Homogenization of linear adsorption in porous media.auroux/IFCAM2012/Harsha_Hutridurga.pdf · Mesoscopic Model Unbounded porous media: Connected Fluid Part Ωf. Solid Part Ωs = Rd

Homogenization of linear adsorption in porous

media.

Gregoire Allaire Harsha Hutridurga

CMAP, Ecole Polytechnique, Palaiseau.

IFCAM, November 2012

Allaire, Hutridurga (Polytechnique) Upscaling Adsorption IFCAM ’12 Nice 1 / 49

Outline

• Model Description.

• Two-scale asymptotic analysis.

• Two-scale convergence with drift.

• Convergence proof.

• Numerical simulation.

• Perspectives.


Objective

• Dispersion of solute in a porous medium in presence of adsorption.

• Approximating the concentration field for a mesoscopic equation.

• Simulation in two dimension.


Mesoscopic ModelUnbounded porous media: Connected Fluid Part Ωf .

Solid Part Ωs = Rd \Ωf .

Flow field (incompressible): b(y) in Ωf and bs(y) on Ωs.In truth, bs(y) should be trace of b(y) (not necessary for analysis).

Solute concentration (unknown): u in Ωf and v on Ωs.

Diffusion matrices (coercive): D(y) in Ωf and Ds(y) on Ωs.

At chemical equilibrium, v = Ku with K equilibrium constant.

Convection-Diffusion in Ωf .

∂u

∂τ+ b(y) · ∇u− div(D∇u) = 0 in (0, ζ)× Ωf .

Convection-Diffusion with linear adsorption reaction on Ωs.

∂v

∂τ+ bs · ∇sv− divs(Ds∇sv) = κ

(

u− v

K

)

= −D∇u · n on (0, ζ)× ∂Ωs.


κ > 0 the reaction rate.

n(y) exterior normal to Ωf .

G(y) = Id− n(y)⊗ n(y) the projection matrix.

∇sv = G∇v the tangential gradient.

divsΨ = div(GΨ) the tangential divergence.

div b = 0 in Ωf , divs bs = 0 and b · n = 0 on ∂Ωf .

The velocity field is given.

Nonlinear isotherms: v = αu/(1 + βu) →Langmuir.

v = γuδ →Freundlich.

α, β, γ, δ (all positive) are experimentally determined.


Periodic porous mediaLength Scales: L→ observation length.

l → mesoscopic length.l/L = ε << 1.

Unit cell Y = [0, 1]d = Y 0 ∪ Σ0 s.t Y 0 ∩ Σ0 = ∅

For j ∈ Zd, we define Y j

ε = ε(Y 0 + j), Σjε = ε(Σ0 + j), Sj

ε = ε(∂Σ0 + j).The periodic porous medium Ωε = ∪j∈ZdY j

ε

The (d− 1)−dimensional surface ∂Ωε = ∪j∈ZdSjε

Y0

Ʃ0

Y∂Ʃ

0

Y = Y0∪ Ʃ

0

Periodic Porous media

ε ℝ2

UnitCell

FluidPart Part

Solid


Scaling

We cast our system on observation length scale and long time scale.

x = εy and t = ε2τ

Parabolic scaling or Diffusion scaling

We define

uε(t, x) = u(τ, y) and vε(t, x) = v(τ, y).


Convection-Diffusion-Reaction system (rescaled)With

bε(x) = b(x/ε) bsε(x) = bs(x/ε)

Dε(x) = D(x/ε) Dsε(x) = Ds(x/ε)

and final time T = ε2ζ.

∂uε

∂t+ 1

εbε · ∇uε − div(Dε∇uε) = 0 in (0, T ) × Ωε,

−1εDε∇uε · n = κ

ε2

(

uε − vεK

)

on (0, T )× ∂Ωε,

∂vε∂t

+ 1εbsε · ∇svε − divs(Ds

ε∇svε) =κε2

(

uε − vεK

)

on (0, T )× ∂Ωε,

uε(0, x) = uinit(x) x ∈ Ωε,

vε(0, x) = vinit(x) x ∈ ∂Ωε.

ε−1 next to the velocity and ε−2 next to reaction term means, theconvection diffusion and reaction are of same order of magnitude at themesoscale.


Another mesoscopic model

• What if the velocity field is not divergence free?

• What if there are reactions in the bulk?

For simplicity, we work without surface concentration.

Convection-Diffusion-Reaction in Ωf .

∂u

∂τ+ b(y) · ∇u− div(D∇u) + r(y)u = 0 in (0, ζ)× Ωf .

Boundary condition on ∂Ωf .

−D∇u · n = κu on (0, ζ) × Ωs.

There are no restrictions on b(y) and r(y): Allaire & Anne-LiseRaphael(07) (Factorization Principle).


Simple case

Before we consider the full system, consider a convection-diffusionequation (parabolically scaled).

∂uε∂t

+1

εbε · ∇uε − div(Dε∇uε) = 0 in (0, T )× R

d.

Idea: Asymptotic expansion (slight variant) for uε in terms of theperiod.

uε(t, x) =

∞∑

i=0

εiui

(

t, x− b∗

εt,x

ε

)

Two-scale asymptotic expansion with drift.

b∗ is unknown to begin with.

b∗ will be computed along the upscaling process.


Plug the expression for uε following the chain rule for differentiation

∂

∂t

(

ui

(

t, x− b∗

εt,x

ε

))

=(∂ui∂t

− ε−1b∗ · ∇xui

)(

t, x− b∗

εt,x

ε

)

∇(

ui

(

t, x− b∗

εt,x

ε

))

=(

∇xui + ε−1∇yui

)(

t, x− b∗

εt,x

ε

)

Identify the coefficients of identical powers of ε.

At order ε−2:

b(y) · ∇yu0 − divy(D∇yu0) = 0 in Y,

y → u0(y) Y − periodic.

We deduce that u0 is independent of y.


At order ε−1: (Eq. for u1 with periodic boundary condition)

−b∗ · ∇xu0 + b(y) · (∇xu0 +∇yu1)− divy(D(∇xu0 +∇yu1)) = 0 in Y.

Above equation is linear in ∇xu0 implies u1(t, x, y) =∂u0∂xi

(t, x)ωi(y)

The cell problem for (ωi)1≤i≤d: (with periodic b.c.)

b(y) · ∇yωi − divy(D∇yωi) =(

b∗ − b(y))

· ei + divy(Dei) in Y.

Fredholm alternative implies the existence of ωi

iff

∫

Y

b(y) dy = b∗

In case of b∗ = 0, the homogenization is classical: Bensoussan, Lions &Papanicolaou(78), Zhikov, Koslov & Oleinik(94).


At order ε0 we get the homogenized equation(after some algebra of course !!!)

∂u0∂t

− div(D∗∇u0) = 0 in (0, T )× Rd,

u0(0, x) = uinit(x) x ∈ Rd.

Where D∗ is the Dispersion Tensor:

D∗ij =

∫

Y

D(

∇yωi + ei

)

·(

∇yωj + ej

)

dy


Convection-Diffusion-Reaction system

We get back to the convection-diffusion-reaction system:

∂uε

∂t+ 1

εbε · ∇uε − div(Dε∇uε) = 0 in (0, T ) × Ωε,

−1εDε∇uε · n = κ

ε2

(

uε − vεK

)

on (0, T )× ∂Ωε,

∂vε∂t

+ 1εbsε · ∇svε − divs(Ds

ε∇svε) =κε2

(

uε − vεK

)

on (0, T )× ∂Ωε,

uε(0, x) = uinit(x) x ∈ Ωε,

vε(0, x) = vinit(x) x ∈ ∂Ωε.

Inspired by our previous calculation, we take

uε(t, x) =

∞∑

i=0

εiui

(

t, x− b∗

εt,x

ε

)

and vε(t, x) =

∞∑

i=0

εivi

(

t, x− b∗

εt,x

ε

)


Formal ResultThe solution (uε, vε) formally satisfy

uε(t, x) ≈ u0

(

t, x− b∗t

ε

)

+ εu1

(

t, x− b∗t

ε,x

ε

)

vε(t, x) ≈ Ku0

(

t, x− b∗t

ε

)

+ εv1

(

t, x− b∗t

ε,x

ε

)

with the effective drift

b∗ =1

Kd

[

∫

Y 0

b(y) dy +K

∫

∂Σ0

bS(y) dσ(y)]

and u0 the solution of the homogenized problem

Kd

∂u0∂t

− divx (A∗∇xu0) = 0 in (0, T )× R

d,

Kd u0(0, x) = |Y 0|u0(x) + |∂Σ0|d−1v0(x) x ∈ R

d.

where Kd = |Y 0|+K|∂Σ0|d−1


The dispersion tensor A∗ is given by

A∗ij =

∫

Y 0

D(y) (∇yχi + ei) · (∇yχj + ej) dy

+ K−1

∫

∂Σ0

DS(y)(

Kei +∇Syωi

)

·(

Kej +∇Syωj

)

dσ(y)

+κ

∫

∂Σ0

(

χi −ωi

K

)(

χj −ωj

K

)

dσ(y)

with (χ, ω) = (χi, ωi)1≤i≤d being the solution of the cell problem suchthat

u1(t, x, y) = χ(y) · ∇xu0(t, x)

v1(t, x, y) = ω(y) · ∇xu0(t, x)

Define uε(t, x) = u0(t, x− b∗

εt). Then, it is the solution of

∂uε∂t

+1

εb∗ · ∇uε −Kd

−1div (A∗∇uε) = 0.


Fredholm alternative

For f ∈ L2(Y 0), g ∈ L2(∂Σ0) and h ∈ L2(∂Σ0), the following system ofPDE’s admit a solution (u, v) ∈ H1

#(Y0)×H1

#(∂Σ0), unique up to the

addition of a constant multiple of (1,K),

b(y) · ∇yu− divy(D(y)∇yu) = f in Y 0,

−D(y)∇yu · n+ g = k(

u− 1Kv)

on ∂Σ0,

bs(y) · ∇syv − divsy(D

s(y)∇syv)− h = κ

(

u− 1Kv)

on ∂Σ0,

y → (u(y), v(y)) Y − periodic.

if and only if∫

Y0

f dy +

∫

∂Σ0

(g + h) dσ(y) = 0.


cascade of equations

At order ε−2:

b(y) · ∇yu0 − divy(D(y)∇yu0) = 0 in Y 0,

−D∇yu0 · n = bS(y) · ∇Sy v0 − divSy (D

S(y)∇Sy v0)

= κ(

u0 −K−1v0)

on ∂Σ0,

y → (u0(y), v0(y)) Y − periodic.

We deduceu0(t, x, y) = u0(t, x)

andv0(t, x) = K u0(t, x)


At order ε−1:

−b∗ · ∇xu0 + b(y) · (∇xu0 +∇yu1)− divy(D(∇xu0 +∇yu1)) = 0

in Y 0,

−b∗ · ∇xv0 + bs(y) · (∇sxv0 +∇s

yv1)− divsy(Ds(∇s

xv0 +∇syv1))

= −D(∇xu0 +∇yu1) · n = κ(

u1 −K−1v1)

on ∂Σ0,


The linearity helps us deduce that

u1(t, x, y) = χ(y) · ∇xu0 and v1(t, x, y) = ω(y) · ∇xu0

The above representation of (u1, v1) results in a coupled cell problem.


Cell Problem

b(y) · ∇yχi − divy(D(∇yχi + ei)) = (b∗ − b(y)) · ei in Y 0,

bs(y) · ∇syωi − divsy(D

s(∇syωi +Kei))

= K(b∗ − bs(y)) · ei + κ(

χi − 1Kωi

)

on ∂Σ0,

−D(∇yχi + ei) · n = κ(

χi − 1Kωi

)

on ∂Σ0,

y → (χi(y), ωi(y)) Y − periodic.

Using the Fredholm result, we get the existence of (χi, ωi) provided

b∗ =

∫

Y 0

b(y) dy +K

∫

∂Σ0

bS(y) dσ(y)

|Y 0|+K|∂Σ0|d−1

• Transport and Chemistry are coupled!Allaire, Hutridurga (Polytechnique) Upscaling Adsorption IFCAM ’12 Nice 20 / 49

At order ε0 (Equation for (u2, v2)):

b(y) · ∇yu2 − divy(D∇yu2) = ϕ1(u0, u1) in Y 0,

−D (∇xu1 +∇yu2) · n = κ(

u2 − 1Kv2)

= bs(y) · ∇syv2 − divsy(D

s∇syv2) + ϕ2(v0, v1) on ∂Σ0,


The compatibility condition yields

Kd ∂tu0 − divx (A∗∇xu0) = 0 in (0, T )× R

d,

Kd u0(0, x) = |Y 0|u0(x) + |∂Σ0|d−1v0(x) x ∈ R

d.

where Kd = |Y 0|+K|∂Σ0|d−1


Bounded domains• Can we work on a similar model in a bounded domain?

In Rd, we cannot expect usual convergence of sequence uε(t, x) in the

fixed frame of reference (convection might dominate diffusion).

We show the translated sequence uε(t, x− b∗t/ε) converges to thesolution of an homogenized parabolic equation.

Convergence in moving coordinates doesn’t work in bounded domain.

Same PDE with uε = 0 on ∂Ω.

In a bounded domain, the initial profile should move rapidly in thedirection of b∗ until it reaches the boundary.

Then, dissipate due to homogeneous dirichlet boundary condition.

Hot Spot problem: If the initial condition uinit has a maximum insidethe domain, how does the maximum of uε evolves with time? Allaire,Pankratova & Piatnitski(12).


Convergence ProofsLet us consider a well-known model problem of heat conduction in aperiodic composite medium.

Let Ω be the material medium.A(y) conductivity matrix and f ∈ L2(Ω) source term.

−div(Aε∇uε) = f in Ω,uε = 0 on ∂Ω.

We take

uε(t, x) =

∞∑

i=0

εiui

(

t, x,x

ε

)

.

Arrive at the cell problem for ωi, solution of the cell problem.Variational formulation of the heat conduction problem:

∫

Ω

Aε∇uε · ∇ϕdx =

∫

Ω

fϕdx for any ϕ ∈ H10 (Ω)

Idea: Pass to the limit, as ε tends to zero, and identify the variationalformulation of the limit function


Difficulty: For a fixed ϕ, we cannot pass to limit in the LHS as itinvolves product of two weakly converging sequences.

Idea: Replace the fixed test function ϕ by a sequence of test functionsϕε.

ϕε(x) = ϕ(x) + εd

∑

i=1

∂ϕ

∂xi(x)ω∗

i (x/ε)

where ω∗i is the solution of the adjoint cell problem.

Integration by parts and the knowledge of the cell problem will help uspass to the limit.

This is called Energy Method (introduced by L. Tartar).

The above method requires a clever way to construct test functions.Also, one needs to do formal calculations at least till the cell problemlevel.


Convergence Results

An alternative to Tartar’s Energy Method → Two-scale convergence.

Two-scale convergence : Nguetseng(89), Allaire(92).

Two-scale convergence on periodic surfaces : Allaire, Damlamian &Hornung(95), Neuss-Radu(96).

A variant : Two-scale convergence with drift.

Two-scale convergence with drift : Marusic-Paloka & Piatnitski(05).

Step 1: Derive a priori estimates.

Step 2: Extract converging subsequences.

Step 3: Pass to limit in the variational formulation.

Step 4: Strong convergence.


Classical Two-scale Convergence:A sequence of functions uε in L2(Ω) is said to two-scale converge to alimit u0(x, y) belonging to L2(Ω× Y ) if, for any function ψ(x, y) inC∞0 [Ω;C∞

# (Y )], we have

limε→0

∫

Ω

uε(x)ψ(

x,x

ε

)

dx =

∫

Ω

∫

Y

u0(x, y)ψ(x, y) dy dx

We denote this convergence by uε2−scale−−−− u0.

Two-scale Convergence on periodic surfaces:A sequence of functions vε in L2(∂Ωε) is said to two-scale converge to alimit v0(x, y) belonging to L2(Ω;L2(∂Σ0)) if, for any function ϕ(x, y) inC[Ω;C#(Y )], we have

limε→0

ε

∫

∂Ωε

vε(x)ϕ(

x,x

ε

)

dx =

∫

Ω

∫

∂Σ0

v0(x, y)ϕ(x, y) dσ(y) dx

We denote this convergence by vε2S−drift−−−−− v0.


Two-scale Convergence with drift:A sequence of functions uε(t, x) in L

2((0, T ) × Rd) is said to two-scale

converge with drift V to a limit u0(t, x, y) ∈ L2((0, T ) × Rd × T

d) if, forany function ϕ(t, x, y) ∈ C∞

0 ((0, T )× Rd × T

d), we have

limε→0

T∫

0

∫

Rd

uε(t, x)ϕ(

t, x−Vtε,x

ε

)

dx dt =

T∫

0

∫

Rd

∫

Td

u0(t, x, y)ϕ(t, x, y) dy dx dt

We denote this convergence by uε2−drift−−−− u0

Similar def. for two-scale Convergence with drift on periodic surfaces.


Proposition:

Let V be a constant vector in Rd and let the sequence uε be uniformly

bounded in L2((0, T );H1(Rd)).

‖uε‖L2((0,T );H1(Rd)) ≤ C

Then, there exists a subsequence, still denoted by ε, and functionsu0(t, x) ∈ L2((0, T );H1(Rd)) and u1(t, x, y) ∈ L2((0, T ) × R

d;H1(Td))such that

uε2−drift−−−− u0

and∇uε

2−drift−−−− ∇xu0 +∇yu1

Similar compactness result for sequences vε on periodic surfaces with

ε

T∫

0

∫

∂Ωε

(

|vε(t, x)|2 + |∇svε(t, x)|2)

dσε(x) dt ≤ C


a priori estimates

• Multiply the bulk equation by uε and then integrate over Ωε.

• Multiply the surface equation by ε 1Kvε and integrate over ∂Ωε.

• Integrals with the convective term vanish in both cases.

• Adding the so got expressions and integrating over (0, T ) results inthe apriori estimates.

‖uε‖L∞((0,T );L2(Ωε)) + ‖∇uε‖L2((0,T )×Ωε)

+√ε(

‖vε‖L∞((0,T );L2(∂Ωε)) + ‖∇svε‖L2((0,T )×∂Ωε)

)

+√ε∥

∥

∥

1

ε

(

uε −K−1vε)

∥

∥

∥

L∞((0,T );L2(∂Ωε))≤ C

(

‖u0‖L2(Rd) + ‖v0‖H1(Rd)

)


ExtractionWe can extract subsequence off bulk and surface concentrations uεand vε which two-scale converges with drift b∗, as ε→ 0, in thefollowing sense

uε2−drift−−−− u0(t, x)

vε2S−drift−−−−− Ku0(t, x)

∇uε 2−drift−−−− ∇xu0(t, x) +∇yu1(t, x, y)

∇svε2S−drift−−−−− KG(y)∇xu0(t, x) +∇s

yv1(t, x, y)

1ε

(

uε − 1Kvε) 2S−drift−−−−−

(

u1 − 1Kv1

)

(t, x, y)

u0(t, x) turns out to be the unique solution of the homogenizedproblem.

It also happens u1 and v1 can be decomposed in terms of cell solutions.Allaire, Hutridurga (Polytechnique) Upscaling Adsorption IFCAM ’12 Nice 30 / 49

Coupled variational formulation

T∫

0

∫

Ωε

[

∂uε∂t

ϕε +1

εbε · ∇uεϕε +Dε∇uε · ∇ϕε

]

dx dt

+ε

T∫

0

∫

∂Ωε

1

K

[

∂vε∂t

ψε +1

εbSε · ∇Svεψε +DS

ε ∇Svε · ∇Sψε

]

dσε(x) dt

+

T∫

0

∫

∂Ωε

[

κ

ε

(

uε −vεK

)

(

ϕε −ψε

K

)]

dσε(x) dt = 0,

where, ϕε = ϕ

(

t, x− b∗t

ε

)

+ εϕ1

(

t, x− b∗t

ε,x

ε

)

,

ψε = Kϕ

(

t, x− b∗t

ε

)

+ εψ1

(

t, x− b∗t

ε,x

ε

)

.

Here ϕ(t, x), ϕ1(t, x, y) and ψ1(t, x, y) are smooth compactly supportedfunctions which vanish at t = T .


• Taking ϕ ≡ 0, pass to limit → coupled variational formulation(u1, v1).

• Coupled variational formulation (u1, v1) → coupled cell problem(χi, ωi).

• Taking ϕ1 ≡ 0, ψ1 ≡ 0, pass to limit → variational formulation u0.

• Variational formulation for u0 is a diffusion equation.

Kd∂u0∂t

− divx (A∗∇xu0) = 0

• Use cell problem for (χi, ωi) to obtain dispersion matrix.

• Dispersion Matrix A∗ is a constant matrix depending on the reaction.


Strong two-scale convergence with drift:Let (uε)ε>0 be a sequence in L2((0, T ) × R

d) which two-scale convergeswith drift to a limit u0(t, x, y) ∈ L2((0, T ) × R

d × Td). Further, if

limε→0

‖uε‖L2((0,T )×Rd) = ‖u0‖L2((0,T )×Rd×Td)

Then, it is said to two-scale converges with drift strongly and it satisfies

limε→0

∫ T

0

∫

Rd

∣

∣

∣

∣

uε(t, x)− u0

(

t, x− Vεt,x

ε

)∣

∣

∣

∣

2

dx dt = 0,

if u0(t, x, y) is smooth, say u0(t, x, y) ∈ L2(

(0, T )× Rd;C(Td)

)

.

For a well-prepared initial data vinit(x) = Kuinit(x), we can show that

uε(t, x)1IΩε

2−drift−−−−→ 1IY 0u0(t, x)

vε(t, x)1I∂Ωε

2s−drift−−−−→ K1I∂Σ0u0(t, x)

If vinit(x) 6= Kuinit(x), then we shall use a time initial layer.


Numerical study

Numerical tests were done using FreeFem++.

Using Lagrange P1 finite elements.

Number of vertices 33586.

Obstacle : circular disk of radius 0.2

The velocity field b(y) is generated by Stokes system:

∇yp−∆yb = ei in Y 0,divyb = 0 in Y 0,b = 0 on ∂Σ0,p, b Y − periodic.

The surface convection bs ≡ 0 for simplicity.


Parameters Values

Radius of the obstacle: r 0.2

Equilibrium constant: K 1

Porosity : |Y 0| = 1− r2π 0.874357

Tortuosity : |∂Σ0| = 2πr 1.25664

Kd factor : |Y 0|+K|∂Σ0| 2.13099

Surface velocity bS 0

Mean velocity∫

Y 0

b(y) dy (0.0385,−2.67 × 10−5)

Drift velocity b∗ (0.0180,−1.25 ∗ 10−5)

Adsorption rate κ0 1

Bulk molecular diffusion D 1

Surface molecular diffusion DS 1

Table: Parameter values


Figure: velocity field b(y) in Y 0.


0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800 900 1000

Lon

gitu

dina

l Dis

pers

ion

Pecler Number

DS=0DS=100DS=200DS=300DS=400DS=500

Figure: Behavior of the longitudinal dispersion with respect to Peloc forvarious values of the surface molecular diffusion DS .


In presence of surface diffusion Ds = 1 and surface convection bs = 0.

IsoValue-0.118263-0.101224-0.0898652-0.0785062-0.0671471-0.0557881-0.0444291-0.03307-0.021711-0.0103520.001007060.01236610.02372510.03508420.04644320.05780220.06916120.08052030.09187930.120277

IsoValue-0.0166337-0.0141745-0.0125351-0.0108956-0.00925613-0.00761666-0.0059772-0.00433773-0.00269826-0.001058790.0005806720.002220140.003859610.005499070.007138540.008778010.01041750.01205690.01369640.0177951

Figure: The cell solution χ1: Left, reference value κ = κ0; Right, κ = 5κ0.


IsoValue-0.00424724-0.00367644-0.0032959-0.00291537-0.00253483-0.0021543-0.00177377-0.00139323-0.0010127-0.000632165-0.0002516320.0001289020.0005094360.000889970.00127050.001651040.002031570.00241210.002792640.00374397

IsoValue-0.0181629-0.0155051-0.0137331-0.0119612-0.0101893-0.0084174-0.00664549-0.00487357-0.00310166-0.001329750.0004421670.002214080.003985990.005757910.007529820.009301730.01107360.01284560.01461750.0190473

Figure: The cell solution χ1: Left, κ = 6κ0; Right, κ = 8κ0.

Similar inversion phenomenon with χ2.


Figure: The reconstructed solution χ1(y) + y1: Left, Reference value κ = κ0;Middle, κ = 6κ0; Right, κ = 19κ0

As κ increases, the presence of surface diffusion results in the transitionof the obstacles from being repulsive to attractive.


coupled cell problem

b(y) · ∇yχi − divy(D(∇yχi + ei)) = (b∗ − b(y)) · ei in Y 0,

−D(∇yχi + ei) · n = κ(

χi − 1Kωi

)

on ∂Σ0,

bs(y) · ∇syωi − divsy(D

s(∇syωi +Kei))

= K(b∗ − bs(y)) · ei + κ(

χi − 1Kωi

)

on ∂Σ0,

y → (χi(y), ωi(y)) Y − periodic.

In the limit κ→ 0, the above system is decoupled.Our value of the drift b∗ doesn’t give the compatibility condition forthe well-posedness of the decoupled equations.Thus, ill-posed in the κ→ 0 limit.


0

20

40

60

80

100

120

140

160

180

200

220

0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05 0.0001

Lon

gitu

dina

l Dis

pers

ion

Reaction Rate 1.55

1.6

1.65

1.7

1.75

1.8

1.85

0 50 100 150 200 250 300

Lon

gitu

dina

l Dis

pers

ion

Reaction Rate

Figure: The variation of effective longitudinal diffusion: Left, κ tending to 0;Right, κ increasing in magnitude


0.7768

0.77681

0.77682

0.77683

0.77684

0.77685

0.77686

0.77687

0.77688

0.77689

0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05 0.0001

Tra

nsve

rse

Dis

pers

ion

Reaction Rate 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 50 100 150 200 250 300

Tra

nsve

rse

Dis

pers

ion

Reaction Rate

Figure: The variation of effective transverse diffusion: Left, κ tending to 0;Right, κ increasing in magnitude


0.828

0.83

0.832

0.834

0.836

0.838

0.84

0.842

0.844

0.846

0 1 2 3 4 5 6 7 8 9 10

Lon

gitu

dina

l Dis

pers

ion

Molecular Diffusion

0.826

0.828

0.83

0.832

0.834

0.836

0.838

0.84

0.842

0.844

0.846

0 1 2 3 4 5 6 7 8 9 10

Tra

nsve

rse

Dis

pers

ion

Molecular Diffusion

Figure: The variation of effective diffusion with DS increasing in magnitude:Left, longitudinal diffusion; Right, transverse diffusion


References

• Allaire G., Brizzi R., Mikelic A., Piatnitski A. Two-scale expansionwith drift approach to the Taylor dispersion for reactive transportthrough porous media, Chemical Engineering Science, Vol 65 (2010).

• Allaire G, Hutridurga H, Homogenization of reactive flows in porousmedia and competition between bulk and surface diffusion, IMA J ApplMath., (2012).

• Marusic-Paloka E., Piatnitski A. Homogenization of a nonlinearconvection-diffusion equation with rapidly oscillating co-efficients andstrong convection, Journal of London Math. Soc., Vol 72 No.2 (2005).


Convection-Diffusion in Ωf .

∂u

∂τ+ b(y) · ∇u− div(D∇u) = 0 in (0, ζ)× Ωf .

Convection-Diffusion with nonlinear adsorption reaction on Ωs.

∂v

∂τ+ bs ·∇sv−divs(Ds∇sv) = κ (f(u)− v) = −D∇u ·n on (0, ζ)× ∂Ωs.

where f(u) = αu/(1 + βu).

In this case, the drift turns out to be

b∗ =1

|Y 0|

∫

Y 0

b(y) dy =1

|∂Σ0|

∫

∂Σ0

bs(y) dσ(y)

Very restrictive!


Multi-component flows:

∂u(1)ε

∂t+

1

εb(1)ε · ∇u(1)ε − div

(

D(1)ε ∇u(1)ε

)

+1

ε2

(

m11u(1)ε −m12u

(2)ε

)

= 0

∂u(2)ε

∂t+

1

εb(2)ε · ∇u(2)ε − div

(

D(2)ε ∇u(2)ε

)

+1

ε2

(

m22u(2)ε −m21u

(1)ε

)

= 0

in (0, T )× Rd. Where the coupling matrix is a cooperative matrix i.e;

mij ≥ 0 for all i, j ∈ 1, 2 and diagonally dominant.In this case, the drift truns out to be

b∗ =

∫

Y

(ψ(1)ψ(1)∗b(1) + ψ(1)D(1)∇yψ(1)∗ − ψ(1)∗D(1)∇yψ

(1)) dy

+

∫

Y

(ψ(2)ψ(2)∗b(2)(y) + ψ(2)D(2)∇yψ(2)∗ − ψ(2)∗D(2)∇yψ

(2) dy

Where (ψ(1), ψ(2)) is the first eigenvector function of a spectralproblem!Homogenization of eigenvalue system: Capdeboscq(98), Allaire &Capdeboscq(00), Capdeboscq(02)


In all our work, we have assumed the velocity field to be purelyperiodic.

bε(x) = b(x/ε)

Our results don’t hold true if the velocity fields are the periodicfunctions with macroscopic modulations, for example,

bε(x) = b(x, x/ε)

with a Y -periodic function b(x, y).

Under strong assumptions: Allaire & Orive(07).


Future work

• 3D numerical simulations in both linear and nonlinear case.

• Multi-Component reactive Flows in porous media.

• Non-Constant Drift Velocity.


Documents

Homogenization of linear adsorption in porous media.auroux/IFCAM2012/Harsha_Hutridurga.pdf · Mesoscopic Model Unbounded porous media: Connected Fluid Part Ωf. Solid Part Ωs = Rd