of the parabolic equation :
According to the lemma 7, it is possible to extract a subsequence of ( u ) ∈ , still noted ( u ) ∈ ,
N N ∞
such that u (resp. ( u ) N∈
) converges weakly towards u (resp. ) in V as well as in L ( V)
weak- .
1
N N
Let us consider v ∈ H ( Q ) , and set v the step function and v
the piecewise affine continuous
function from [0 , T ] to V, built from the sequence v = v( n t ) as introduce in (8) .
Step 1 : energy equality
According to (6), for any n, one has :
0
T
2at
2
e E ( B b( u, ϕ)) dx dt.
N
T
T
2at 2
2at
N 2
+ e E ( B b( u, ϕ)) dx dt + lim e E|| ϕ ϕ || dt.
0
n 1
v +1 v
unvn u0v0 dx t
uk t
∇ ∇
n 1
n 1
t ( u ) v dx + t ( u u ) v d =
k=0
0 2
Following (12), one nds that lim L
Thus,
0
k+1 k+1
k=0
n
1
∞
k+1 k+1 k+1
v +1 v
ϕnvn ϕ0v0 dx t
ϕk t
N N
dx+
∇ ∇
nt N
N N N
N
u ( n t) v ( n t) u0v (0) dx u
0
+
nt n 1
N N
( ) +
N
( k+1 k)
+
∂v
∂t
dx ds
u v dx ds t u u ∂v
∂t dx
0
nt
N ∞,N
N
( u u ) v d ds =
N N
ϕ ( n t) v
( n t) dx
nt N
n 1
N
N
N
ϕ v (0) dx ϕ + ( k+1 k)
∂v
∂v
dx ds t ϕ ϕ
∂t ∂t dx.
0
t [0 , T ] n ( n 1)t < t nt,
t
N N N
u ( t) v ( t) u0v (0) dx
N
N
u +
∂v
Therefore, for any in and for such that we get,
∂t
dx ds
0
n
0
k=0
k=0
0
k=0
0
0
14
0
0
0
k k
k k
k=0
dx.
N N
(14)

Since
t t
N N
N ∞,N
N
∇( u ) ∇v dx ds + ( u u ) v d ds =
0 0
t N
N N N
N
ϕ ( t) v ( t) ϕ0v (0) dx ϕ
0
+
n N N
[ ( ) ( )]
n N N
[ ( ) ( )] +
∂v
∂t
dx ds
u v t v n t dx ϕ v t v n t dx
1
+1
1
+1
n t
N
n
t
n t
n
t
+
( )
( )
N
u
n
N
k k
n
N
k k
∂v
∂t
∂v
dx ds ϕ
∂t
dx ds
t u u ∂v
∂t
dx t
∂v
ϕ ϕ
∂t dx
=0
k
=0
k
nt nt
n N
n ∞,n
N
∇( u ) ∇v dx ds ( u u ) v d ds.
∞,n
V
√ √
n
|| || 2 || n|| 2
n
V
V + || || V
1
+1
+1
=0
n
vk vk
t E ( uk uk) t
k
k=0
k=0
0 0
0
nt N
n N N n ∂v
u [ v ( t) v
( n t)] dx = u
( s) ds dx;
∂t
according to (4), one has
nt
E
nt
n n
∇( u ) ∇v dx ds E
n ∞,n
n
( u u ) v d ds
Moreover, using lemmas 4 and 2, we get,
t || v ||
[2 E|| u || + || u
|| ]
t v t E u C C t v .
0
0
∞ ∞
0
dx
1 1
2
+1
|| 2
+1 || || ||
n
n
vk vk
t
E uk uk
t
C t|| (respectively with ); and then, for any ,
||
t N
N N N
N
( ) ( ) 0
(0)
0
+
t t
N N
∇ ( ) ∇ +
N ∞,N
N
( ) =
∂v
∂t
ϕ t
C t
u t v t u v dx u ∂v
∂t
dx ds
u v dx ds u u v d ds
where ε( t) goes to 0 with t.
t
t
n V
t N
N N N
N
ϕ ( t) v ( t) ϕ0v (0) dx ϕ + ( )
∂v
dx ds ε t
∂t
Let us consider, as proposed in Grecksch et al [8], r ∈ L (0 , T ) and b ∈ L () . Then
n V
N
T
T
E
N N
u ( t) v
( t) r( t) b dx dt E
N
u v
(0) r( t) b dx dt
15
t
t
t

A : V → V
and A ∈ V :
∞
∞
T t
N
E u ∂v
∂t
′
′
0 0
∞
′
V ′ ,V
V ′ ,V
∞
0 0
∞ ′ ∞ ∞
∈
=
T t
N N
r( t) b dx ds dt + E ∇( u ) ∇v
r( t) b dx ds dt+
T t T
E
N ∞,N
N
( u u ) v r( t) b d ds dt = E
N N
ϕ ( t) v ( t) r( t) b dx dt
0 0
T
T t N
N
N
E ϕ v (0) r( t) b dx dt E ϕ ( )
∂v
r t b dx ds dt
∂t
0
T
+ E ε( t) r( t) b dt.
T T
E u( t) v( t) r( t) b dx dt E u v(0) r( t) b dx dt
T t T t
E u ( ) + ∇ ∇ ( ) +
0 0 0 0
T t T
∞
( ) ( ) = ( ) ( ) ( )
∂v
r t b dx ds dt E v r t b dx ds dt
∂t
E u u v r t b d ds dt E ϕ t v t r t b dx dt
T T t
E ϕ v(0) r( t) b dx dt E ϕ ( )
∂v
r t b dx ds dt.
∂t
the monotone operator :
< Au, v > = ∇( u) ∇v
dx + uv d,
< A , v > = ∇ ∇v
dx + uv d.
t t
∞
u( t) u + A ds = F ds + ϕ( t) ϕ ,
where F V is dened by < F , v > u v d.
N
N
0
Thus, it follows that u ( T ) converges weakly in H towards some U and
T T
∞
U = u A ds + F ds + ϕ( T ) ϕ .
∞
0 0
But, since one can choose u continuous, it ensures that u ( T ) ⇀ u( T ) in H.
2at
2
Then, by applying the Itô formula to (15) with f( t, x) = e || x||
and since
N
0
0 0
0 0
Then, passing to the limit to innity with N,
one obtains :
Let us note :
0 0
0 0 0
0 0
0
0
t t
ϕ( t) ϕ(0) = J ds + B dw( s ) ,
N
0
0 0
0
Then A is the weak limit in V of ( Au ) and one has,
0 0
16
0
0
(15)

2at
2
one has : e E|| u( t) || || u || =
t t
2 aE
2as 2
e || u( s) || ds 2 E
2as ∞
e < A , u > ds+
t t t
2 E
2as e
∞
u u dds + 2 E
2as e J u dxds + E
2as
e
2
B dxds.
Step 2 : discrete energy inequality
1
+
2
1
Let us look for a similar formula in the discrete case. For this, remark that using the proof of
lemma 4, we get
2 2
2
|| un+1|| || un|| || un+1 un|| + t ∇( un+1 ) ∇un+1
dx+
2
t u d t u u d =
∞,n+1
n+1
t J( u ) (1 ϕ ) u dx + b( u , ϕ ) u dx ( w w ) .
T
2aT N 2
at N
( ) 0 2
2
2 2
0
N
1
2akt 2
a( k+1)t 2as ∞ 2as
2as 2 2aT
2 || ||
2aT
N 2
N
N∈
Therefore, using a similar demonstration as the one proposed for the sequence ( ϕ
) , one has :
e E|| u T || E|| u || a e E|| u || dt
T T
2
2at e
N N
∇( u ) ∇u dx dt 2
2at e
N ∞,N
N
( u u ) u d dt+
2C t ( e e ) + C t+
T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) u dx dt + e E b( u , ϕ ) dx dt.
0
Step 2 : lower semi continuity
Let us set
2aT N 2 2aT
2
= e lim E|| u ( T ) || e E|| u( T ) || .
Remark that 0 and that
0
0
0
0 2
0
2
n+1
n n n+1 n n n+1 n+1 n
0
k=0
0
t t
+ E|| u || 2 aE 2as 2
e || u( s) || ds 2E
2as ∞
e < A , u > ds
0 2
0
t
t
+2 E e u u d ds + 2 E e J u dx ds+
0
t
E e B dx ds = + e E u( T ) =
0
e lim E|| u ( T ) ||
T T
where, = E|| u || + 2 2at e E ∞ u u d dt + lim { 2a
2at
N 2
e E|| u || dt
0 2
0
T T
2at N N
2at
N 2
2 e E ∇( u ) ∇u dx dt 2 E e ( u ) d dt+
0
0
0
0
0
17
0
0
(16)

T T
2at N N N
2at
N N 2
2 e E J( u ) (1 ϕ ) u dx dt + e E b( u , ϕ ) dx dt } .
Since the sequences
0
0
T
T
2at
N 2
2at
N N
( e E|| u || dt ) , ( e E ∇( u ) ∇u
dx dt ) ,
0
N∈
T T
( E
2at
e
N 2
( u ) d dt ) , (
2at
e E
N N N
J( u ) (1 ϕ ) u dx dt)
0
N∈
T
2at
N 2
and ( e E|| u u || dt)
are bounded in R,
a subsequence can be found, still indexed by N,
such that the above sequences
converge and consequently,
Let us set
N∈
T
2at ∞
= E|| u || + 2 e E u u d dt
N∈
T
T
2at N 2
2at
N N
2a lim e E|| u || dt 2 lim e E ∇( u ) ∇u dx dt
0
T T
2 limE
2at e
N 2
( u ) ddt + 2 lim
2at
e E
N N N
J( u ) (1 ϕ ) u dxdt
0
T
2at
N N 2
+ lim e E b( u , ϕ ) dx dt.
T T
= 2a 2at N 2
e E|| u u || dt 2
2at
N N
e < Au Au , u u > dt+
N
0
T
2
2at
e E
N N N
[ J( u ) (1 ϕ ) J( u) (1 ϕ)]( u u) dx dt+
0
T
T
2at N N
2
2at
N 2
e E ( b( u , ϕ ) b( u, ϕ)) dx dt + e E|| u u || dt.
0
T
Remark that c( J, , b) 2at
N 2
e E|| ϕ ϕ || dt+
Moreover, since
N
0
0
0 2
0
0
0
0
0
T
2at
N 2
(1 2 a + c( J, , b)) e E|| u u || dt.
0
T T
= 2a 2at N 2
e E|| u || dt 2
2at
N N
e E < Au , u > dt+
N
0
18
0
0
0
0
N∈
(17)

converges and lim =
Since,
one has,
T T
2
2at e E
N N N
J( u ) (1 ϕ ) u dxdt +
2at
e E
N N 2
( b( u , ϕ )) dxdt+
0
T T
E|| u || + 2 e E u u ddt E|| u || 2 e E u u ddt+
0 2
0
2 ∞
0
∞
2
at
2at
0
T T
2
2at e E < Au, u > dt + 2
2at
N
e E < Au, u > dt+
0
T T T
2
2at N
e E < Au , u > dt + 4a 2at e E
N
uu dxdt 2a
2at
2
e E|| u|| dt
0
0
T T
2at N 2
2at
2
+ e E|| u u || dt + e E ( b( u, ϕ)) dx dt
0
0
T T
2
2at e E
N N
b( u , ϕ ) b( u, ϕ) dxdt 2
2at
e E
N N
J( u ) (1 ϕ ) u dxdt
0
N N
T
2 2at
e E
N
J( u) (1 ϕ)( u u) dx dt,
0
T T
E|| u || 2
2at e E
∞
u u d dt + 2
2at
N
e E < Au , u > dt+
0 2
0
T T T
2 a
2at 2
e E|| u|| dt + lim
2at N 2
e E|| u u || dt +
2at
e E
2
( b( u, ϕ)) dxdt
0
0
T T
2 2at e E B b( u, ϕ) dx dt 2
2at
e E J u dx dt.
0
T T
2at e E
2
( b( u, ϕ)) dx dt 2
2at
e E B b( u, ϕ) dx dt =
0
T T
2at e E
2
( B b( u, ϕ)) dx dt
2at
e E
2
B dx dt,
0
T
lim = E|| u || 2
2at e E
∞
u u d dt+
0 2
T T
2
2at N
e E < Au , u > dt
2at
e E
2
B dx dt
0
N
19
0
0
0
0
0
0
0
0
0
0
0

5.3.3 Conclusion
So, according to (16), lim
T T
2 2at e E J u dx dt + 2 a
2at
2
e E|| u|| dt+
0
T T
lim
2at N 2
e E|| u u || dt +
2at
e E
2
( B b( u, ϕ)) dx dt.
0
T
T
2at 2
2at
N 2
+ e E ( B b( u, ϕ)) dx dt + lim e E|| u u || dt.
0
In order to conclude, using (13) and (17 ), we get : L
⎡
T T
⎤
(1 2 a + 2 c( J, , b)) ⎣ 2at N 2
e E|| u u || dt +
2at
N 2
e E|| ϕ ϕ || dt ⎦ .
Then, for a 1 + c( J, , b ) , (14) and (18) lead to :
N N
T
2at
2
0 + + 2 e E ( B b( u, ϕ)) dx dt+
T T
lim
2at N 2
e E|| ϕ ϕ || dt + lim
2at
N 2
e E|| u u || dt.
0
So ( u ) and ( ϕ ) converge to u and ϕ in H and B = b( u, ϕ)
.
0
As , and J are Lipschitz functions, it is possible to identify ( u ) , (1 ϕ) and J( u)
as the
N N limits of the sequences ( ( u )) N , ( (1 ϕ )) N
and H respectively such that :
N and ( J( u )) N . Then, there exists u and ϕ in V
∀v ∈ V, ∀t ∈ [0 , T ];
t t
( u( t) u ) v dx + ∇( u) ∇v dx ds +
∞
( u u ) v d ds =
0
N
In order to prove that ( u, ϕ) is a solution, we have to show that 0 ϕ 1 i.e. 0 ϕ( t)
1.
2
So, considering a C ( R, R)
function , the Itô formula gives :
Remark 2 Thanks to lemma (6), if u ∈ V then G( u) ∈ L ( , H ( Q))
.
0
0
0 0
t t
ϕ( t) ϕ = (1 ϕ) J( u) ds + b( u, ϕ) dw( s ) .
0
t t
′
2
[ ( ( ))] [ ( 0)]
= (1 ) ( ) ( ) + 1
E ϕ t E ϕ E ϕ J u ϕ ds E b ( u, ϕ) ”( ϕ) ds.
2
0
0
0
0 0
0 0
0
0
N N
( ϕ( t) ϕ ) v dx
Thus, if we suppose that ( x) = 0 for any x in [0, 1] , E[ ( ϕ( t))] = 0 for any t.
Then, for any t in [0 , T ] , 0 ϕ( t ) 1 , and ( u, ϕ)
is a solution.
20
+
0
2 1
(18)

6 Uniqueness
Assume that ( u, ϕ) and ( u, ϕ
) in V H are two solutions. Then,
t
t
( u u )( t) v dx + ∇[ ( u) ( u )] ∇v dx ds + ( u u ) v d ds =
0 0
0
0 0
0
0
0
0
0
0
0
0
2 2
V
0
0 0
0
0
( ϕ ϕ )( t) v dx;
t t
ϕ( t) ϕ ( t) = [(1 ϕ) J( u) (1 ϕ ) J( u )] ds + [ b( u, ϕ) b( u, ϕ )] dw( s ) .
Since 0 ϕ, ϕ 1,
∀v ∈ V, ∀t ∈ [0 , T ];
By applying the Itô formula (the stochastic energy equality Grecksch et al [8]), it follows that,
and
Thus,
t
2
E|| ( u u )( t) || + 2 E ∇[ ( u) ( u )] ∇[ u u ] dx ds+
t
t
2
2
2 E ( u u ) d ds = E [ b( u, ϕ) b( u, ϕ )] dx ds+
t
2 E [(1 ϕ) J( u) (1 ϕ ) J( u )][ u u ] dx ds.
t
2
2
E|| ( ϕ ϕ )( t) || = E [ b( u, ϕ) b( u, ϕ )] dx ds+
t
2 E [(1 ϕ) J( u) (1 ϕ ) J( u )][ ϕ ϕ ] dx ds.
t
2
E|| ( u u )( t) || + 2 E || ( u u )( s) || ds + E|| ( ϕ ϕ )( t)
||
t
t
2
2 E [ b( u, ϕ) b( u, ϕ )] dx ds + 2 E [ ϕ ϕ] J( u)[ u u ] dx ds+
t
2 E [ J( u ) J( u)] ϕ [ u u ] dx ds+
t
t
2 E [ ϕ ϕ] J( u)[ ϕ ϕ ] dx ds + 2 E [ J( u ) J( u)] ϕ [ ϕ ϕ ] dx ds.
t
2
E|| ( u u )( t) || + 2 E || ( u u )( s) || ds + E|| ( ϕ ϕ )( t)
||
0
2 2
V
t t
C
2
E|| ( u u )( s) || ds + C
2
E|| ( ϕ ϕ )( s) || ds.
Then, the Gronwall lemma leads to the uniqueness of the solution as well as to the continuity of
the solution with respect to the initial conditions.
21

7 Numerical Simulations
Let us start with a presentation of the experimental context and of the experimental observations
(cf. Dumas et al [6]) :
Several emulsions have been used. For example, Octadecane in water, glycerol and TWEEN 80
( P = 0, 50 ); water in motor oil ( P = 0, 25 and P = 0, 50),
where P indicates the mass fraction. The
experimental cell (see gure 1) is a vertical metallic tube closed by two isolated caps. This cylinder
contains a cage which supports 12 thermocouples. They are located regularly in a horizontal plane
D, at different radii, for r = 0 for the thermocouple (1) to r = 27, 5 mm for the number (12) .
Then, the thermocouple number (13) is located on the outer surface and the number (14) in the
bath, where the cell is immersed. This bath is cooled between 60 c and 40 c at a constant rate
( 5 c/h to 30 c/h).
Figure 1: Experimental cell
On gure 2, we can see the experimental temperature curves, of the different thermocouples,
versus time during the cooling (for water, hexadecane and octadecane emulsions). We observe,
at the beginning of the cooling, before any crystallization appears, a linear decreasing of the
temperature only due to the heat conduction inside the emulsion (the temperature for r = 0 being
always the highest). Crystallizations are detected when the curves deviate from linearity.
22

As expected, the droplets near the inner side of the tube rst crystallize; but a part of the
released energy heats up the axis region of the cylinder, delaying the crystallizations of the droplets.
Generally, the result is that the temperature near the axis region is practically constant over a
lapse of time ( depends on P and ).
For the numerical simulations, we consider a linear heat equation and a space nite difference
method.
u and ϕ being given, we calculate the following sequences ( u ) ( ϕ ) :
0 0
Figure 2: Experimental curves
u +1 u a +1 ϕ +1 ϕ
u =
t 2 t
n+1 n n+1 n
a ∂u + u u + u ∞,n+1
= ( u ) ,
2 ∂
2
ϕ ϕ = tJ( u )(1 ϕ ) + coef.b( u , ϕ )( w w )
n+1 n n n n n n+1 n
12
N (0, 1) U 6,
i=1
n n
n n n n n n
i
a
+ u
,
2
where we use the following estimation for the N (0, 1) Gaussian law :
U i being uniform laws on [0, 1] .
One can see in gures 3, 4 and 5 some numerical curves. coef represents a multiplicative
coefficient put in front of b in order to control the effect of the stochastic perturbation. If coef = 0,
the model is a deterministic one (as if b = 0 cf.
dependence of with respect to coef.
Vallet [16]) and one can see in gure 6 the
23

Figure 3: Deterministic case
Figure 4: Stochastic case (small coef)
24

Figure 5: Stochastic case (greater coef)
Figure 6: Dependance of
We see that the shapes of the curves are rather identic to the experimental ones.
On gures 7, 8 and 9, we give, for each value of r, the function ϕ versus time. We can see
that the proportion of crystallized droplets ϕ is very small when the temperature is stabilized. ϕ
increases sharply only at the end of the temperature plateau and the temperature decreases again
only just after the crystallizations of all the droplets.
25

Figure 7: Deterministic case
Figure 8: Stochastic case (small coef)
26

Figure 9: Stochastic case (greater coef)
It is shown in Vallet [16] that in the deterministic case, the model gives nonincreasing temperature
curves only. With coef = 0,
the numerical curves give globally the same behavior. This
behavior is generally the one observed in the experimentation. But, in one very particular case,
the temperature may rise, instead of the plateau (see gure 10). This rising occurs only with water
emulsions in motor oil and when P = 0, 5.
Figure 10: Water emulsion with P = 0. 5
It is known that a frozen water droplet takes a place more important than a liquid one. Moreover,
during the crystallization, dendrites appear. If a liquid droplet, with temperature under TF
is touched, it crystallizes instantaneously. This crystallization excess induce some more heat in the
cylinder, from which temperature rises.
As, in our model, wn+1 wn
is with mean 0,
that is, in mean, there are as many crystallizations
as in the deterministic case. In order to take into account the above remark, we propose two
numerical propositions :
27

the rst one is to use | w w | instead of w w in the computing (see gure 11),
n+1 n n+1 n
Figure 11: With | w w |
n+1 n
the second one is to consider that wn+1 wn
has a positive mean (instead of 0,
see gure 12).
That is, to introduce a derive vector to the Wiener process i.e. to consider the equation
instead of
∂ϕ
∂t
∂w
= (1 ϕ) J( u) + b( u, ϕ)( cte + ) , in
∂t
∂ϕ
∂t
= (1 ϕ) J( u) + b( u, ϕ)
∂w
, in .
∂t
Figure 12: With positive mean
The shapes of the temperature curves are the same as the experimental one (cf. gure 10).
But, we do not prove any mathematical results for this stochastic context.
An other way to take into account this phenomenon is to modify the deterministic model by
introducing some convection terms, in the above equation, for example. To our knowledge, such a
model does not actually exist.
28

References
[1] D. Bainov, P. Simeonov, Integral Inequalities and Applications,
Mathematics and its
Applications, Vol. 57 , Kluwer Academic Publishers (1992) .
[2] A. Bensoussan, Some Existence results for Stochastic Partial Differential Equations,
in Stochastic
Partial Differential Equations ans Applications, G. Da Prato and L. Tubaro Eds,
Pitman Res. Notes Math. Ser., 268,
(1992) , pp.37-53.
[3] A. Bensoussan, R. Temam, Equations Stochastiques du Type Navier-Stokes,
J. of Func. Anal.
13,
(1973) , pp.195-222.
[4] H. Breckner, W. Grecksch, Approximation of solutions of stochastic evolution equations by
Rothes method,
report of the Institute of Optimization and Stochastic, Halle University,
n 13,
(1997) .
[5] M. Capiński, S. Peszat, Local Existence and Uniqueness of strong solutions to 3-D stochastic
Navier-Stokes Equations,
NoDEA 4,
(1997) , pp.185-200.
[6] J.P. Dumas, M. Strub, M. Krichi, Etude thermique des changements de phases dans une
émulsion, Rev. Gén. Therm. Fr., 341,
Mai (1990) , p.267-273.
[7] A. Friedman, Stochastic Differential Equations and Applications,
Vol.1 & Vol.2, Academic
Press, New York, (1975) .
[8] W. Grecksch, C. Tudor, Stochastic evolution equations,
Akademie Verlag, Berlin, (1995) .
[9] H. Holden, B. Øksendal, J. Uboe, T. Zhang, Stochastic Partial Differential Equations :
A Modeling, White Noise, Functional Approach,
Birkhäuser, Berlin, (1996) .
[10] H. Holden, N.H. Risebro, Conservation Laws with a random Source,
Applied Mathematics
and Optimization, Vol 36 - No 2 (1997) pp. 229-239.
[11] N.V. Krylov, On Lp
theory of stochastic partial differential equations in the whole space,
SIAM
J. Math. Anal., Vol. 27,
n 2 , (1996) , pp.313-340.
[12] H.H. Kuo, J. Xiong, Stochastic Differential Equations in White Noise Space, Inn. Dimens.
Anal. Quantum Probab. Relat. Top. 1,
No.4, (1998) , pp.611-632.
[13] P.L. Lions, P.E. Souganidis, Fully Nonlinear Stochactic Partial Differential Equations,
C. R.
Acad. Sci. Paris, t. 326,
Série 1, (1998) , pp.1085-1092.
[14] B. Øksendal, Stochastic differential equations,
Springer-Verlag, Berlin, (1989) .
[15] 1) E. Pardoux, Equations aux dérivées partielles stochastiques non linéaires
monotones, Etude de solutions fortes de type Ito,
Thèse, Université de Paris-Nord,
Nov. (1975) .
2) Stochastic partial differential equations and ltering of diffusion processes,
Stochastics, Vol.
3,
(1979) , pp.127-167.
[16] 1) G. Vallet, Analyse mathématique des transferts thermiques dans des systèmes dispersés
2
subissant des transformations de phases, M AN, Vol. 27,
n 7,
(1993) , pp.895-923.
2) G. Vallet, Mathematical modelization of the phase change heat transfer in an emulsion,
in
Progress in partial differential equations : the Metz surveys 2,
M. Chipot (editor), Pitman
Research Notes in Mathematics, n 296,
(1993) , pp.55-66.
[17] J.B. Walsh, An introduction to stochastic partial differential equations,
In Ecole dété de
Probabilité de Saint-Flour XIV, P.L. Hennequin editor, Lecture Notes in Mathematics 1180,
Springer Verlag, Berlin, (1986) .
29