Homogenization of reactive flows in porous media

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1 Homogenization of reactive flows in porous media HUTRIDURGA RAMAIAH Harsha Abstract In this article we study reactive flows through porous media. We use the method of two-scale asymptotic expansions with drift to get the upscaled model. This article is a slight variant from an already published result. We have added Laplace-Beltrami operator on the boundary. An attempt has been made to study the effect of introducing the Laplace Beltrami operator on the dispersion tensor we get in our Homogenized equation. Introduction This project report is divided into various sections. Section gives a general introduction to what has been done in the rest of the report. Section is dedicated to the description of the microscopic equations. Section 3 is dedicated to the apriori estimates and the existence results for the microscopic problem. Section 4 gives a general description of the classical Two scale expansion method and has been applied to our problem to arrive at the Homogenized equation. Section 5 is dedicated to the convergence result where we have employed the Two Scale Convergence with drift to show the convergence of the sequence of the solutions to the solution of the homogenized equation. Section 6 deals with some numerical tests that were done during the project and contains some of the interesting observations done during that period. Section 7 concludes the report. Description of the problem Let Ω be a bounded open set in R. Let Y =,] be the unit periodicitycell which is encountered in the classic periodic homogenization theory. This unit periodicity cell is identified with T = R /Z. Let Y = Σ where Σ is the solid part which in our case is a discfor simplicity of radius less than and is the fluid part. Σ is a circle. Now, for a sequence of positive numbers going to zero, we define a perforated domain Ω by Ω = {x Ω x } We shall also define a -dimensional space, Ω as Ω = {x Ω x Σ} The microscopic model we intend to study is a convection-diffusion system in the fluid phase and a diffusion-reaction system on the pore surface. u t + x b x u D u =. in Ω. Centre de Mathématiques Appliquées, CNRS UMR 764, Ecole Polytechnique, 98 Palaiseau, France hutridurga@cmap.polytechnique.fr

2 D u n = v t α D S S v = κ u v ]. on Ω. u,x = u x, v,x = v x 3 Where, D and D S represent molecular diffusionspositive constants, κ represents the rate constant for adsorption and represents the linear adsorption equilibrium constant. Also κ and are positive. The velocity field, by, is assumed to be periodic satisfying by L R, div y by =, by ny = on Σ It should be noted here that the divergence free assumption and the no-penetration condition on the velocity field is not necessary as one can still homogenize the system in their absence using factorization principle to start with as is done in The initial data are chosen such that u x L R v x H R We shall take α = in the rest of the calculations to follow. 3 A Priori Estimates We first derive the energy estimate for --3. To derive the same, we shall multiply by u and then integrate over Ω. u t u dx+ bx xu u dx+ D u u dx D u nu dσx = Ω Ω Ω Ω d u dx+d u dx+ dt Ω Ω κ Ω u u v dσx = 4 As rightly noticed, the integral with the convective term has disappeared. That is due to our choice of the velocity field b being divergence free and b n =, where n is the normal to the boundary. bx xu u dx = x divb u dx = u x b ndσx = Ω Ω Ω Let us consider v t DS S v = κ u v ] on Ω. Multiply the above equation with v and integrate over Ω. This results in d v D S dσx+ S v κ v dσx+ dt u v dσx = Ω Ω Adding 4 and 5 results in the energy estimate for - given below: d u d dx+ v dσx dt dt Ω Ω Ω 5

3 +D Ω u dx+ D S Ω S v dσx+ κ Ω v u v dσx = Integrating the above energy estimate over the time interval,t results in the estimate given below: u L,T;L Ω + v L,T;L Ω + u L,TXΩ + S v L,TXΩ C u L R n + v H R n 6 The a priori estimate 6 implies the existence and uniqueness of solution u,v to --3 in L,T;H Ω X L,T;H Ω. 4 Classical Two Scale Expansion We take the ansatz We take the ansatz u = v = Consider y = x. Then we have: i= i= i u i, x i v i, x ] t φ ǫ, x ǫ = φ t N b j j= ǫ ] x j φ ǫ, x ǫ = x j φ+ y j φ ] φ x j ǫ, x ǫ ] ǫ,y j {,} We substitute the two ansatz 7 and 8 in our equations and. u t = i= iu i ] t i b x u i b x x u = ] i= i by x u i + i by y u i D u = D ] i= i x u i + i div y x u i + i div x y u i + i y u i D u n = D i= i x u i n+ i y u i n ] v t = i= iv i t i b x v i ] D S S v = D S i= κ u ] v = κ i= i u i i div x G x v i + i div x G y v i + i div y G x v i + i y S v i ] i vi ] This is the point where we identify the co-efficients of the various powers of in to get the systems of PDE s given below. Co-efficients of. b y u D y u = in, D y u n = D S S v = k ] u v on Σ, y u y,v y periodic, 3

4 Co-efficients of. b y u D y u = b b] x u +Ddiv x y u in, D y u n D x u n = D S S v D S div y G x v D S div x G y v b x v = k ] u v on Σ, y u y,v y periodic, Co-efficients of. b y u D y u = b b] x u +Ddiv x y u u t +D x u in, D y u n D x u n = D S S v D S div y G x v D S div x G y v b x v + v t DS x G x v = k ] u v on Σ, y u y,v y periodic, 3 Where G is the inverse of the metric tensor. The matrix tensor is a symmetric matrix. Hence we have G which is symmetric. This symmetric property of G plays a very interesting role when we try to symmetrize the Dispersion Tensor in our Homogenized equation. Also we have S yu = G y u. That is, G projects the usual gradient on the surface. Being a projection, we can say that G = G. We shall address these two observations at a later stage. Our goal at hand is to solve the above system of p.d.e s. The below given lemma plays a very crucial role in solving the cascade of sytems of p.d.e s we have. Lemma. For f L, g L Σ and h L Σ, the following system of p.d.e. s admit a solution u,v HperY H Σ, unique up to the addition of a constant multiple of,, b y u D y u = f in, D y u n+g = D S S v h = k u v on Σ, 4 y uy,vy periodic, if and only if f dy + g +hds =. 5 Σ Proof. The variational formulation of 4 is D S b y uφdy+d y u y φdy+ S y v S y ψds+ Σ fφdy + Σ gφ+ h ψ ds. Σ k u v ] φ ψ ds = ] Taking φ,ψ =, we find the necessary condition 5. The left hand side of the variational formulation is coercive on the space H per H Σ ]/R,] where the space R, is the set of constant vectors of the type C,C when C runs into R. 4. Solving for u,v On identifying with 4 we get { f = in, g = h = on Σ 6 4

5 So, 5 is trivially satisfied. Hence lemma assures the existence and uniqueness of u,v H per H Σ ]/R,]. Now, let us try to analyse the dependence of u,v on the variable y. Consider the variational formulation for u,v by taking φ,ψ as their corresponding test functions. b y u φ dy+d y u y φ dy+ D S S yv S yψ ds+ Σ Taking φ,ψ = u,v in the above variatioanl form yields b y u u dy+d y u D S dy+ S y v ds+ Σ Σ k We shall show that the first term in the above equality is zero. div y u by = by yu +u div yby Σ k u by y u = divy u by u div yby ] u by y u dy = div y u bydy u div ybydy Using Divergence theorem we shall have u by y u dy = u by nds u div ybydy Σ u v ] φ ψ ] ds = u v ] ds = By the very hypothesis on by, we will have that the RHS is zero. Hence in the variatioal form we are left with D y u D S dy + S yv ds+ k u v ] ds = Σ Σ This implied that y u = a.e in. Also v = u. y u = a.e in implies that u = u t,x This helps us get rid of the term, Ddiv x y u and D S div x G y v in. 4. Solving for u,v On observing, we can say that and u t,x,y = χy x u v t,x,y = ωy x u The above representation of u,v results in the following coupled cell problem, for i {,}. by y χ i y D y χ i y = b i y b iy in, D y χ i y n De i n = D S S y ω i D S div y Ge i b i = κ ] χ i ωi in Σ, y χ i,ω i periodic, 7 5

6 The coupled variational formulation for χ i y,ω i y to 7 is by y χ i φ dy +D y χ i y φ dy + D S S y ω i S y ψ ds+κ ] ] χi ωi φ ψ ds Σ Σ = b i φ dy b i φ dy D e i nφ ds+d S div y Ge i ψ ds+ b i ψ ds Σ Σ Σ 8 On identifying 7 with 4 we get f = b i y b iy in, g = De i n on Σ, 9 h = D S div y Ge i +b i on Σ According to our lemma, we shall have the existence and uniqueness of u,v Hper H Σ ]/R,] provided the below equality holds. b i b i dy D e i nds+ D S div y Ge i ds+ b i ds = Σ Σ Σ By divergence theorem D e i nds = D Σ Also, by integration by parts Thus we will have Σ D S div y Ge i ds = D S b i div y e i dy = Σ Ge i S y ds = + Σ ] = b i dy Thus the compatibilty condition for the existence and uniqueness of χ i,ω i has aided us in finding the drift velocity b used in our ansatz for u,v. b y = + Σ bydy 4.3 Solving for u,v On identifying 3 with 4 we get f = b b] x u +Ddiv x y u u t +D x u in, g = D x u n on Σ, h = D S div y G x v +D S div x G y v +b x v v t +DS x G x v on Σ According to our lemma, we shall have the existence and uniqueness of u,v Hper H Σ ]/R,] provided the below equality holds. b u b] x u dy +D div x y u dy dy +D x u dy t D x u nds+d S div y G x v ds+d S div x G y v ds Σ Σ Σ 6

7 + b x v ds+d S x G x v ds Σ Σ We have from Divergence theorem D x u nds = D div y x u dy = D Σ Also from integration by parts D S Σ div y G x v ds = D S Σ Σ v t ds = div x y u dy Σ G x v S y ds = The above two observation reduces the compatibility condition to b b] x u dy +D div x y u dy +D x u dy +D S div x G y v ds+ b x v ds+d S x G x v ds Σ Σ Σ v = t ds+ u t dy Substitutingforu,v intermsofχ,ωandv intermsofu intheaboveequation results in + Σ ] u u u = χ j dy χ j dy t x i x j x i x j +D x i y i + b i ] u χ j dy+d u +D S x j x i x j Σ b i x i u ω j ds+d S x j Σ Σ x i b i We write the above equation in a compact form as below + Σ ] u t Where, A is given by A ij = b i χ j dy b i χ j dy +D +D S Σ Ge i S yω j ds+b i x i ] u G iβ ω j ds y β x j u G ij ds x j = div x A x u Σ ω j ds+d S χ j y i dy +D δ ij Σ Ge i e j ds We will symmetrize the above dispersion tensor A as the antisymmetric part doesn t contribute as div x A a x u =, where A a is the antisymmetric part. We shall continue to denote the symmetric part by A. A ij +A ji = b i χ j dy +b j χ i dy b i χ j dy + b j χ i dy 7

8 + χ j χ i D dy +D dy +D δ ij + b i ω j ds+b j ω i ds y i y j Σ Σ + D S Ge i S yω j ds+d S Ge j S yω i ds +D S Ge i e j ds Σ Σ Σ Now, we shall turn our attention to the the coupled variational formulation of our cell problem. We shall at first test the variational formulation for χ i,ω i with χ j,ω j as test functions and then viceversa. We shall then add the two variational fomulations so got and divide by. These calculations result in D S D y χ i y χ j dy + S yω i S yω j ds+κ χ i ω ] i χ j ω ] j ds Σ Σ + χ j χ i D dy +D dy + D S Ge i S y i y j yω j ds+d S Ge j S yω i ds Σ Σ = b i χ j dy +b j χ i dy b i χ j dy + b j χ i dy + b i ω j ds+b j ω i ds Σ Σ Using the above expression in the symmetrized A ij gives us a new expression for A. A ij = D y χ i y χ j dy +D χ j y i dy +D χ i y j dy +D δ ij +κ ] ] χi ωi χj ωj ds Σ + DS S yω i S yω j ds+d S Ge i S yω j ds+d S Ge j S yω i ds+d S Ge i e j ds Σ Σ Σ Σ 3 Practically speaking, if we put D S = in our expression for the dispersion tensor, we should get back the dispersion tensor of the article. Actually, we do. The first line in the above definition of the dispersion tensor is same as that in the article. The only question is about the second line. In the absence of D S, the cell problem can be easily decoupled. We get an expression for ω i pretty easily. It is given by ω i = χ i + κ b i Then the second line in the above definition of A ij : κ Σ χ i ω i ] χ j ω j ] ds = b i b j Σ κ Taking D S = O and replacing the above expression for the second line, we shall recover the same expression as in the article. At last, we shall write the dispersion tensor of 3 in a symmetric form: A ij = D y χ i +e i y χ j +e j dy +κ ] ] χi ωi χj ωj ds Σ +D S Σ G S y ωi +ei S y ωj 8 +ej ds 4

9 Let us see how our last term accounts for some terms in our original expression: D S S ωi y +e i S ωj y +e j ds Σ G = DS S y ω i S y ω jds+d S Ge i S y ω jds Σ Σ +D S Ge j S y ω ids+d S Ge i e j ds Σ Σ The third term in the RHS is due to the fact that G is symmetric. The first term in the RHS does appear because G is known to be a projection. G S y ω i S y ω j = GG y ω i S y ω j = G y ω i S y ω j = G y ω i S y ω j = S y ω i S y ω j 5 Two scale convergence with drift Let us start this section by defining two scale convergence with drift. Definition 5. Let µ be a constant vector in R. We say that a sequence of functions U t,x L,T R two scale converges in moving co-ordinates t,x t,x µt to a function U t,x,y L,T R T if U L,T R C and for any φt,x,y C,T R T T lim U t,xφt,x R µt, x T dxdt = We shall denote this convergence by U drift U. In the spirit of a result in ], we have the following result: R T U t,x,yφt,x,ydydxdt Proposition 5. Let µ be a constant vector in R and let the sequence U be uniformly bounded in L,T;H R. Then, there exists a subsequence, still denoted by, and functions U t,x L,T;H R and U t,x,y L,T R ;H T such that U drift U and U drift x U + y U In the next proposition, we try to extend the concept of two-scale convergence with drift to sequences defined on surfaces. Proposition 5.3 Let µ be a constant vector in R and let W be a sequence in L,T Ω such that T Ω W t,x dσ xdt C 9

10 Then, there exists a subsequence, still denoted by, and a function W t,x,y L,T R ;L Σ such that W t,x two-scale converges in moving co-ordinates to W t,x,y in the sense that T lim Ω W t,xφt,x µt, x T dσ xdt = for any φt,x,y C,T R ;C pery Proposition 5.4 Let W t,x L,T;H Ω be such that and T T R Ω W t,x dσ xdt C Ω S W t,x dσ xdt C Σ W t,x,yφt,x,ydσydxdt then there exists W t,x L,T;H R andw t,x,y L,T R ;H Σ such that drift W W t,x S W drift Gy x W t,x+ S yw t,x,y where, Gy represents the projection matrix that projects the full derivative on the tangent plane. We just provethe last proposition. The proofof the first one can be found in ] and that of the second one can be found in 3]though the proof is not for the two-scale convergence with drift, the proof can be immitated to fit in our case. However, the same is true with the proof that we are going to provide for the third proposition. Proof By the very definition of the two-scale convergence with drift on the boundary,wehavetheexistenceofw t,x,y L,TXR ;L Σ andm t,x,y L,T R ;L Σ ] such that and W drift W t,x,y S W drift M t,x,y Let us choose test function Ψt,x,y C,T R ;C per Y]. Integration by parts yield: T T = Ω S W t,x Ψt,x W t,xdivy S Ψ Ω T W t,xdivx S Ψ Ω µt, x dσ xdt t,x µt, x ] dσ xdt t,x µt, x ] dσ xdt

11 Passing to the two-scale limit as tends to zero results in: = T R Σ W t,x,ydiv S y Ψt,x,ydσydxdt = W t,x,y = W t,x Now, let us choose the test function Ψ such that div S yψ =. Then, T lim = Ω W t,xdiv x GyΨt,x,y] T R, x dσ xdt Σ W t,xdiv x GyΨt,x,y dσydxdt Since div S yψ =, we can integrate by parts in the LHS of the above equation and then apply the two-scale limit as follows: Hence we have T T lim R T = Ω S W t,x Ψt,x R µt, x dσ xdt Σ M t,x,y Ψt,x,ydσydxdt Σ M t,x,y Gy x W t,x] Ψt,x,ydσydxdt We know that the orthogonal of the divergence free functions coincide exactly with the gradients. Thus, we havethe existence of W t,x,y L,T R ;H Σ such that M t,x,y = Gy x W t,x+ S yw t,x,y Now, we shall turn our attention on to our problem and try to use the above propositions to arrive at the two-scale limit with drift. Let us choose the drift b to bejustification shall be given later b = + Σ bydy It would be better to recall the estimates that we have derived in section3 on u,v as we shall be using them extensively in obtaining their two-scale limits. T Ω v t,x dσ xdt+ u L,T;H R T Ω S v t,x dσ xdt C From Proposition5., we have the existence of ut,x L,T;H R and u t,x,y L,T R ;H T such that u drift ut, x

12 and u drift x ut,x+ y u t,x,y From Proposition5.4, we have the existence of vt,x L,T;H R and v t,x,y L,T R ;H Σ such that v drift vt, x S drift v Gy x vt,x+ S yv t,x,y If we take w = u v, we do have an estimate on w given below: T w t,x dσ xdt C Ω Thus, by the very definition of the two-scale convergence on boundary, we have the existence of qt,x,y L,T R ;L Σ such that u v From the estimate on w it follows that drift qt,x,y u v L,T Ω C This implies that the two-scale limits of both u and v should be the same. Thus we have ut,x = vt,x Thus, we shall take the test function corresponding to v as that of u multiplied by. In fact, we can show that qt,x,y = u v The following lemma helps us immensely to arrive at this result. Lemma 5.5 Let φt,x,y L,T R ;L Σ suchthat φt,x,ydσy = Σ for every t,x,t R. Then there exist θt,x,y L,T R ;H T ] and Θt,x,y L,T R ;H Σ ] such that div y θ = in, θ n = φ on Σ, 5 div S yθ = φ on Σ, Proof The existence of periodic solutions θ and Θ for 5 is a straight forward result due to our choice of φ. Let us choose φ as in the previous lemma as our test function. Our aim is to calculate T lim b t w t,xφt,x, x dσ xdt Ω

13 We shall try to realise these calculations term by term. = T T Ω u φ, x T dσ xdt = Ω divu θt,x Ω T drift T R b t, x T dxdt = R Ω Ω u θt,x b t, x ndσ xdt u θ, x +u div x θ, x ] dxdt x u+ y u θ +udiv x θ] dydxdt Σ u θ ndσydxdt = T R Σ u φdσydxdt Now, let us turn our attention to the term that involves v. T v φ, x T dσ xdt = v div S y Θ, x dσ xdt T = T = Ω Ω T drift R Σ T = Ω v div S Θ, x x div x G Θ, x ] dσydxdt Θ, x x S v div x G Θ, x ] v dσydxdt R Σ Thus, we have shown that T R for all φ such that Θ Gy x v + S yv divx GyΘv ] dσydxdt T v divy S Θdσydxdt = Σ qt,x,yφdσydxdt = Σ φdy =. Thus, T R Σ R Σ qt,x,y = u v +lx v φdσydxdt u v φdσydxdt for some function l depending only on x. Since, u, v are also unique upto addition of a function solely dependant on x, we can get rid of lx. Let φ,ψ be the test functions corresponding to u,v. Let us now write down the coupled variational formulation for -. T + Ω T Ω u t φ + ] bx xu φ +D u φ dxdt v t ψ + DS S v S ψ + κ 3 u v φ ψ ] dσ xdt =

14 The choice of the test functions are as below: φ = φ +φ, x ψ = φ +ψ, x We shall resolve the variational formulation term by term by substituting the chosen test functions. T u T ] t φ φ dxdt = u dxdt+ T u b ] x φ dxdt t Ω Ω Ω T T + Ω Ω u b x φ bx xu φ dxdt = T +, x ] dxdt u xφ,xdx+o Ω Ω b T u b Ω x x u φ t,x b t, x ] x x φ t,x b t dxdt dxdt It should be noted that the integration by parts was done only in the term involving φ and not the one involving φ. T + D S T Ω T T Ω T D Ω u φdxdt = D T +D Ω u y φ v T t ψ dσ xdt = v b x ψ Ω, x T Ω u φ ] dxdt, x ] dxdt+o Ω v φ T t dσ xdt+ ] dσ xdt Ω Ω v b x φ ] dσ xdt v xφ,xdσ x+o Ω T ] S v S ψ dσ xdt = D S S v S φ dσ xdt Ω Ω T + DS S v S yψ, x ] dσ xdt+o Ω κ u v φ ψ T κ dσ xdt = u v φ ψ dσ xdt 4

15 References ] Allaire G. Homogenization and two-scale convergence, SIAM J. Math. Anal., Vol 3 99, No.6. ] Marusic-Paloka E, Piatnitski A. Homogenization of a nonlinear convectiondiffusion equation with rapidly oscillatig co-efficients and strong convection, In Journal of London Math. Soc., Vol 7 5, No.. 3] Allaire G, Damlamian A, Hornung U. Two-scale convergence o periodic surfaces and applications. In Mathematical modelling of flow through porous media. Vol 983. Springer-Verlag, Berlin, 9. 5