A Numerical anlysis of Resin Transfer Molding

(1)

A Numerical anlysis of Resin Transfer Molding

R. Aboulaich

Ecole Mohammadia d’ingénieurs, Laboratoire d’Etude et de Recherche en Mathématiques Appliquées, B.P 765, Rabat-Agdal, Maroc.

S. Boujena

Université Hassan II, Faculté des sciences Ain-Chok, B.P 5366 Maârif Casablanca MAROC J. Pousin

Mathematical modelling and scientific computing Laboratory, UMR CNRS 5585, National Institute of Applied Sciences in Lyon,

20 av. Einstein, F-69621 Villeurbanne Cedex France

The aim of this work is to study a mathematical model, based on the pseudo-concentration function model, for the filling of shallow molds with polymers. The proposed model is 2-D, the chemical reactivity of the fluid is accounted with the conversion rate satisfying a Kamal-Sourour model, and the temperature is not considered. We prove the existence of a solution of the proposed mathematical model.

I. INTRODUCTION

Resin Transfer Molding (R.T.M.) is a very fast indus- trial process for direct production of thin components of complex shape from low viscosity monomers or oligomers.

It consists in low pressure injection of a reactive compound in a hot shallow mold. The shallow mold will be represented by a bounded open space Ω of R² which is its projection on a mean plan in the third direction, and T will be the time required for a complet filling, Π de- noting the domain ]0, T[×Ω. As time elipses domain Ω will be split in sub-domains Ω1(t) and Ω2(t) sepa- rated by a free boundary interface Γl(t) and such that Ω1(t)uΩ2(t) = Γl(t) and Ω = Ω1(t)tΩ2(t)tΓl(t) which can be schematically represented by the following figure:

!

#"

$

&%

'

Ω1(t) represents the mold fraction filled with injected reactive fluid and Ω2(t) is the mold fraction still full of air. It is experimentally well known that in mold process inertia terms are negligible compared to viscosity terms in the equations discribing fluid flow[1]. In each domain Ωk(t) , for k=1,2, the fluid flow speed uk and the fluid pressurepk satisfy Navier-Stokes law of viscous flows for viscosityηk at any timet∈[0, T]. Thusη1is the viscosity of the compound being injected and η2 the viscosity

of the air. In this simple mathematical model temperature variations are not taken into account and chemical reactions are represented by the conversion rateα (see [10]). Note that the functionαis defined only in domain Ω1(t), and a function of pseudo-concentrationS (see [1]) such that{(t, y)∈Π/ S(t, y) = 1} will be used as char- acteristic of domain Ω1(t) and{(t, y)∈Π/ S(t, y) = 0}

of domain Ω2(t). The mathematical model proposed for the R.T.M. process is a generalization of the two-fluids model studied by Nouri and Poupaud [12].

A. Functional spaces

Let Ω be a bounded open set of R² the boundary of which is piecewiseC¹ . If

H¹(Ω) ={u / u∈L²(Ω) and

∂u

∂xi ∈L²(Ω);i= 1,2} with ∂u

∂xi

the derivative in a weak sense, the trace application fromH¹(Ω) on H¹²(∂Ω) is denoted γ and the following spaces are defined

L²₀(Ω) =n

p∈L²(Ω)/R

Ωp dx= 0o

; H₀¹(Ω) ={u∈H¹(Ω)/(γu)(x) = 0

∀x∈∂Ω}.

Then the space

V={ϕ∈(D(Ω))²/divϕ= 0}

is considered, and we write H = V^(L²^(Ω))² and V = V^(H¹^(Ω))² . Then

V ={ϕ∈(H₀¹(Ω))²/divϕ= 0}

and the space H is equipped with the usual scalar product of (L²(Ω))² denoted (·,·). The norm associated to this 5 4°2001 The Morrocan Statistical Physical and Condensed Matter Society^c

(2)

scalar product is denoted|.|. SpaceV is equipped with the following scalar product

((u, v)) = X2

i,j=1

Z

Ω

DijuDijv dx

Diju=1 2(∂ui

∂xj +∂uj

∂xi) for anyu, v∈V . The associated norm of this scalar product is denotedk.kand is equivalent to the norm of (H¹(Ω))². Note thatV ⊂H ⊂V⁰ with compact and continuous injections. Moreover for any 1≤p≤ ∞and for any Hilbert spaceB, we set L^p(0, T;B) ={v: (0, T)→ Bsuch thatv

is measurable and kv(.)kB∈L^p(0, T)}.

Then for any u0 given in (H¹²(∂Ω))² such that Z

∂Ω

u0.n dσ = 0 where n is the exterior normal to ∂Ω , we consider the following space

U ={u∈L^∞(0, T; (H¹(Ω))²)/

div(u) = 0, γ(u) =u0}.

If moreover we suppose that Ω is lipschitzian and that its boundary is comprised of three connex components Γe,Γ0 and Γs, then it is possible to define the following space

T ={ϕ∈ D(R³)/ϕ(t, x) = 0, ∀x∈Γs,

∀t∈(0, T)}.

Defining now the domains

Π1={(t, x)∈Π/x∈Ω1(t)};

Π2={(t, x)∈Π/x∈Ω2(t)};

L={(t, x)∈Π/x∈Γl(t)}, we have Π =L∪Π1∪Π2 .

For 1 ≤ k ≤ 2 , we set: V₀(Π_k) = {p |_Π_k /;p ∈ L²(0, T;L²(Ω))};

V1(Πk) ={u|Πk∈L^∞(0, T; (H¹(Ω))²)}.In the following, we will denote by Γk the trace application fromV1(Πk) with values inH¹²(∂Πk). The polymerization

reaction is represented by a function f :R+×R+−→R verifying the hypothesis:

H1)∃ 1 < p and ∃ 1 < k, odd such that the function z 7−→f(t, z)∈C^k+p(R+, R+)∀t∈R+. Moreover,

∃ 0 < α0 ≤ 1 such that f(t,0) = f(t, α0) = 0

∀t ∈ R+; 0 ≤ f(t, z) if 0 ≤ z ≤ α0 and f(t, z) ≤ 0 if α0 ≤ z ∀t ∈ R+. Furthermore,

∂_z^kf(., α0) ∈ L¹(0, T) with ∂_z^kf(t, α0)≤ µ <0 on [0, T] and∂^m_z f(t, α0) = 0∀m∈ {1,2, ..., k−1}.

H2) f is a continuous and∂zf is bounded on subsets of R+×R+.

A functionU0∈L^∞(0,+∞; (C₀¹(IR²))²) verifying the following hypothesis will also be needed:

H3)







divU₀= 0;

U0.n <0 ifx∈Γe andt≥0;

U0.n= 0 ifx∈Γ0 andt≥0;

U0.n >0 ifx∈Γs andt≥0.

Hypotheses (H1),(H2) are related to the function f of the conversion rate and are limiting conditions but are satisfied by the Kamal-Sourour model for instance. The functionU0 is needed for extending the inside and out- side speeds on Γeand on Γswhich is crutial for defining renormalized solutions to the transport problem with a trace on Γs.

II. MATHEMATICAL MODEL

For representing the process of filling the mold leads towe propose the coupled following problem:

foru0, β0, S0 andf given, find (u, p, S, α) verifying:

(1)





divx(η(t, x)²(u(t, x))) =5p in Ω divx(u) = 0 in Ω

γ(u) =u0 on∂Ω

(2)









∂tS(t, x) + (~u. ~5x)S(t, x) = 0

∀(t, x)∈Π;

S(0, x) =S0(x) ∀x∈Ω;

S(t, x) = 1 ∀t∈]0, T[; ∀x∈Γe;

(3)





∂tα=f(t, α)S(t, x) t∈]0, T];

α(0) =β0, withη(t, x) =g(S(t, x);α(t)).

Let 0 < m ≤M be two constants, the viscosity of the mixture in Π is described with the function η : Π −→

[m, M] where

(4)g: [0,1]×R₊ −→[m, M] (ξ, v) 7−→g(ξ, v) is a given bounded function such that

g∈ C⁰([0,1]×R+; [m, M]) ; g(0, v) =η2 and g(1, v) = η1 for all v ∈R+. By setting g1(.) =g(1, .) we will as- sume thatg1is a bounded increasing function. This last hypothesis expresses that during the polymerization process,η1 the polymer viscosity is a continuous increasing function with respect toαsuch that

η1(t, x) =g(1, α(t)) for anyt∈[0, T] and any fixed x in Ω such that

S(t, x) = 1 . The air viscosityη2is a constant. The main result we obtained for Problem (1)−(3) is given by:

Theorem 1Let Ωbe a lipschitzian open bounded subset included in IR², the boundary of which consists in three

(3)

connex partsΓe,Γ0 andΓssuch that there exist two open sets Ωe andΩs verifying

Γe⊂Ωe, Γs⊂Ωs and Ωe∩Ωs=∅ .

Let U0 ∈ L^∞(0,+∞;C₀¹(R²)) be a function satisfying (H3).

Let S0 ∈ L^∞(Ω;{0,1}) be given function, Let β0 be a given scalar and let

f : R+×R+ −→ R be a function verifying (H1) and (H2) . Then Problem (1), (2) and (3) has at least a weak solution(u, p, S, α)inL^∞(0, T; (H¹(Ω))²)×L²₀(Ω)×

L^∞[(0, T)×Ω]×L^∞[(0, T)].

The proof will requires to solve independently Problem (1), Problem (2) and Problem (3) which is the issue of the following. In the subsections (2.1); (2.2) and (2.3) we presente briefly the study of problem (1); problem (2) and problem (3).

A. Stokes problem (1)

In this subsection, we give a formulation of Stokes problem over the complete domain Ω. Assume η to be a continuous function with respect to time, positive bounded from above and from below, measurable with respect to x and let t to be fixed. While t is fixed we still denote the function η(t,·) by η. For givenu0 find u∈(H¹(Ω))² andp∈L²₀(Ω) verifying:

(5)





div(η²(u)) = 5pin (D⁰(Ω))²; div(u) = 0 a. e. in Ω;

γ(u) = u0 in (H¹²(∂Ω))²

To give a weak formulation of problem (5) we have to extend the boundary conditions in the domain Ω.

Lemma 1 The following proposal are equivalent:

i)u0∈(H¹²(∂Ω))² and Z

∂Ω

u0.n dσ= 0.

ii)There exists U0 ∈ (H¹(Ω))² such that γ(U0) = u0 and divU0= 0.

Proposition 1 Let η: Ω −→R be a measurable func- tion such that there exist two mandM reals verifying 0 < m ≤ η ≤ M. Let u0 ∈ (H¹²(∂Ω))² such that Z

∂Ω

u0.n dσ= 0andU0as defined by lemma1 Then prob- lem (4) is equivalent to the following

(6)











Find v∈V such that v=u−U0

and for allw∈V; Z

Ω

η²(v) :²(w)dx+

Z

Ω

η²(U0) :²(w)dx= 0.

For the existence of solutions to problem (6) we have the classical following result:

Proposition 2Within the hypotheses of proposition 1, problem (6) has a unique solutionvsuch that u=v+U0

is the unique solution to (5). Moreover we have:

kuk ≤( 2M

m c(Ω) + 1)kU0k wherec(Ω) is the poincar´e constant.

We deal now with a function η which is a measurable function defiened in Π such that 0 < m ≤ η ≤ M a.e in Π. Then a weak solution to problem (1) is a couple (u, p)∈ U ×L^∞(0, T;L²₀(Ω)) such that

v=u−U0 verifying for allw∈V

(7)





 Z

Ω

η²(v) :²(w)dx+

Z

Ω

η²(U0) :²(w)dx= 0 a.e t∈(0, T)

We prouve (see [2]) that propositions (1) and (2) are still valid when problem (6) is replaced by problem (7) for whicht∈(0, T) and equations (5) by equations (1)

B. The transport problem (2)

In the following Ω is assumed to be an open bounded and lipschitzian and that boundary is made of three connex components Γ0, Γeand Γs. The transport problem is defined by: for givenu∈ U,

S0∈L^∞(Ω), findS∈L^∞(Π) verifying:

(8)









∂tS(t, x) + (~u. ~5x)S(t, x) = 0

∀(t, x)∈Π

S(0, x) =S0(x) ; ∀x∈Ω S(t, x) = 1 ∀t∈]0, T[ ; ∀x∈Γe.

Theorem 2Let Ωbe an open bounded and lipschitzian ofR²and that its boundary consists in three connex com- ponents Γ0,Γe and Γs, and let S0∈L^∞(Ω) andu∈ U. Then the transport problem (8) has a unique renormal- ized solutionS with a traceSΓs overΓswhich belongs to L^∞((0, T)×Γs)and verifying:

Z _T

0

Z

Ω

S(t, x)(∂tϕ+ (u.5)ϕ)(t, x)dxdt+ Z _T

0

Z

Γe

ϕ(t, x)|U0.n|dσ(x)dt= (9) Z _T

0

Z

Γs

SΓs(t, x)ϕ(t, x)|U0.n|dσ(x)dt

− Z

Ω

S0(x)ϕ(0, x)dx.

for anyϕ∈ D([0, T]×R²).Moreover Z _T

0

(T−t) Z

Γs

S_Γ²_s(t, x)|U0.n|dσ(x)dt+ Z _T

0

Z

Ω

S²(t, x)dx= (10)

(4)

Z _T

0

(T−t) Z

Γe

|U0.n|dσ(x)dt + T

Z

Ω

S₀²(x)dx

andS∈L^∞(Π;{0,1}) if S0∈L^∞(Ω;{0,1}).

Remark II.1 It is important to consider the renormal- ized solutions to get Equation (9) (which will be crutial to obtain strong convergence) and to get solutions with values in {0,1}. Renormalized solutions have a trace in L²(Γs;|U0.n|dσ); which is not valid for the weak solutions of problem (8) ( cf [3]).

For a proof of theorem 2 the reader is referred to [12].

C. The polymerization Problem

It is assumed that the polymerization rate of the monomer is derived from empirical model such as Kamal - Sourour [10], i.e. there exists a functionf :R+×R+−→

Rsatisfying hypotheses (H1) and (H2) and such that for β0∈Rgiven , the functionα: [0, T]−→R+ verifies

(11)





∂tα = f(t, α); t∈]0, T];

α(0) = β0.

The results which are going to be derived for Prob- lem (11) still remain valid for polymerisation problem (3), since function S ∈ L^∞, thus the function z 7→

f(·, z)S(·, x) is continuous and locally Lipschitzian with respect to z independently of time and space. So the required hypotheses of Carath´eodory Theorem are satisfied, the fact thatt, z, x7→f(t, z)S(t, x) is not continuous with respect to time is not a restriction. For Problem (3) we use the theorem of carath´eodory (see remark p. 60 of [7] or [6]) and we get the existence. For the uniqueness we use Theorem 3.6 p. 64 of [7] which is still valid in the case where the functionf, the right hand side of the ordinary differential equation is not a continuous function with respect to time (in the proof of the theorem 3.6 of [7], integrate the equation verified byϕ). We have existence and uniqueness ofα∈L^∞(]0, Tx[×Ω, R^∗₊), continuous with respect to time, a weak solution to Problem (3) defined by

α(t, x) =β0+ Z _t

0

f(s, α(s, x))S(s, x)ds

∀t∈[0, Tx] and a. e. x∈Ω. (12) The main result of this sub-section is given by:

Proposition 3 Let f : R+×R+ −→ R be a function verifying hypotheses (H1) and (H2) and let β0 given.

Then Problem (11) has a unique maximal solution α ∈ L^p([0, T], R^∗₊)(1 ≤ p < +∞).The function α called a weak solution of(11)is defined by

α(t) =β0+ Z _t

0

f(s, α(s))ds (13) a.e. t∈[0, T].

For the proof see [2]

III. FIXED POINT AND EXISTENCE RESULT

LetS₀∈L^∞(Ω;{0,1}) and

f : R+×R+−→R

verifying hypotheses (H1) and (H2) and Let U0 ∈ L^∞(0, T,(C₀¹(Ω))²) be such that divx(U0) = 0 and β0

be a given scalar. We setγ(U0) =u0 over (0, T)×∂Ω.

Introduicing the following setC C={ϕ∈L^∞((0, T)×Ω)/ 0≤ϕ(t, x)≤1 a. e.}.

The proof of the existence of the fixed point (theorem 1) needs lemma2 and lemma3

Lemma 2If S in L^∞(Π,{0,1}), f verifying hypotheses (H1)and(H2),β0∈R^∗₊are given and ifS^²is a sequence of functions such thatkS^²k_L^∞_(Π) ≤1 and verifying

S^²−→S inL²(Π).

²→0

Then the sequenceα^² of the solutions of problem





∂tα^²=f(t, α^²)S^²(t, x)t∈]0, T];

α^²(0, x) =β0,

is converging inL²(Π;R₊^∗), towards α(t) =β0+

Z _t

0

f(r, α(s))S(r, x)dr (14) a weak solution to problem (11).

For a proof the reader is referred to [2]

Defining uk = u |Πk for 1 ≤ k ≤ 2 it is noted that theorem 2 leads to confirm that solutionS has its values in {0,1}. Then equation (3) represents the evolution of the reaction rate levels, andα corresponds to the polymerization problem.

The operator Λpolym can be defined using the Lemma 2 Λpolym:L^∞(Π;{0,1})−→L^∞(Π)

such that Λpolym(S) =α defined by (15)

In the following an operatorFdefined overCwith values in C is built such that the solution of problem (1)-(2)- (3) is a fixed point of F in C. To prove that F ad- mits a fixed point, the sequential compactness of F as an operator from (C,k.k_L²_(Π)) in (C,k.k_L²_(Π)) is used. It

(5)

remains to demonstrate that for any sequence (Sn)n∈N

weakly converging in (C,k.k_L²_((0,T)×Ω)) we can derive a sub-sequence strongly converging.

Building an operator F over C:

It is possible to define [12]:

Λstok:L^∞(Π;R+∩ {m≤z≤M})−→ U such that Λstok(η) =u a weak

solution to problem (1).

With theorem 2 it is possible to introduce the operator:

Λtrans:U −→L^∞(Π;{0,1});

such that Λtrans(u) =S

renormalized solution to problem (2).

For a givenSⁿ∈C, thenαⁿ;ηⁿ;uⁿandSⁿ⁺¹are defined by:

(16)







αⁿ= Λpolym(Sⁿ)ηⁿ=g(Sⁿ, αⁿ) ( where g is given by (4)) uⁿ= Λ_stok(ηⁿ)

and finally Sⁿ⁺¹= Λtrans(uⁿ)

The operatorF :C−→CassociatesSⁿ⁺¹toSⁿ. A weak formulation of problem (1)-(2)-(3) will be a quadruplet (u, p, S, α)∈L^∞³

0, T;¡

H¹(Ω)¢₂´

×L²₀(Ω)×L^∞[(0, T)× Ω]×L^∞[(0, T)×Ω] such that

(17)F(S) =S , α= Λpolym(S) η=g(S, α); u= Λstok(η) andpis defined by proposition 1.

The existence of a weak solution of problem (1)-(2)-(3) is then derived of the existence of a fixed point ofF in C.

Starting with a technical result concerning the sequence of renormalized solutions of problem (2).

Lemma 3 If (uⁿ)n∈N is a sequence of L²(0, T; (H¹(Ω))²) bounded in

L^∞(0, T; (H¹(Ω))²) such that uⁿ * u in L²(0, T; (H¹(Ω))²) with γ(uⁿ) = γ(U₀) and divuⁿ = 0 for any n, and such that (Sⁿ)n∈N is the sequence of solutions to transport problems associated to uⁿ with S0 ∈ L^∞(Ω;{0,1}) given by theorem 2. Then the sequence (Sⁿ)n∈N converges in L²((0, T)×Ω) towards the fonction S renormalized solution of the transport problem associated to uand given by theorem 2.

For a proof the reader is referred to [12]. The key point for the strong convergence of (Sⁿ)_n∈N in L² is to work with the space X = L²((0, T) × Ω) × L²(0, T;L²(Γs,|U0.n|dσ(x)))

and providingX of the following norm

|||(δ, µ)|||= h

kδk²_L2((0,T)×Ω)+ Z _T

0

(T−t) Z

Γs

µ²|U0.n|dσ(x)dt i¹

2.

Proof 1 (Proof of theorem 1) If (Sⁿ, αⁿ, ηⁿ, uⁿ) a sequence defined by the recurrent relations (17) and the

FIG. 1: position of the front of polymer as a function of time

fonction g defined by (4) is bounded, it follows that the sequenceηⁿ is uniformly bounded. It is deduced [12] that:

kuⁿk ≤( 2M

mc(Ω)+ 1)kU₀k.

The weak sequential compactness of a reflexive Banach space unit ball leads to the existence of u∈ U such that uⁿ converges weakly towards u in L²(0, T; (H¹(Ω))²).

Lemma3 implies that the sequence Sⁿ is strongly con- vergent inL²(Π)towards S∈L^∞(Π,{0,1})a renormal- ized solution of problem (2) and Lemma 2 implies that the sequenceαⁿ converges towardsα a weak solution of problem (3) and thatuis a weak solution of (1).

SinceSⁿ is strongly convergent towardsS inL²(Π, R^∗₊), neglecting the extraction of a sub-squence again noted (Sⁿ, αⁿ) the sequence (Sⁿ, αⁿ) is convergent towards (S, α) almost everywhere in Π. The continuity of g im- plies that ηⁿ = g(Sⁿ, αⁿ) is convergent almost every- where towardsg(α, S).

It remains to show that u is a weak solution of prob- lem (1). For any(t, x)∈Π, we set g(S(t, x), α(t, x)) = η(t, x). Then(ηⁿ)nconverges towardsηinL²((0, T)×Ω).

Let now the subsequences (Sⁿ^p)p∈N of (Sⁿ)n∈N and (αⁿ^p)p∈N of (αⁿ)n be such that for any p∈ N and for any t ∈ (0, T), uⁿ^p(t, .) is the solution of (1) associ- ated to ηⁿ^p(Sⁿ^p(t, .);αⁿ^p(t, .)). Then ηⁿ^p converges to- wardsη(S, α) inL^∞(Π) andfor anyw ∈ Vand for any t ∈(0, T):

Z

Ω

η(t, x)²(v(t, x)) :²(w)dx+ Z

Ω

η(t, x)²(U0(t, x)) :²(w)dx= 0

u(t,·) =v(t, .) +U0(,·)is a solution of (1) associated to η(t,·). Theorem 1 is proved.

Let us end this paper with some comparaisons between experimental data and numerical results for a case where the function α is given. Numerical results have been obtained with a finite element approximation of Problem (1)-(2). The transport equation (2) is numerically solved with a least square space time integrated method (see [15], [16]).

A time marching strategy has been implemented, and for each time step, the algorithm used is the one sug- gested by relations (17) with just one step of the itera- tion functionF. A heated shallow rectangular mold with a thickness of 10⁻³m, a length of .036m and a with of

(6)

FIG. 2: position of the front of polymer as a function of time

0.1m is filled with a polymer with an injection velocity

of 10⁻¹m/s. The viscosity of the air is 1.7310⁻⁶pa and the viscosity of the polymer has an order of magnitude of 10⁻¹pa. In figure 1 the velocity field is presented at the beginning of the filling, the free boundary is character- ized by the points having a non null velocity component in y direction in the mold. In figure 2, the position of the interface between polymer and air is plotted as a function of time with squares for experimental data and with balls for numerical simulations. A time step of 0.1 and a mesh size of 0.6 has been used.

[1] J. F. Agassant, Y. Demay and A. Fortin, ‘Prediction of stationary interfaces in coextrusion flows ’. Polymer engineering and science, July 1994 , Vol. 34 , No. 14 . [2] R. Aboulaich, S. Boujena and J. Pousin, ’A math-

ematical model for Resin Transfer Molding’, Annales Math´ematiques Blaise Pascal, Vol. 8, N 2, pp 115-136 (2001)

[3] C. Bardos, ’Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels;

théorèmes d’approximation; application à l’équation de transport’, Ann. Scient. Ecol. Norm. Sup. 4êmesérie, t.

3, p. 185-233 1970.

[4] H. Br´ezis, Analyse fonctionnelle. Masson Paris 1983.

[5] P. Carrier, Y. Demay and A. Fortin, ‘Numeri- cal simulation of coextrusion and film casting ’ , Int.j.numer.methods in fluids , 20 , 31-57 (1995).

[6] E. A. Coddington and L. Norman, ‘Ordinary differential equations’. Mcgran-hill book compagny, New-York;

Toronto; London; 1955 .

[7] M. Crouzeix and A. Mignot, ‘Approximation des

´equations ordinaires’. Masson, Paris; 1992.

[8] R. T. Dipperna, P. L. Lions, ‘Ordinary differential equations transport theory and Sobolev spaces’ , In- vent.math.98 , 511-547 , 1989.

[9] K. Friedrichs,‘symetric positive systems of differential equations ’ ,comm.pure.appl.math , 7 , 1954 , 345-392 .

[10] M.R. Kamal, S. Sourour, ‘Kinetics and thermal charac- terization of thermoset cure . Polymer Engineering and science , 1973 , Vol.13 ,pp.59-64.

[11] J.L. Lions, ‘Quelques méthodes de résolution des problèmes aux limites non linéaires ’ , Gautier-Villard , 1969.

[12] A. Nouri, F. Poupaud, ‘An existence theorem for the mul- tifluid Stokes problem ’, Quart.Appl.Math., 55, 1997, No.

3, 421-435 .

[13] S. Trenoguine, ‘Analyse fonctionnelle’, Mir-Moscou, 1985 .

[14] F. Verhulst, ‘Nonlinear differential equations and dynam- ical=20 systems’ , second revised and expanded edition , Springer-Verlag , Berlin , 1996.

[15] P. Azerad, P. Perrochet, J. Pousin, ’Space-Time Inte- grated Least-Square: a simple,stable and precise finite element scheme to solve advection equations as they were elliptic. Progress in Partial Differential Equations’, he Metz surveys 4.M. Chipot and I. Shafrir Editors. Pitman Reasearch Notes in Mathematics series 345 LONGMAN p. 240-252 1996.

[16] P. Azerad, J. Pousin, ’Inégalité de Poincaré courbe pour le traitement variationnel de l’équation de transport’, Comptes rendus de l’Académie des Sciences Paris , t.

322 serie 1 p. 721-727. 1996.