Multicomponent flow in a porous medium. Adsorption and Soret effect phenomena : local study and upscaling process

Mod´elisation Math´ematique et Analyse Num´erique Vol. 35, N 3, 2001, pp. 481–512

MULTICOMPONENT FLOW IN A POROUS MEDIUM.

ADSORPTION AND SORET EFFECT PHENOMENA: LOCAL STUDY AND UPSCALING PROCESS

Serge Blancher

, Ren´ e Creff

, G´ erard Gagneux

, Bruno Lacabanne

, Fran¸ cois Montel

and David Trujillo

Abstract. Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions on state functions.

Then, we present a numerical scheme based on a “mixed finite element” method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an efficient modelling of Soret effect and adsorption in a porous medium at a macroscopic scale.

R´esum´e. On ´etudie un syst`eme d’´equations paraboliques non lin´eaires mod´elisant l’´evolution des fractions massiques d’un fluide multiconstituant dans un ´ecoulement convectif forc´e sous l’influence d’un gradient thermique. Des r´esultats d’existence et d’unicit´e sont donn´es sous des hypoth`eses relatives aux fonctions d’´etat des ´equations. On propose ensuite une m´ethode num´erique de type “´el´ements finis mixtes” aboutissant `a un sch´ema “volumes finis” dont on effectue l’analyse et la pr´esentation des premiers r´esultats. Nous appliquons enfin une technique d’homog´en´eisation aux ´equations ´etudi´ees dans le but d’obtenir une mod´elisation macroscopique fid`ele des ph´enom`enes d’adsorption et d’effet Soret en milieu poreux.

Mathematics Subject Classification. 35K55, 35K60, 65N30.

Received: November 14, 2000. Revised: February 7, 2001.

Introduction

We present the mathematical study of a model describing the coupling between adsorption and second order terms such as molecular diffusion and thermodiffusion occurring in a multicomponent fluid inside a capillary column. The results of this study have a very wide application field which extends from the modelization of a chromatographic column to an oil layer. In the example of a chromatographic column, when sorption effects

Keywords and phrases. System of nonlinear parabolic equations, Soret effect, separation by thermal diffusion, mixed finite element, finite volume scheme, homogenization, two scale convergence.

1 Laboratoire de Transferts thermiques, Universit´e de Pau et des Pays de l’Adour, Technopole H´elioparc, av. du Pdt Angot, 64000 Pau, France.

2Laboratoire de Math´ematiques Appliqu´ees, ERS 2055, I.P.R.A., Universit´e de Pau et des Pays de l’Adour, 64000 Pau, France.

e-mail:bruno.lacabanne@univ-pau.fr

3 Elf E.P., Centre Scientifique et Technique Jean Feger, av. Larribau, 64000 Pau, France.

c EDP Sciences, SMAI 2001

are only considered, the system is then described by first order hyperbolic equations (cf.particularly [17]), but in practice, it is necessary to take into account diffusive transport and dispersive phenomena. In the case of an oil layer, thermodiffusion and adsorption effects which are often neglected in production, are essential when one has to describe the initial distribution of hydrocarbons at the field scale (cf. [20]). The mass conservation equation for thek^thspecie, in the absence of chemical reaction, is then given by,

∂t(Nk+ϕk(N)) + div(J~k) = 0, (1≤k≤n) (0.1)

whereJ~k is the mass flux associated with the weight fractionNk of componentkandϕk(.) is a function of the vectorial variable, called “adsorption isotherm”, describing the quantity of matter present in the porous matrix of the column. This function allows then to reduce the unknowns number since we are only interested in the species present in the mobile phase, the fluid. The aspect of adsorption that we favor here is the thermodynamic equilibrium, i.e.a state where component segregation between the two phases (the stationary and the mobile one) is established and does not depend on the initial conditions. The expression of the functions family (ϕk)_k is obtained using statistical thermodynamic and does not take into account any kinetic factor (the delay termϕk does not depend directly on time). Separation results thus from the coupling between forced motion and retention, i.e.between thermodynamic and hydrodynamic phenomena. The total matter flux, sum of the diffusive (Fick’s law), convective and thermodiffusive fluxes is finally given by

J~k=− Xn j=1

Dkj∇N~ j+NkU−D^k_θNk( Xn j=1

Nj−Nk)∇θ.~

One can recognize in this formulation the diffusive term (which includes the expressions of the “cross”-diffusion terms) to which we add two first order terms (in comparison with the unknowns), the forced convective term (along a known velocity fieldU) and the thermodiffusive one or “Soret effect” term (which is mainly character- ized by the positive coefficients S_tⁱ

and the term “∇~θ”, representative of the internal thermal gradient). The goal is to prove that the considered problem is a well posed one (in the Hadamard ’s sense): given a “regular velocity” profile, we consider the following equations:











U·∇θ~ −κ∆θ= 0

∂t

Ni+kⁱ(N₁, ..., Nn)

+U·∇N~ i− Xn j=1

Dij∆Nj−div(S_tⁱNi( Xn j=1

Nj−Ni)∇θ) = 0~ (1≤i≤n), in a rollQ= ]0, T[×Ω, Ω⊂R³

(0.2)

associated to initial and boundary conditions in accordance with experiments, where Ω is a bounded connex open part ofR³, with a Lipschitz border Γ, divided according to the rule

Γ =∂Ω = Γe∪Γl∪Γs∪∂Γl, L²−meas (Γe∪Γs)>0,

where Γe(resp. Γs) denotes the inlet (resp. the outlet) section of the fluid, and Γl the watertight and adiabatic wall of the considered domain. We add to these equations some boundary conditions of non homogeneous Dirichlet and homogeneous Neumann conditions and an initial condition, for the complete formulation of the problem on the parabolic border. In order to enlight the mathematical structure of the model and to keep control of the analysis in the case of possible correction of state laws, we consider a more general formulation.

We introduce the following system of partial derivative equations with the boundary conditions

(Ei)























∂tNi+ Xn j=1

∂kⁱ(N)

∂xj ∂tNj− Xn j=1

Dij∆Nj+∇(µ(N~ i)).U+ div(σⁱ(N)∇θ) = 0~ on ]0, T[×Ω, under the following boundary conditions:

Ni(0, .) =N_i⁰, Ni=N_i^bordon Γe∪Γs, ∂Ni

∂n = 0 on Γl,U.^→n= 0 and ∂θ

∂n = 0 on Γl

U and∇θ~ being given steady and regular vector fields, such as:

∇→θ∈(L^∞(Ω))³, ∆θ∈L²(Ω), div(U) = 0, 1≤i≤n,

(0.3)

whereµdenotes a Lipschitz function fromRtoRsuch asµ(0) = 0 and (σⁱ) denotes a functions family fromRⁿ toRsuch asσⁱ(N₁, ..., Ni−1,0, Ni+1, ..., Nn) = 0, extended in such a way thatσⁱ(N) = 0 on{N∈Rⁿ, Ni≤0}.

We propose here the following framework:

First, we prove the existence of a solution of the problem, by establishinga priori estimate from a linearized problem. Then we use a fixed point method (Schauder Tychonov), usually used for the treatment of quasi-linear parabolic conservative laws in porous media (cf. Gagneux and Madaune-Tort [14]).

Secondly, we prove that this solution verifies physical criterions in particular cases, that are useful for the experiments such as a space dimension equal to 1 (case of the capillary tube).

Then we set up a uniqueness and a continuity dependence result for somead hoc topologies of the solution against the initial state and some parameters of the convective transport by considering an auxiliary prob- lem, following the transposition method introduced by Antontsev and Domansk¨ı [4] (cf. on this point too Gagneux [13] or Holmgren [25], pp. 66–68).

Having proved the well-posedness of this problem, we present the numerical analysis of a computation scheme in it’s mixed formulation.

A last part is dedicated to the application of homogenization techniques to the upscaling of equations de- scribing thermodiffusion and effects in porous media. Two extreme situations are studied: the irreversible and full reversible cases.

We introduce in a classic way the functional Hilbert space V=

v∈H¹(Ω),trace(v) = 0 on Γe∪Γs .

The steady dynamical and thermal profiles are supposed to be givena priori, with the regularity given above. We consider an incompressible laminar flow (we know that thermal diffusion and viscosity are altered by turbulence).

Remarks.

In the following part, bold letters will denote vectorial lengths.

The considered problem gives a non homogeneous Dirichlet condition on (Γe∪Γs) which is treated with the help of the introduction of a variational vectorial inequation (cf.[7] , p. 143, [12] or [18]) relative to the closed convex partKof H¹(Ω)n

, defined by

K= Yn i=1

Ki, where Ki=

v∈H¹(Ω), v|_Γe∪Γs =N_i^b andV=Ki−Ki.

One has to take care that each of Ki is nonempty, which is supposed here, i.e. precisely, each function N_i^b is sufficiently regular (N_i^bord∈H¹²(Γe∪Γs),defined as the space of the restrictions to (Γe∪Γs) of the functions in H¹²(Γ) (cf. on this point, [8], Vol. 4, Chap. VII,§2)).

In order to write scalar equations under a vectorial form, we introduce the following notations: N=



 N₁ ...

Nn



 the weight fractions vector, andV=



 V₁ ...

Vn



the “test” functions vector, takena priori in (V)ⁿ.We introduce the following matrixes:

K(x˜ ₁, ..., xn) =



 K₁₁(x) ... K1n(x) ... ... ...

Kn1(x) ... Knn(x)



, where Kij(x) = δij + ∂kⁱ

∂xj(x), for (1 ≤ i, j ≤ n) and for

x∈(R⁺)ⁿ, then forx∈Rⁿ, afterad hoc extension (whereδij denotes the Kronecker’s symbol). The diffusive tensor is defined byD= [Dij]_1≤i,j≤n, Dij ∈R.We will give assumptions easy to verify in practice that will be sufficient for the mathematical analysis of the model.

By multiplying each equation (Ei) by the correspondent component Vi in V and by integrating on Ω, we obtain, with the help of Green’s formula, the vectorial variational problem, in order to define a strong solution:

Find N∈(L^∞(0, T;H¹(Ω))∩H¹(Q))ⁿ, N(t, .) ∈Ka.e.in t , such as,a.e. t∈]0, T[,∀V∈(L²(0, T;V))ⁿ, (P)

( (K(N)∂˜ tN,V) +d(N,V) +U(N,V) +σ(N, θ,V) = 0 N(0, .) =N⁰

where we will have first definedd, U, σ et(., .) in the following way, by using the sum convention on the repeated suffix:

d(N,V) =Dij

Z

Ω

∇N~ j·∇V~ idx , U(N,V) =− Z

Ω

µ(Ni)U·∇V~ idx σ(N, θ,V) =−

Z

Ω

σⁱ(N)∇θ~ ·∇V~ idx , (K(N)∂˜ tN,V) = Z

Ω

Kij(N)∂tNjVidx.

We are induced to study the existence of a strong solution of a coupled system of nonlinear parabolic evolutive equations, with transport and nonlinear thermal convection terms and with mixed boundary conditions. One has to prove that this formulation is a well posed one, in the Hadamard sense.

Assumption 1. The matrix K(x) and the diffusive tensor have a dominant diagonal,˜ i.e. they verify the

“pseudo-ellipticity” conditions, independent ofx:

(H)





∃α >0,∀x∈Rⁿ, ∀ζ∈Rⁿ, (K(x)˜ ζ,ζ)≥αkζk²,

∃d >0,∀x∈Rⁿ,∀ζ∈Rⁿ, (D(x)ζ,ζ) ≥dkζk². Assumption 2. We consider moreover that the diffusive tensor is a symmetrical onei.e.

∀(i, j)∈ {1, .., n}² Dij=Dji.

This last assumption, which is in agreement with physical experiments, is commonly adopted in the literature (Bia and Combarnous [6]; Duvaut and Lions [12]).

First, for technical arguments, we treat the problem for the following regularity conditions:

µ∈C¹(R)∩W^1,+∞(R), µ(0) = 0, σⁱ∈C¹(Rⁿ)∩W^1,+∞(Rⁿ).

Then, considering a density argument, we will be able to free ourselves from the differentiability assumption, and to keep the only Lipschitz feature of the state function (which is closer to the reality).

1. An existence result

In a first time and for more writing conveniences, we define the following functions family:

µ^k_i ≡ ∂σⁱ

∂xk· Considering a vector f = (f₁, ..., fn)^> taken in H¹(Q)n

and the problem (P_lin(f)) (with the Einstein’s con- vention)



























Find N∈(L^∞(0, T;H¹(Ω)))ⁿ, such as ∂N

∂t ∈ L²(Q)n

,

solution of the paralinearized Cauchy’s problem inQ= ]0, T[×Ω

∂tNi+∂kⁱ(f)

∂xj ∂tNj−Dij∆Nj+µ⁰(fi)U·∇N~ i+σⁱ(f)∆θ+

µ^k_i(f)∂Nk

∂xj

∂θ

∂xj = 0 Ni(0, .) =N_i⁰,

associated to boundary conditions : Ni=N_i^bordon Γe∪Γs, ∂Ni

∂n = 0 on Γl,U·^→n= 0 and ∂θ

∂n = 0 on Γl (1≤i≤n)

(1.1)

which owns a unique solution, with the help of the Lions theorem, applied to vectorial equations ([18] Chap. 3; [19])) and under the previous assumptions, it is clear that the solutionN belongs to (L^∞(0, T;H¹(Ω))∩H¹(Q))ⁿ.

1.1. A priori estimates in an appropriated functional frame

Proposition 1.1. The analysis of the paralinearized problem(P_lin(f))allows to establish the following a priori estimates:

∃C1>0, such that, ∀f ∈ H¹(Q)n

,kNk_(L∞(0,T;H¹(Ω)))ⁿ≤C₁ (1.2)

∃C₂>0, such that, ∀f ∈ H¹(Q)n

,k∂tNkL²(0,T;L²(Ω)ⁿ)≤C₂ (1.3) these constants depending only on N⁰, d, α, on the various Lipschitz constants of the state functions and on kUk_(L∞(Q))ⁿ, T,∇θ~

(L^∞(Q))ⁿ,|∆θ|_L2(Q).

Proof. By multiplying each equation by∂tNi (that can be justified by replacing∂tNi by a differential quotient) and by summing oni, we obtain the inequality, after having integrated successively on Ω and on [0, τ] where τ ∈[0, T]:

Z Z

Ω×[0,T]Kij(f)∂tNj∂tNidxdt− Z Z

Ω×[0,T]Dij∆Nj∂tNidxdt≤ Z Z

Ω×[0,T]

µ⁰r(fi)∇~Ni.U∂tNidxdt+ Z Z

Ω×[0,T]

σⁱ(f)∂tNi∆θdxdt +

Z Z

Ω×[0,T]

µ^k_i(f)∂Nk

∂xj∂tNi ∂θ

∂xj

dxdt.

We denote bykNk²_n =X

i



Z

Ω

∇N~ i²dx



, and observing that the expression

"

kNk²_n+X

i

Z

Γe∪Γs

(trace (Ni))²dσ

#¹2

defines onH¹(Ω)ⁿ an equivalent norm to the usual one; it comes Z τ

0

(K(x)∂˜ tN, ∂tN)dt≥α Z τ

0

|∂tN|²_ndt,

and, by considering that the diffusive tensor is symmetrical and permanent, we can write Z Z

Ω×[0,T]

Dij∇~Nj·∇~(∂tNi)dxdt≥d

2kN(τ)k²n−X

i,j

Dij

2 kN(0)k²n. (1.4) Moreover, thanks to the Young’s inequality, and for a positive constantε, we have,

Z

Ω

µ⁰(fi)∇N~ i.U∂tNidx ≤ C(ε)kNk²_n+ε 3|∂tN|²_n Z

Ω

σⁱ(f)∂tNi∆θdx ≤ C⁰(ε)|∆θ|²_L2(Ω)+ε 3|∂tN|²_n Z

Ω

µ^k_i(f)∂Nk

∂xj ∂tNi ∂θ

∂xj

dx ≤ C”(ε)kNk²_n+ε 3|∂tN|²_n

whereC”(ε) depends onµⁱ_k

L^∞(Ω),∇θ~

L^∞(Q),and C(ε) depends on∇µ~

(L^∞(Ω))ⁿ, kUk_(L∞(Ω))n. Thus we obtain the inequality, for allτ in [0, T],

α Z τ

0

|∂tN|²ndt+d

2kN(τ)k²n≤C₁(N⁰, θ) +ε Z τ

0

|∂tN|²ndt+C(ε) Z τ

0

kN(s)k²nds (1.5)

whereC(ε) =C

ε,µⁱ_k

L^∞(Ω), ∇~θ

L^∞(Q),∇~µ

(L^∞(Ω))ⁿ,kUk_(L∞(Ω))n . Then we chooseεsmall enough and obtain the following inequality:

∀τ∈[0, T], kN(τ)k²_n ≤C₁(N⁰, θ) +C(ε₀) Z τ

0

kN(s)k²_nds.

According to the Gronwall’s lemma and the remark on the norms equivalence, the element N of K remains in a fixed bounded part of L^∞(0, T; H¹(Ω))n

and a fortiori, there exists a constant ˜C₁ such that, for all p∈]1,+∞]

kNk₍L^p(0,T;H¹(Ω)))ⁿ≤C˜₁. Then we use again inequality (1.5) in order to obtain (1.3).

1.2. A fixed point method

The previous estimates prove that the functions setN=N(f₁, f₂, ..., fn) remains in a fixed bounded part of X = (L^∞(0, T; H¹(Ω))∩H¹(Q))ⁿ, independently off,when f covers (H¹(Q))ⁿ.

Notations and strategy

In order to use a fixed point method, we search a Banach space with separable dual space (in order to use Th. III.25⁰, pp. 50 of [7]). Moreover, in order to have compacity result for the weak topology, we introduce the reflexive space for p₀ fixed in [2,+∞[ Xp₀ =

L^p0(0, T; H¹(Ω))∩H¹(Q) ⁿ and K =

V∈X_p0,kVk_X

p0 ≤C,where C=

qC˜₁²+C₂², V(0, .) =N⁰, V(t, .)∈Ka.e.int

, the bounded part relative to the previous lemma.

From the consideration of the application=_p0 :

X_p0 →X_p0 f →N

and from the previous calculus, one can easily prove that the setKis stable by the application=p₀. Moreover,Kis a convex closed bounded nonempty set of Xp₀ and thus weakly compact ofXp₀.

Proposition 1.2. The application =_p0 admits a fixed point which is solution of the problem (0.3).

Proof. We consider the application=p₀ whenXp₀ is endowed with a weak topology σ(Xp₀,(Xp₀)⁰), that gives a structure of topological locally convex separated vectorial space. In order to use a “fixed point method”, one hast just to prove that=p₀ is a weakly-weakly sequentially continuous fromKtoK. So we consider a sequence (f^q)q∈N of elements fromXp₀ that weakly converges inXp₀ to f.

The functionsN^q and∂tN^q remaining respectively in some bounded fixed parts of L^p0(0, T; H¹(Ω))n

and of (L²(Q))ⁿ, we can extract some weakly convergent subsequences of suffixqk. Thus

∃N∈Xp₀,tel que N^qkX*^p0 N etN∈ K.

It remains to prove that

N==p₀(f).

Now, for allqk∈N,the linear associated problems (P_lin(fq_k)) are written under their variationnal formulations byN^qk∈ Kand∀V∈ Vⁿ, a.e.int,











 Z

Ω

Kij(fq_k)∂tN_j^qkVidx+ Z

Ω

Dij∇N~ _j^qk·∇V~ idx+ Z

Ω

µ⁰((fq_k)i)Vi∇N~ _i^qk·Udx +

Z

Ω

σⁱ(fq_k)Vi∆θdx+ Z

Ω

µ^l_i(fq_k)∂N_l^qk

∂xj

Vi ∂θ

∂xj dx= 0.

With the help of the compacity of the embedding fromH¹(Q) toL²(Q) (Rellich-Kondrachoff’s theorem) fq_k L⁴−a.e.inQ

−−−−−−−−→f

and, as ˜Kis a matrix of regular functions, we have:

Kij(f_qk)Vi L⁴−a.e.inQ

−−−−−−−−→Kij(f)Vi.

Using Lebesgue’s dominated convergence theorem, and considering in L²(Q) the product of strong and weak convergences, we obtain:

Z

Ω

Kij(fq_k)∂tN_j^nkVidx^k−→^→∞

Z

Ω

Kij(f)∂tNjVidx.

We use the same argument for the other integrals with the essential fact here that the state functions

µ⁰, ∂σⁱ

∂xk,etc.

are bounded and continuous. The trace application in t = 0 being strongly con- tinuous fromXp₀ to L²(Ω) and linear, it is weakly-weakly continuous and thus

N(0, .) =N⁰.

We obtain, passing to the limit, for allV inVⁿ and almost everywhere on ]0, T[: N∈ Kand















 Z

Ω

Kij(f)∂tNjVidx+ Z

Ω

Dij∇N~ j·∇V~ idx+ Z

Ω

µ⁰(fi)Vi∇N~ i·Udx +

Z

Ω

σⁱ(f)Vi∆θdx+ Z

Ω

µ^l_i(f)∂Nl

∂xj

Vi ∂θ

∂xj dx= 0 N(0, .) =N⁰(.)

It follows that, according to the uniqueness property,

N==p₀(f)

The cluster point =p₀(f) being independent of the extracted subsequence (because of the uniqueness of the solution of the linearized parabolic problem (P_lin(f)), we deduce that the entire sequence (N^q)_q∈N converges weakly inXp₀ toN==p₀(f). Then, the application=p₀ is weakly-weakly sequentially continuous fromKinK, which is weakly compact. According to the Schauder - Tychonov lemma,=p₀ admits a fixed point noted byN.

Hence, because for alli∈ {1, ..., n}







σⁱ(N)∆θ+

µ^k_i(N)∂Nk

∂xj

∂θ

∂xj = div

σⁱ(N)∇θ~ , µ⁰(Ni)U·∇N~ i= div(µ(Ni)U), car div(U) = 0

which is a justificationa posteriori of the formulation of the paralinearized problem, we can state:

∃N∈(L^p0(0, T; H¹(Ω))∩H¹(Q))ⁿ,N(t) ∈ Ka.e.int,such that,a.a. t∈[0, T],∀V∈(L²(0, T;V))ⁿ,

(P)





(K(N)∂˜ tN,V) +d(N,V) +U(N,V) +σ(N, θ,V) = 0 N(0, .) =N⁰.

Then, forp₀ taken in [2,+∞[,

=p₀(Xp₀)⊂X ⊂Xp₀

and thus,a fortiori, all the fixed points of the application=p₀, and in this case,Nremains in (L^∞(0, T; H¹(Ω)))ⁿ.

The result of the Proposition 1.2 holds in the following regularity frame:

µ∈W^1,+∞(R), σⁱ ∈W^1,+∞(Rⁿ).

Indeed, we know that, using mollifiers, each function inW^1,+∞(R) can be approached by sequence of elements in C^∞(R)∩W^1,+∞(R). The result of the proposition 1.2 remaining true for this type of functions, one can approach µ (resp. σⁱ

) by a sequence of elements in C¹(Rⁿ)∩W^1,+∞(Rⁿ)) that have the same Lipschitz module. We conclude, passing to the limit, and observing that, according to the proposition 1.1, thea priori estimates remain true, as they can be bounded independently of the considered state functions by this way of regularization.

2. Physical admissibility of the solution

Assumption 3. We suppose in this section that the following assumption is satisfied:

(H)







∀(i, j),∀(x₁, ..., xi−1,0, xi+1, ..., xn)∈(R+)ⁿ, ∂kⁱ

∂xj (x₁, ..., xi−1,0, xi+1, ..., xn) = 0,

∀(i, j), Dij=Dijδij (diagonal diffusive tensor).

Proposition 2.1. Under the previous assumption(H), the solution of the problem(P)verifies the property of physical admissibility:

∀τ∈[0, T],∀i∈ {1, ..., n}, Ni(τ, .)≥0, £³− a.e. inΩ.

The proof is based on the fact that the space is stable by null truncature at the origin and allows the choice of the test function V=−N⁻.

Remark 2.1. For some less elaborated models where the sum Xn j=1

Nj is takena priori as constant, equal toN0

(in order to fix ideas), the same method leads to the property

∀τ ≥0, ∀i∈ {1, ..., n}, Ni(τ, .)≤ N0 a.e.in Ω.

Generally, each method developed herein holds when the Soret coefficientsS_tⁱ are “Soret laws” (as functions of some weight fractions), as soon as there is a separated Lipschitz dependence on each variableNi.

In the mono-dimensional case (capillary column), the Ascoli Lemma allows to set a more precise result; this is done in

Proposition 2.2 (Unidirectional moves, capillary tube case). WhenΩ is a bounded part ofR, i.e. Ω = ]0, L[

andΓe={0},Γs={L},Γl=∅, we prove that

∀i∈ {1, ..., n},Ni∈C⁰([0, T]×[0, L]) =C⁰( ¯Q).Consequently,

∃M ≥0, ∀i∈ {1, ..., n},∀(t, x)∈Q,¯ 0≤Ni(t, x)≤M.

Thus, the solutions are physically admissible by this property, deduced from the model.

3. Uniqueness and stability of the solution

In this part, we prove the uniqueness of the solution using a duality technique,i.e.by looking for test-functions that allow to conclude. This technique, recently used by Diaz for Boussinesq-like problems (cf.[10] and [11]), reduces the uniqueness question to the study of the existence of a solution to the dual problem. We detail this transposition method here.

Proposition 3.1. In order to prove uniqueness of the solution of the problem(P), one has to prove the existence of a solution of a dual linear problem(P⁰).

Proof. Considering one by one each term of the equation, we denote by N=



 N₁ ...

Nn



et Nˆ =



 Nˆ₁ ...

Nˆn



 two possible solutions of (P). We subtract the two equations verified by each solution to obtain:





K˜_i(N)∂tNj−K˜_i(N)∂ˆ tNˆj, ζ

+

di(N, ζ)−di(N, ζ)ˆ +

Ui(N, ζ)−Ui(N, ζ)ˆ

+

σi(N, θ, ζ)−σi(N, θ, ζ)ˆ

= 0

(3.1)

where ζ is a test function of L²(0, T;V)n

to be precised, and applications d, U, σ are taken without being summed oni. Let’s examine one to one each term of this equation, after having integrated them on [0, T]:

Intⁱ₁ = Z T

0

K˜_i(N)∂tNj−K˜_i(N)∂ˆ tNˆj, ζ

dt

Green

= −

Z

Q

h

κⁱ(N₁, .., Nn)−κⁱ( ˆN₁, ..,Nˆn) i

∂tζidxdt

for each functionζverifyingζi(T) = 0 and because h

κⁱ(N₁, .., Nn)−κⁱ( ˆN₁, ..,Nˆn) i

(0) =κⁱ(N⁰)−κⁱ(N⁰) = 0.

Thus,

Intⁱ₁=− Z

Q

X

j∈{1..n}

κⁱ( ˆN₁, ..,Nˆj−1, Nj, .., Nn)−κⁱ( ˆN₁, ..,Nˆj, Nj+1, .., Nn) ∂tζidxdt.

Writing

κⁱ( ˆN₁, ..,Nˆj−1, Nj, .., Nn)−κⁱ( ˆN₁, ..,Nˆj, Nj+1, .., Nn) = (Nj−Nˆj)κⁱ_j(t, x) we define on the£⁴-measurable setsEj =

n

(t, x)∈Qsuch thatNj6= ˆNj

o

andQ\ Ej

κⁱ_j(t, x) =









κⁱ( ˆN₁, ..,Nˆj−1, Nj, .., Nn)−κⁱ( ˆN₁, ..,Nˆj, Nj+1, .., Nn)

(Nj−Nˆj) onEj

∂κ^i∗

∂xj

Nˆ₁, ..,Nˆj, Nj+1, .., Nn

onQ\ Ej

(3.2)

and ∂κ^i∗

∂xj is a bounded borelian representative (in it’s Lebesgue class) of the derivative (in the classic sense), defined almost everywhere, of the Lipschtiz function xj → κⁱ(..., xj, ...) (Rademacher’s theorem). Thus, we obtain the following expression forIntⁱ₁:

Intⁱ₁=− Z

Q

X

j∈{1..n}

(Nj−Nˆj)κⁱ_j(t, x)

∂tζidxdt.

Then, by summing the equations relative to each component,i.e. by summing oni∈ {1..n},we obtain:

Int₁=− Z

Q

X

j∈{1..n}

(Nj−Nˆj)h X

i∈{1..n}

κⁱ_j(t, x)∂tζi

i dxdt.

We use the same technique for the third and the fourth terms of the equation (3.1), having first defined the functions

µ^∗(t, x) =





µ(Ni)−µ( ˆNi) Ni−Nˆi

onEj

µ^0∗(Ni) onQ\ Ej

and

S_jⁱ(t, x) =









σⁱ( ˆN₁, ..,Nˆj−1, Nj, .., Nn)−σⁱ( ˆN₁, ..,Nˆj, Nj+1, .., Nn) Nj−Nˆj

onEj

∂σ^i∗

∂xj

Nˆ₁, ..,Nˆj, Nj+1, .., Nn

onQ\ Ej

µ^0∗ and ∂σ^i∗

∂xj being, according to the same principle, some bounded borelian representatives of the derivatives ofµand ofσⁱ. Let’s have a look to the second term of the equation (3.1):

Intⁱ₂ = X

j∈{1..n}

Dij

Z

Ω

∇~

Nj−Nˆj

·∇ζ~ idx

= − X

j∈{1..n}

Dij

Z

Ω

Nj−Nˆj

∆ζidx+ X

j∈{1..n}

Dij

Z

∂Ω

Nj−Nˆj

∂ζi

∂ndσ

= − X

j∈{1..n}

Dij

Z

Ω

Nj−Nˆj

∆ζidx

as soon as ∂ζi

∂n

Γl

= 0, because Nj−Nˆj

Γe∪Γs

= 0 by definition of the problem (P). Then, we obtain, after having summed oni∈ {1..n}and integrated on [0, T],

Int₂=− Z

Q

X

j∈{1..n}

(Nj−Nˆj) X

i∈{1..n}

Dij∆ζi

dxdt.

We obtain the following equation:

(†)











 Z

Q

X

j∈{1..n}

Nj−Nˆj

h− X

i∈{1..n}

κⁱ_j(t, x)∂tζi− X

i∈{1..n}

Dij∆ζi

+µ^∗(t, x)U·∇ζ~ i+ X

i∈{1..n}

S_jⁱ(t, x)∇θ~ ·∇ζ~ i

i

dxdt= 0

Let us consider in the following backward problem of unknown ζ= (ζi)_i :

(P⁰)

















− X

i∈{1..n}

κⁱ_j(t, x)∂tζi− X

i∈{1..n}

Dij∆ζi+µ^∗(t, x)U·∇ζ~ i+ X

i∈{1..n}

S_jⁱ(t, x)∇θ~ ·∇ζ~ i =Nj−Nˆj in Q

ζj(T) = 0 in Ω

ζj= 0 on Γe∪Γs, ∂ζj

∂n = 0 on Γl,(1≤j≤n).

The functionsκⁱ_j, µ^∗, S_jⁱ are inL^∞(Q) as soon as the state functions are Lipschitz ones against each variable, assumption that we will consider to be true. One can easily prove with the help ofa priori estimates and the Lions theorem Lions ([7], p. 218) applied to vectorial equations that the problem (P⁰) admits a unique solution.

The dependence of the solution during time on the initial state or on some parameters linked to the convective transfer is studied in the following proposition:

Proposition 3.2. The problem (P) has a unique solution. Furthermore, the (nonlinear) application which associatesNin

L^∞(0, T; H¹(Ω))∩H¹(Q)n

toN⁰ inH¹(Ω)ⁿ is a local Lipschitz one fromL²(Ω)ⁿ toL²(Q)ⁿ. In the same way, the (nonlinear) application that associatesN to{Stⁱ}1≤i≤n

h

resp. ~∇θi

is a local Lipschitz one fromRⁿ toL²(Q)ⁿ

resp. f romL^∞(Ω)ⁿto L²(Q)ⁿ .

Proof. Taking precisely for test functionζthe solution of the associated backward system (P⁰) and introducing the equation verified byζ in the expression (†), we obtain immediately, according to the solutions regularity,

∀t∈[0, T], N(t, .) =N(t, .)ˆ £³−a.e.in Ω.

In order to prove the stability of the solution of the problem (P), let us consider N (resp. N) the solutionˆ relative to N⁰ (resp. Nˆ⁰). Then, using again the previous method and for the same choice ofζ, we get

N−Nˆ²

(L²(Q))ⁿ= Xn

i=1

N_i⁰−ki(N⁰)−Nˆ_i⁰+ki(Nˆ⁰), ζⁱ(0)

L²(Ω). (∗)

As, a.e. intand a.e. in Ω, and for i∈ {1, ..., n}

N_i⁰−Nˆ_i⁰−

ki(N⁰)−ki(Nˆ⁰)≤N_i⁰−Nˆ_i⁰+ Xn

j=1

N_j⁰−Nˆ_j⁰ ∂ki

∂xj

L^∞(Rⁿ)

(∗∗) and, with the help of general results on the linear parabolic equations on the continuous dependence of the solutions on the data (cf.Dautray and Lions [8], Vol. 8, Chap. XVIII,§.3 and 4), (hereζ is a solution relatively to an initial vanishing data and a sink term equal toN−N), there exists a constantˆ C^∗, dependinga priori on Ω, T,U, θ,and on the state functions, such that:

|ζ(0)|L²(Ω)ⁿ ≤ |ζ|C⁰([0,T];L²(Ω)ⁿ)≤C^∗N−Nˆ²

(L²(Q))ⁿ. (∗ ∗ ∗)

Thus, according to (∗),(∗∗), and (∗ ∗ ∗) and due to the Cauchy-Schwarz inequality, there exists a constant C (which depends mainly on the Lipschitz constant C^∗ and on the Lipschitz modules of the partial functions xj−→ki(..., xj, ...)), such as

N−Nˆ

(L²(Q))ⁿ≤CN⁰−Nˆ⁰_L2

(Ω)ⁿ. The other properties are obtained with the same transposition method.

4. A numerical scheme

In the following part, nc will denote the number of unknowns (i.e. the number of components in the fluid) in order to avoid any confusion between this one and the time discretization suffix.

We present here a numerical scheme based on a mixed formulation [22, 23] of the problem for which we give numerical analysis.

First, we describe briefly the adopted mesh and present the different spaces and notations used in the analysis.

Then we prove the results on the approximation error made in the stationary problem. Finally we study the evolutive problem for which we demonstrate existence and uniqueness of a solution and present convergence and consistence results for the numerical scheme.

The main difficulty in this numerical study remains in the verification of three Inf-Sup conditions associated with the used mixed formulation. The choice of the discretization spaces likely to verify these conditions must be done with caution. Moreover, we treat a system and no more a scalar equation. We will see that under ad hoc assumptions on diffusive and adsorption tensors (which are close to the reality), it is possible to prove the method’s convergence. Finally, we link the proposed discretization with the finite volumes method.

Remark 4.1. Here is detailed the numerical analysis of a scheme in the case of low Peclet numbers (situation where the diffusive phenomena dominate the convection, which is true for viscous fluids submitted to natural convection in deposits).

In a first time, we are interested in the following problem:

−divD˜˜(∇N)^>

+KN˜˜ =f(N) ∆θ which is rewritten,∀i∈ {1..nc},

− X

1≤l≤n_c

Dil∆Nl+ X

1≤j≤n_c

KijNj =fi(N, x).

We define then ˜˜p=D˜˜

gradN˜ _>

. An equivalent problem is given by,∀i∈ {1..nc},

−div (pi) + X

1≤j≤n_c

KijNj=fi(N, x)

4.1. Notations

We modelize the medium by a planar rectangular column, and construct a regular rectangular mesh.

Let us introduce the following spaces:

W₁= (H(div,Ω))^nc, W₂= ( L²(Ω)₂

)^nc, M₁= (H¹₀(Ω))^nc, M₂=L²(Ω) endowed with their usual norms, to which we associate the discrete spaces

W1h⊂W₁, W2h⊂W₂, M1h⊂M₁, M2h⊂M₂. Then, we define the spaces

• P₀(K) the space of constant functions onK

• P₁,0(K) the space of linear functions (against the first space variable) onK

• P₀,1(K) the space of linear functions (against the second space variable) onK

• Q₁(K) the space of linear functions (against each space variable) onK.