A mathematical model for resin transfer molding

The known pseudo-concentration model is a generalization of the classical model of two immiscible fluids when the interface between the two fluids is not a sufficiently regular curve. Besides, it provides efficient and robust numerical methods. The aim of this article is to prove existence of solutions to a mathematical model, based on the pseudo-concentration function model, for the filling of shallow molds with polymers. Numerical methods and numerical simulations with comparison with experimental results have been presented in [6]; [7]. 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 renormalized solution to the mathematical model, and an analysis of time stability is carried out illustrating that the proposed model is suitable for describing the polymer state.


Introduction
Resin Transfer Molding (R.T.M.) is a very fast industrial process for direct production of thin components of complex shapes from low viscosity monomers or oligomers. It consists in low pressure injection of a reactive lThe first author is partially supported by by a Action intégrée CNR-CNRS grant AI 95/0845 2The second author is partially supported by Action intégrée CNR-CNRS grant AI 95/0845 3The third author is partially supported by Action intégrée CNR-CNRS grant compound in a hot shallow mold. The shallow mold will be represented by a bounded open space S2 of R2 which is its projection on a mean plan in the third direction, and T will be the time required for a complete filling, II denoting the domain ]0, T[ 03A9. The reader is referred to [6] for a justification of the domain reduction. As time ellipse domain Q will be split in sub-domains and S22(t) separated by the interface and such that 03A91(t) ~ 03A92 (t) = 0393l(t) and SZ = 03A91(t)03A92(t)0393l(t) which can be schematically represented by the following figure: The domain represents the mold fraction filled with injected reactive fluid and 02(t) is the mold fraction still full of air. It is experimentally well known that in mold process inertia terms are negligible compared to viscous terms in the equations describing fluid flow (see [1] or [6] for a formal justification issued from a perturbation development). In each domain flk(t) , for k=1,2 , the fluid flow speed uk and the fluid pressure pk satisfy Stokes law of viscous flows for viscosity ~~ for any t E [0, T] . Thus r~I is the viscosity of the compound being injected and r~2 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 rate a (see [8] for a justification). Note that the function a is defined only in domain S~i (t), and a function of pseudo-concentration S (see [1]) such that {(t, y) E TI / S(t, y) = 1 } will be used as characteristic of domain and ~ (t, y) E II / S(t, y) = 0~ of domain !12(t). We apply the result of Nouri and Poupaud [10] to the mathematical model for proving existence. To determine the speed u and pressure p for the R.T.M. process, solutions of a Stokes problem, but also the pseudo-concentration function S, solution of a transport equation, and the conversion rate function a solution of a differential equation have to be considered. We will study firstly the existence and stability of the solution of this differential equation and, the existence of a fixed point solution of the coupled problem. In the case where domains 03A91(t) and 03A92(t) are smoothly variable (the interface separating and S~2 (t) is a curve), the model proposed becomes equivalent to the classical model with a free boundary. The cornerstone for proving the existence is to deal with renormalized solutions for the transport equation.
The outline of this work is the following. Firstly, we introduce the functional spaces and hypotheses needed. Then in section 2, we describe the proposed mathematical model to be investigated. Some results required for the study of the model are quoted. And in section 3, we show the result of existence of the proposed problem. is denoted, and the following spaces are defined Then the space V = { ~ (D(03A9))2 / div03C6 = 0 } is considered, and we write H = 03BD(L2(03A9))2 and V' -03BD(H1(03A9))2. Then V = ( 03C6 ~ (H10(03A9))2/div03C6 = 0 } and space H is equipped with the usual scalar product of (L2(S2))2 denoted (., .). The norm associated to this scalar product is denoted [ . I. Space V is equipped with the following scalar product where Diju = 1 2 (u i xj + ũj xi ) for any u, v E V ((u, v)) = / DijuDijv dx. S2 We denote = i~i~2 and we also write 1~j~2 ((u, v)) = 0 3 A 9 ( u ) : dx .
Note that V C H C V' with compact and continuous injections. Moreover for any 1 p oo and for any Hilbert space B, we set = {v: : (O,T) -> B ; v is measurable and E Lp(0,T)}.
If moreover it is assumed that 03A9 has a Lipschitz continuous boundary and that this one is comprised of three connected components 0393e, 03930 and 0393s, then it is possible to define the following space T: Hypotheses (J~i), (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 function Uo is needed for extending the inside and outside speeds on re and on TS which is crucial for defining renormalized solutions to the transport problem with a trace on rs.

Mathematical Model
The mathematical model we propose for representing the process of filling the mold leads to the coupled following problem: for u0 , 03B20 , S0 and f given, find (u, p, S, a) verifying: The viscosity of the mixture in domain II is described with the function 1} : n -3 defined by 1](t, x) = g(S(t, x), a(t)). By setting gl(.) = g(i, .) we will assume that gi is a boundeckincreasing function. This last hypothesis expresses that during the polymerization process, ~71 the polymer viscosity is a continuous increasing function with respect to a such that r~l (t) = g(1, a(t)) for any t E [0, T]. The air viscosity r~2 is a constant. Please remark that similar problems to problem (1)-(2) settled in JR2 are considered as quasistationary problems in [10]. The main result we obtained for solutions (see The proof will be given in section 3, and requires to solve independently Problem (1), Problem (2) and Problem (3) which is the issue of the following sections.

2.1
Stokes Problem (1) In this subsection, we give a formulation of Stokes problem over the complete domain S2. Assume ~7 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 ~(t, .) by ~.
ii)There exists U0 ~ (H1(03A9))2 such that 03B3(U0) = u0 and divU0 = 0 . Proposition 2.1 Let r~ : S2 -~ lf8 be a measurable function such that there exist two rn and NI reals verifying 0 G m ~ G M. Let uo E (11 z (r7S2))2 such that 03 A 9 u0.n do = 0 and U0 as defined by lemma 1 Then problem (2.5) is equivalent to the following Find v E V such that v = u -Uo and for all w E V; For 0 n, let pn be the classical mollifier function, we defined the regularized function ~7~ by convolution We have, ~n --> ~ in L1(II) when n goes to infinity. We define un as the solution to problem (2.6) when 7y is replaced by Propositions (2.1), (2.2) apply, and for every t E [0, T] we get, the existence and uniqueness and a uniform bound with respect to n for u" in L°°(O,T; verifying T0 03A9 03C6(t)~n(t,x)~(un(t, Since u" is uniformly bounded with respect to n in we deduce the existence of u E L°°(O, T; such that, up to a subsequence un ~ u in weak star. Since 03C6~n(t,x)~(w(x)) -03C6~(t, x)~(w(x)) in we pass to the limit in the previous equation and we get p2(t) for all cp E D((0, T)) and w E V. From this we get that (u, P) is a weak solution to Problem (1). The uniqueness of weak solution is straightforward from its definition. Thus we deduce the existence of u E such that, up to a subsequence un -3 u in weak star.
The weak formulation of the Stokes problem, over the complete domain S2 is equivalent to the weak formulation of the Stokes problem in each sub-domains 03A9k(1 k 2) with matching conditions on the interface 0393l(cf [10]). In each domain IIk ((1 k 2)) , the speed uk E V1(03A0k) and pressure pk E V0(03A0k) verify for a. e. t in the interval (0, T] : And we have also the linking conditions at the interface 0393l(t) . The constraint tensor 03C3k is defined by 03C3k = ~k~(uk) -pkId and we set Uk = (1,uk)T and Ek = then the continuity conditions for the constraints tensor, the speed trace continuity and immiscibility over L are: Ei.N = over L; (2.10) ui = u2 over rl; ; (2.11) Ui.N = U2.N = 0 over L, (2.12) where N is normal to L defined by N = Nl = -N2 and Nk for k = 1, 2 , , is the normal to ~03A0k B 852.
2.2 The transport problem (2) In the following, S2 is assumed to be an open domain bounded and that boundary is made of three connected Lipschitz continous components 03930 , 0393e and 0393s. Let U0 E L~(0, +~, (C10(IR2))2) be a function verifying the hypotheses given by (H3 Definition 1 If u E U then a renormalized solution of (2.13) is a function S E L~(II) such that for any function 03B2 E C1(R) ; 03B2(S) is a weak solution of (2.13) with ,3(So) and ,Q(1) as initial and boundary conditions. Theorem 2 Let S2 be an open bounded domain with Lipschitz continuous boundary of R2 and that its boundary consists in three connected components To, re and Ts, and let So E L°°(S2) and u E U. Then the transport problem (2.13) has a unique renormalized solution S with a trace Sr, over rs which belongs to L~((0,T) x 0393s) and verifying: T0 0 3 A 9 S(t, x)(~t03C6 + (u.~)03C6)(t, x) dxdt + 0 3 9 3 e 03C6(t, x)I |U0.n| d03C3(x)dt = T 0 0 3 9 3 s Srs(t, x)03C6(t, x)|U0.n| d03C3(x)dt -J n x) dx. The method used for proving existence of solutions to Transport problem consists in introducing solutions to the problem extended to R2 and applying the results of Diperna-Lions to the transport equation. For a proof of theorem (2) the reader is referred to [10] or to [9]. Remark  interface 0393l is a curve, one can prove that Transport problem(2.13) and immiscibility condition U.N = 0 over 0393l are equivalent (see [10]). When the boundaries of domains S2k are not Lipschitz continuous, Transport problem(2.13~ is a generalization of the immiscibility condition. ~

The polymerization Problem (3)
It is assumed that the polymerization rate of the monomer is derived from empirical model such as Kamal (2.16) The function a does not depend on x. This means that the polymerization reaction starts at the beginning of the filling process. We study the stability  [4]) and we get the existence. For the uniqueness we use Theorem 3.6 p. 64 of [5] which is still valid in the case where the function f, 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 ~5j, integrate the equation verified by We have existence and uniqueness ofae L°° (j o, Tx[ 03A9, R*+), continuous with respect to time, a weak solution to Problem (3) defined by a(t, x) = 03B20 + t0 f (s, a(s, x))S(s, x) ds Vt E [0, Tx] and a. e. x E 03A9.
(2.17) Finally, accounting for the asymptotic stability results of the lemma 2.3, we get that a defined by (2.17) verifies Q; E L~(03A0,R*+).
In order to be more readable, the results concerning the stability are not given with the function f (t, z)S(t, x) but only with the function f (t, z). Nevertheless, these results are still valid for the functionj(t, z)S(t, x). The main result of this sub-section is given by: Proposition 2.3 Let f : R+ x R+ ~ R be a function verifying hypotheses (Hl) and (H2) and let Qo a non negative constant given. Then Problem (2.16) has a unique maximal solution a continuous with respect to time. The function a called a weak solution of Problem (2.16) is defined by a(t) = ~o + it f(s, a(s)) ds for t E ~0, TJ.
~~ -. i~M oreover, ao the solution at equilibrium of the differential equation (2.16) over [0, +oo(, as defined by hypothesis (Hl), is asymptotically stable, whereas 0 the other solution at equilibrium of the equation (,~.1 B~ is not asymptotically stable. Proof. The theorem (3.1) of (5~ brings us to assert that Problem (2.16) has a unique continuous in time solution over a given interval [0, tl~ for any initial condition /3o at time 0.
In order to demonstrate that this solution is maximal and study its stability and asymptotic stability, we start by the verification of these properties for the solution of the following associated problem to (2.16).
Lemma 2 If f : ~ R is a function verifying hypotheses (Hl), (H2) and yo E R*+ the initial condition, Problem (2.20) has a unique solution which is maximal y and the solution at equilibrium 0 is asymptotically stable. Moreover if 0 Yo, then 0 y :::; Yo and if y0 0 then y0 ~ y 0.
Proof. If 0 yo the solution of (2.20) y is given by: for any t > 0. And if yo 0 the solution of (2.20) y is given by: o Now it is possible to give a similar result for the non-linear problem(2.16). Lemma 3 If f : R+ x R+ ~ R is a function verifying hypotheses (H1), and if 03B20 ~ R*+ are given then the Problem (2.16) has a unique maximal solution a over ~0, +oo(. Moreover if ao ~io then ao a ~io and if 03B20 03B10 then 03B20 ~ a 03B10. Proof. If a is the solution of Problem (2.16) that the proposition (2.3) confirms the existence and uniqueness over [0, T] , as a is continuous, that shows that there exists Ti E]0, T[ such that ao a and due to the sign of f i. e. (H2) a /~o for any t E [0, Ti] if /30 belongs to a neighborhood of ao with ao ~3o and a ao for any t E [0, Ti] if /3o belongs to a neighborhood of ao with /30 ao.. Now considering the stable point 0. If we take as initial condition 03B20 close to 0 with 0 ,Qo then a(t) ao for any t E (0, t1]. Effectively supposing that there exists a real Tl such that Ti E]0, tl( and ao a(Ti) then there exists a real 6 such that 03B1(T1) > b > 03B10 > 03B20 and there exists a real t2 ~]0, T1[ such that 03B1(t2) = b. Thus a is strictly increasing after a given time t3 E [t2, T1[. -ztf(s, Z(S)) dS (2.22) which is contradictory with (H2) (f(., z) 0 ao z). Since we have established that a E (0, max (03B10, 03B20)] we deduce that the solution can't blow up, so it is a maximal solution.
o The stability result for Problem (2.16) can be written as: Lemma 4 Let f : R+ x R+ -1EF be a function verifying hypotheses (H1), and /~o a non negative constant, then the solution at equilibrium ao of the differential equation (2.16) over [0, +oo( is asymptotically stable whereas the solution at equilibrium 0 of the differential equation (2.16) over [0, +ocĩ s not asymptotically stable.
Proof. If 0 e then for any initial condition /?o such that 1/30 -ao~ I e the solution a of problem (2.16) given by lemma 3 with the initial condition /~o verifies Ja(t) -ao) e for any t E (0,+00~. Thus the equilibrium solution a = ao of (2.16) over [0,-}-oo[ is stable. Moreover as f(t, a) in the neightbourhood of ao is written as: (Hl) by ao) 0 ) then there exists a real 0 6 (6 = min (v, eo) from the proof of lemma 3) such that if |03B20 -03B10| 8, then a verifies la(t) -03B10| b for any t E [0, +~[. And thus for any condition 0 03B20 such that ao b we get: ao) + (a -ao )PR( t, a, ao ) 2 for any t e [0, +~o(. For a given Po we set y(t) = a(t)ao, then (2.16) can be written: For a proof the reader is referred to [10] corollary 5.1. The key point for the strong convergence of (Sn)n~N in LZ is to work with the space |U0.n|d03C3(x))) and providing l~ of the following norm Lemma 5 applies and we get that an converges towards a = polym(S) a weak solution of problem (25). Since is strongly convergent towards (S, a) in L2(03A0, R*+), neglecting the extraction of a sub-sequence again noted an) Theorem 4.9 in [2] claims that the sequence is convergent towards (S, a) almost everywhere in iI. The continuity of g implies that r~n = an) is convergent almost everywhere towards g(a, S). The function g defined by (2.4) is bounded, it follows that the sequence r~n is uniformly bounded. It is deduced of proposition 2.  l i m p + T 0 0 3 A 9 n p ( t , x ) ( v n p ( t , x)) : E(w) dx dt + T 0 0 3 A 9 n p ( t , x ) ( U 0 ( t , x ) ) : E(w) dx dt = T0 03A9~(t, x)~(t, x)) : ~(w) dx dt + x)) : ~(w) dx dt.
So v is solution of Problem (2.6) and thus u = v + Uo is a solution of (2.5) then a solution of (1) associated to r~. Moreover u is unique, so it is all the sequence which converges, and we have u = !~stokes(~ = (~)). On the other hand, by using Corollary 5.1 of [10] we have that S = lim F(Sn) = trans(u) since un weakly converges to u in T; (H1(03A9))2). Thus we have S _ F(S). Finally let us establish the compactness of F. We start with (Sn)n~N an arbitrary bounded sequence in C. The weak sequential compactness of the unit ball of L2(03A0) implies the existence of S E L2(03A0) and of a sub-sequence Snp S in L2(03A0) weak. Let an = = stokes(~n = g(Sn, 03B1n)) which is uniformly bounded in (Hl(Q))2). The weak sequential compacity of the unit ball of the Banach space U implies the existence of u E U and a sub-sequence unp --~ u in (H1(S~))2) weak. Corollary 5.1 of [10] applies, and we have in L2 (II)..Arguing as before we prove that Snp converges strongly in toward S = stokes(~ = g(8, a) ) .
Theorem 1 is proven.