Les lois d’interface pour la pression et pour la vitesse ... · He used a statistical approach to...

Les lois d’interface pour la pressionet pour la vitesse pour des

écoulements visqueux sur un litporeuxAndro Mikeli c

Institut Camille Jordan, Departement de Mathematiques, Universite Lyon 1, FRANCE

Joint work with Anna Marciniak-Czochra (IWR and BIOQUANT, Universitat Heidelberg)

Expose au GT Methodes Numeriques, Laboratoire Jacques-Louis Lions, Universite Paris 6, le 12 mars 2012 – p. 1/49

P0

This research was partially supported by theJoint-Programme Programme Inter Carnot Fraunhofer PICFFPSI Filt: Modeling of fluid interaction with deformable por ousmedia with application to simulation of processes in indust rialfilters , 2012-2014 and by the Romberg professorship atIWR, Universität Heidelberg, 2011-1013.

1. INTRODUCTION

Finding effective boundary conditions at the surfacewhich separates a channel flow and a porous mediumis a classical problem.


P1

Supposing a laminar incompressible and viscous flow,we find out immediately that the effective flow in aporous solid is described by Darcy’s law. In the freefluid we obviously keep the Navier-Stokes system.Hence we have two completely different systems ofpartial differential equations :

−µ∆u+∇p = f (1)

div u = 0 (2)

in the free fluid domain ΩF and in the porous medium Ωp

−µvF = K(f −∇p) (3)

div vF = 0 (4)


P1a

The orders of the corresponding differential operatorsare different and it is not clear what kind of conditionsone should impose at the interface between the freefluid and the porous part.

We search for the correct interface conditions betweena porous medium Ωp and a free fluid ΩF .(Navier-Stokes Darcy)

Pression and the filtration velocity in a porous mediumare the averages over REVs. Consequently oneshouldn’t apply directly the first principles to obtain theinterface laws.


P3

""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

porous medium

channel


P4

Interface condition of Beavers et Joseph (J. Fluid Mech. 1967 )They considered a 2D Poiseuille’s flow over a naturally

permeable block, i.e. a laminar incompressible flow througha 2D parallel channel formed by an impermeable upper wallx2 = h and a permeable lower wall x2 = 0, defining aninterface between the porous medium and the free flow in ahorizontal channel.

A uniform pressure gradient (p0 − pb)/b is maintained inthe longitudinal direction x1 in both the channelΩ1 =]0, b[×]0, h[ and the permeable materialΩ2 =]0, b[×]−H, 0[.

Problem : Find the effective flow in Ω1 ∪ Σ ∪ Ω2.


P5

Σ

Ω1

x =−L

x =h

b

2

2

0x1

x2

p =p0

p =pb

ε

εu =0

εu =0

ε ε


P6

Beavers et Joseph proposed (and confirmedexperimentally) the following law

∂u1∂x2

(x1, 0) =α√K

(

u1(x1, 0)− vF1 (x1, 0))

(5)

where K is the permeability and vF = (K/µ)∇p is thefiltration velocity.

Analogously to the Poiseuille flow, u2 = 0 and

u1 =p0 − pb2µb

1

1 + αh/√K

·(

(1 + αh/√K)x22−

α√Kx2(h

2/K − 2)− h√K(h

√K + 2α)

)

(6)


P8

The mass flow rate M per unit width through channel isthen

M = −(1 + Φ)h3ρ

12µ

p0 − pbb

; σ =h√K

; Φ =3(2α + σ)

σ(1 + ασ)(7)

The agreement between the measured values in theexperiment by Beavers and Joseph and the predictedvalues for Mexp/(µb) was good, with over 90% of theexperimental values having errors of less than 2%

A ” theoretical ” justification of the Beavers and Josephlaw , at a physical level of rigor, is in the article of P.G.Saffman (Studies in Applied Maths 1971) : He found


P9

that the tangential velocity on Σ is proportional to the shearstress i.e.

u1 =

√K

α

∂u1∂x2

+O(K) (8)

He used a statistical approach to extend Darcy’s law tonon-homogeneous porous media and in order to deduce(8),made an ad hoc hypothesis about the representation of the averagedinterfacial forces as a linear integral functional of the velocity, with anunknown kernel.

In the article of G. Dagan (Water Resources Research 1979) wehave the same conclusion. He supposed Slattery’s relation betweenthe pressure gradient and the 1st and 2nd ordre derivatives of thefiltration velocity in order to get the law (8).


P9A

A numerical study of the hydrodynamic boundarycondition at the interface between a porous and a plainmedium is in Sahraoui and Kaviany (Int. J. Heat MassTransfer 1992).They calculated the slip coefficient andthey found that the Brinkman extension do notsatisfactory model the flow field in the porous medium.

Next we have the articles by J.A. Ochoa-Tapia and S.Whitaker (Int. J. Heat Mass Transfer, Vol. 14 (1995),2635 - 2655 and J. Porous Media 1998). Using thevolume averaging they obtained at the interface (a)continuity of the velocity and (b) the continuity of the ”modified ” normal stress. In order to perform theaveraging they had to suppose the Brinkman’s flow inthe porous part and a transition layer between twodomains.


P10

Laws proposed by Ene, Levy and Sanchez-Palencia (J.de Mécanique and Int. J. Eng. Sci. 1975):Case A: The velocity of the free fluid u is much larger than the

filtration velocity vF in the porous medium. Then [p] = O(√K) on

Σ.Case B: The free fluid velocity and the filtration velocity are of the

same order. Then u · ν = vF · ν (continuity of the normal velocitieson ) Σ and p = constant on Σ.

WHAT ARE THE TRUE INTERFACE CONDITIONS ?

Is it possible to find the interface conditions on Σ in the limit whenthe characteristic pore size ε → 0 ?

If yes, are we able to prove convergence, i.e. to find a relationbetween the ε-problems and the effective problem when ε → 0?


P12

JUSTIFICATION OF THE LAW BY BEAVERS AND JOSEPH

We will present the justification for the physical situation presentedin the article by Beavers and Joseph, for a periodic porous medium.It was published in the article[JM1] W.J ager, A.Mikeli c : On the interface boundary conditions byBeavers, Joseph and Saffman, SIAM J. Appl. Math. , 60 (2000), pp.1111 - 1127.

Construction is based on the results in[JM2] W.J ager, A.Mikeli c : On the Boundary Conditions at theContact Interface between a Porous Medium and a Free Fluid, Ann.Sc. Norm. Super. Pisa, Cl. Sci. - Ser. IV, Vol. XXIII (1996), Fasc. 3,p. 403 - 465.


P13

Constants in the law by Beavers and Joseph arecalculated in [JM3] W.J ager, A.Mikeli c, N.Neuß : Asymptoticanalysis of the laminar viscous flow over a porous bed, SIAM J. onScientific and Statistical Computing , Vol. 22 (2001), p. 2006-2028.Also a numerical evidence for the pressure jump law is given.

All these paper contain a derivation of the law by Beavers andJoseph in the setting of the their experiment work, based on aperturbation of the Poiseuille flow. The rigorous derivation of theBeavers and Joseph law for a general flow regime and of thepressure jump in[AMCAM] Anna Marciniak-Czochra, Andro Mikeli c : Effectivepressure interface law for transport phenomena between anunconfined fluid and a porous medium using homogenization,accepted for publication in SIAM: Multiscale Modeling andSimulation, 2012.


P13a

Formal derivation of the law using a 2-scalesasymptotic expansion is explained in the articleW. Jager, A. Mikeli c : Modeling effective interface laws for transportphenomena between an unconfined fluid and a porous mediumusing homogenization, Transport in Porous Media, Volume 78,Number 3, 2009, p. 489-508.

There is a recent unpublished work on the numerical confirmation ofthe law by Beavers and Joseph and on the pressure interface law:T. Carraro, C. Goll, A. Marciniak-Czochra and A. Mikeli c :Numerical comparison of different interface couplings in t heStokes-Darcy model, preprint, Universitat Heidelberg, March 2012,and I will present some direct simulations of the ε-problem.


J1

We suppose a periodic porous medium, obtained bytranslations of the cell Y ε = εY , where the squareY = (0, 1)2 contains an open Lipshitz set Z∗, strictlyincluded in Y .

Let YF = Y \ Z∗and let χ be the characteristic function

of YF , extended by periodicity to IR2. We setχε(x) = χ(xε ), x ∈ IR2, and define Ωε

2 byΩε2 = x | x ∈ Ω2, χ

ε(x) = 1. In addition,Ωε = Ω1 ∪ Σ ∪ Ωε

2 is the fluid part of Ω = Ω1 ∪ Σ ∪ Ω2. Wesuppose that (b/ε, L/ε) ∈ IN2. Consequently, our porousmedium contains a large number of channelsperiodically distributed and of the characteristic size ε,being small compared with the characteristic length ofthe macroscopic domain.


J2

Σ

Ω1

Ω2ε

−L

H

x=0y=0 x=b

ε

Y

Y*Z*


J3

A uniform pressure gradient is maintained in thelongitudinal direction in Ωε, as in the experiment byBeavers and Joseph. More precisely, for a fixed ε > 0,vε, pε are defined by

−µ4vε + ρ(vε∇)vε +∇pε = 0 in Ωε, (9)

div vε = 0 in Ωε, (10)

vε = 0 on ∂Ωε \ ∂Ω, (11)

vε = 0 on (0, b)× (−L ∪ h), (12)

vε2 = 0 on (0 ∪ b)× (−L, h), (13)

pε = p0 on 0 × (−L, h)

and pε = pb on b × (−L, h), (14)


J4

where µ > 0 is the viscosity and p0 and pb are givenconstants.Is there a solution for the problem (9)-(14)? Is it unique ? Isit possible to get uniform a priori estimates with respect toε?

Let us note that the classical Poiseuille flow in Ω1,satisfying the no-slip condition on Σ, is given by

v0 =

(

pb − p02bµ

x2(x2 − h), 0

)

for 0 ≤ x2 ≤ h,

p0 =pb − p0

bx1 + p0 for 0 ≤ x1 ≤ b.

(15)

We extend this solution to Ω2 by v0 = 0.


J5

Suppose that the Reynolds number Re= |pb−p0|µ2

h2

8 is notlarge.Then the Poiseuille flow is a unique solution forthe ball

B =

z ∈ H1(Ω1)2| ‖z‖L4(Ω1)2 ≤

µ

4ρ 4√2bh

(1 +h

b√2)−1/2

Our idea is to construct a solution for the system(9)-(14) as a non-linear perturbation of the Poiseuille’sflow (15).

Proposition 1. Suppose that the Reynolds number is not large andε small. Then the problem (9)-(14) has a solution

vε, pε ∈ H2(Ωε)2 ×H1(Ωε) satisfying

‖∇(vε − v0)‖L2(Ωε)4 ≤ Cµ|pb − p0|√ε. (16)


J6

In addition, all solutions contained in the ball

B0 =

z ∈ H1(Ωε)2| ‖z‖L4(Ωε)2 ≤ Cµ (17)

are equal to vε, pε.Proposition 2. For the solution to problem (9)-(14), satisfying (16), wehave the following a priori estimates :

‖vε‖L2(Ωε2)2 ≤ Cε

√ε (18)

‖vε‖L2(Σ)2 + ‖vε − v0‖L2(Ω1)2 ≤ Cε (19)

‖pε − p0‖L2(Ω1) ≤ C√ε (20)

Consequently, in the 1st approximation the free flow doesn’t see theporous medium .


General case 1

In the papers [JM1] and [JM3] from 2000-2001 only theperturbed Poiseuille flow corresponding to theexperiment by Beavers and Joseph was considered.Also the invariance of the law with respect to the choiceof the interface position was proved only for a perturbedPoiseuille flow.

The above result is valid in much more general setting(a general force term, non-constant pressure drop, . . . ).Nevertheless, we are obliged to impose a moderateReynolds number, in order to have uniqueness for thestationary Navier-Stokes system.

For simplicity we will deal with the Stokes operator. Alsothe realistic boundary conditions lead to outer boundarylayers,


General case 2

which don’t influence the interface condition, but causeimportant pollution in the estimates. We include them intothe force term and suppose the periodicity conditions at theinlet/outlet boundaries.

Question of the stress interface conditions was notanswered rigorously.

Answering all these questions was undertaken in therecent article [AMCAM]Anna Marciniak-Czochra, Andro Mikeli c : Effective pressureinterface law for transport phenomena between anunconfined fluid and a porous medium usinghomogenization, accepted for publication in SIAM:Multiscale Modeling and Simulation, 2012.


Beavers and Joseph 1

In this part of the talk I will try to shortly explain thederivation of the law by Beavers and Joseph and in somedetails why the effective pressure has a jump at theinterface. For simplicity we will deal with the Stokesoperator.

Let us recall the derivation of the law by Beavers andJoseph:

1. STEP Our first approximation is the impermeable interface

approximation , where we had v0 = 0 on Σ. In [AMCAM] the

case of the general force was studied and the resultsanalogous to Propositions 1 and 2 proved. For this stepuniqueness is important. After it, the expansion naturallylinearizes the next order problems. We subtract the 1stapproximation from the solution and study the new problem.



2. STEP At the interface Σ we obtain the shear stress

jump equal to −∂v01∂x2

|Σ. Contrary to the pressure difference,

which could be easily set to zero by the appropriate choiceof the effective porous medium pressure p0, the shearstress jump requires construction of the correspondingboundary layer. The natural stretching variable is given bythe geometry and reads y = x

ε . The correction w, pw isgiven by

−4yw +∇ypw = 0 in Ω1/ε ∪ Ωε2/ε, (21)

divyw = 0 in Ω1/ε ∪ Ωε2/ε, (22)

[

w]

(·, 0) = 0, [pw](·, 0) = 0, and[∂w1

∂y2

]

(·, 0) = ∂v01∂x2

|Σ on Σ/ε.

(23)Expose au GT Methodes Numeriques, Laboratoire Jacques-Louis Lions, Universite Paris 6, le 12 mars 2012 – p. 25/49


Using periodicity of the geometry and independence of∂v01∂x2

|Σ of the fast variable y, we obtain

w(y) =∂v01∂x2

|Σβbl(y) and pw(y) =∂v01∂x2

|Σωbl(y), (24)

where βbl, ωbl is a boundary layer given by


CL 2

−4yβbl +∇yω

bl = 0 in Z+ ∪ Z− (25)

divyβbl = 0 in Z+ ∪ Z− (26)

[

βbl]

S(·, 0) = 0 on S (27)

[

∇yβbl − ωblIe2

]

S(·, 0) = e1 sur S (28)

βbl = 0 on ∪∞k=1 (∂Z

∗ − 0, k), (29)

βbl, ωbl is y1 − periodic, (30)

where S = (0, 1)× 0, Z+ = (0, 1)× (0,+∞),Z− = (0, 1)× (−∞, 0) \ ∪∞

k=1(Y∗ − 0, k) and

ZBL = Z+ ∪ S ∪ Z−.


CL 3

S

Z+

Z−

Z1

Z0

Z−1

Z−2

Zbl

y1

y2


CL 4

The theory developed by Jäger et Mikelic in Ann. Sc.Norm. Sup. Pisa 1996 guarantees the existence ofγ0 ∈ (0, 1), Cbl

1 et Cblω such that

eγ0|y2|∇yβbl ∈ L2(ZBL)

4, eγ0|y2|βbl ∈ L2(Z−)2,

eγ0|y2|ωbl ∈ L2(Z−) and

| βbl(y1, y2)− (Cbl1 , 0) |≤ Ce−γ0y2 , y2 > y∗

| ωbl(y1, y2)− Cblω |≤ Ce−γ0y2 , y2 > y∗. .

(31)

In addition, constants Cbl1 and Cbl

ω are given by

Cblω =

∫ 10 ωbl(y1, a) dy1, ∀a ≥ 0

∫ 10 βbl

1 (y1, 0)dy1 =∫ 10 βbl

1 (y1, a)dy1 = Cbl1 < 0

(32)


Numerical simulation of the boundary layer

Numerical simulation for βbl1 , βbl

2 and ωbl from the paper byJäger, Mikelic, Neuss, SIAM Sci Comp 2001



Now we set

βbl,ε(x) = εβbl(x

ε) and ωbl,ε(x) = ωbl(

x

ε), x ∈ Ωε,

(33)

3. STEP We obtained that that βbl − (Cbl1 , 0) and ωbl − Cbl

ω

are exponentially small for y2 > 0.Stabilization of βbl,ε towards a nonzero constant velocityε(

Cbl1 , 0)

, at the upper boundary, generates a counterflow. Itis given by the following Stokes system in Ω1:

−4zσ +∇pσ = 0 in Ω1, (34)

div zσ = 0 in Ω1, (35)

zσ = 0 on x2 = h and z

σ =∂v01∂x2

|Σe1 on x2 = 0, (36)



In the setting of the experiment by Beavers and Joseph, zσ

is proportional to the two dimensional Couette flowd = (1− x2

h )e1 and pσ = 0.

Now we expect that the approximation for the velocityreads

vε = v0 − (βbl,ε − ε(Cbl

1 , 0))∂v01∂x2

|Σ − εCbl1 z

σ + O(ε2). (37)

A formal calculation on Σ gives, at order O(ε),

∂vε1∂x2

=∂v01∂x2

(

1− ∂βbl1

∂y2(x

ε))

and1

εvε1 = −βbl

1 (x

ε)∂v01∂x2

(38)



Averaging gives the law by Beavers and Joseph

ueff1 = −εCbl1

∂ueff1

∂x2on Σ (39)

where ueff is the average of vε and Cbl1 is given by (32).

We’ll rigorously justify (39).Concerning the pressure, there are additional complicationsdue to the stabilization of the boundary layer pressure toCblω , when y2 → +∞. Consequently, the correction in Ω1 is

ωbl,ε −H(x2)Cblω∂v01∂x2

|Σ.


Pressure interface conditions 1

At the flat interface Σ, the normal component of the normalstress reduces to the pressure field. Subtraction of thestabilization pressure constant at infinity leads to thepressure jump on Σ:

[pε]Σ = p0(x1,+0)−p0(x1,−0) = −Cblω∂v01∂x2

|Σ+O(ε) for x1 ∈ (0, L).

(40)Therefore the pressure approximation is

pε(x) = p0H(x2) + p0H(−x2)−(

ωbl,ε(x)−H(x2)Cblω

)∂v01∂x2

|Σ−

εCbl1 p

σH(x2) + O(ε). (41)

where the effective porous media pressure p0 satisfies



div(

K(f(x)−∇p0))

= 0 in Ω2 (42)

p0 = peff + Cblω

∂ueff1

∂x2(x1, 0) on Σ;

K(f(x)−∇p0)|x2=−H · e2 = 0. (43)

Let us define the errors in velocity and in the pressure:

Uε(x) = vε − v0 + (βbl,ε − εCbl

1 e1H(x2))

∂v01∂x2

|Σ + εCbl1 z

σ (44)

Pε(x) = pε − p0H(x2)− p0H(−x2) +(

ωbl,ε(x)−

H(x2)Cblω

)∂v01∂x2

|Σ + εCbl1 p

σH(x2). (45)



Theorem 1 (see [AMCAM]). Let Uε be defined by (44) and Pε by

(45). Let p0 be a smooth function satisfying the interface condition (40).Then, the following estimates hold

ε‖∇Pε‖H−1(Ωε) + ε‖∇Uε‖L2(Ωε)4 + ‖Uε‖L2(Ωε2)2+

ε1/2‖Uε‖L2(Σ)2 ≤ Cε2 (46)

Idea of the proof: We use the variational formulation for Uε,Pε.Contrary to the setting of the experiment by Beavers and Joseph, for ageneral forcing term Uε isn’t divergence free. By several additional

corrections we make it of order O(ε5/2). Then we use Tartar’s restrictionoperator, test by the modified Uε and obtain (46).Corollary 1 (see [AMCAM]). The following estimate holds

√ε‖Pε‖L2(Ω1) + ‖Uε‖H1/2(Ω1)2 ≤ Cε3/2. (47)


CL 11

Now we introduce the upscaled problem

−4ueff +∇peff = f in Ω1, (48)

div ueff = 0 in Ω1,

∫

Ω1

peff dx = 0, (49)

ueff = 0 on (0, L)× h; ueff and peff are L− periodic in x1,

(50)

ueff2 = 0 and ueff1 + εCbl1

∂ueff1

∂x2= 0 on Σ. (51)


CL 12

Proposition 3 Let f ∈ C∞(Ω1)2 and L-periodic in x1. Let

ueff , peff be the solution of the problem (48)-(51)and letM eff =

∫

Ω1

ueff1 dx be the effective mass flow rate through thechannel. Then we have

‖vε − ueff‖L2(Ω1)2 + |M ε −M eff | ≤ Cε3/2 (52)

‖vε − ueff‖H1/2(Ω1)2 + ‖pε − peff‖L1(Ω1)

+‖∇(vε − ueff )‖L1(Ω1)4 ≤ Cε, (53)


CL 13

Under the hypotheses of experiment by Beavers andJoseph, the upscaled problem has a unique solution

ueff =

(

pb−p02L

(

x2 − εCbl1 h

h−εCbl1

)(x2 − h), 0

)

0 ≤ x2 ≤ h,

peff = p0 0 ≤ x1 ≤ L.

(54)

The mass flow rate is then

M eff = L

∫ h

0

ueff1 (x2) dx2 = −pb − p012

h3h− 4εCbl

1

h− εCbl1

(55)


CL 14

Beavers−Joseph prof ilex =H2

0

− (H − )11

H−ε C H1bl

ε C Hbl

H2

4−11

H−ε C H1bl

ε C Hbl

14

x 1

p −pb 02bµ

p −pb 02bµ

Poiseuilleprofile

01

p −pb 02bµ 2 2v = x (x −H )

x 2

u (x ) =eff,01 2

(x −H) (x − )2 211

H−ε C H1

ε C Hbl

bl

p −pb 02bµ


Interface position 1

DO WE HAVE A LIBERTY IN CHOOSING THE INTERFACE?

Let Ωaε = (0, L)× (aε, h) for a < 0 and let ua,eff , pa,eff be asolution for (48)-(51) in Ωaε, with (51) replaced by

ua,eff2 = 0 and ua,eff1 +εCa,bl1

∂ua,eff1

∂x2= 0 on Σa = (0, b)×aε.

(56)Problem (48)-(50), (56) has a unique smooth solutionua,eff , pa,eff, its derivatives are bounded independently ofε and a simple calculation with the boundary layer givesCa,bl1 = Cbl

1 − a. Then we have



0 = ua,eff1 (x1, εa) + εCa,bl1

∂ua,eff1

∂x2(x1, εa) =

ua,eff1 (x1, 0) + εCbl1

∂ua,eff1

∂x2(x1, 0)+

(εa)2

2(∂2ua,eff1

∂x22(x1, ξ1) +

∂2ua,eff1

∂x22(x1, ξ2)), for ξ1, ξ2 ∈ (0, εa).

Therefore, a perturbation of the interface position for anO(ε) implies a perturbation in the solution of O(ε2) inHk(Ω1). Consequently, there is a freedom in fixing positionof Σ. It influences the result only at the next order of theasymptotic expansion.



The physical permeability Kphys is proportional to ε2. Ourresult on the influence of the interface position on theeffective slip is in agreement with the observation ofKaviany. In fact, it has been noticed by Larson and Higdonthat changes of O(1) in the slip coefficients are possible,after the change of order O(

√

Kphys) of the interfaceposition. Therefore, the exact position of Σ does not poseproblems, since it influences the solution only at orderO(Kphys).



Next we study the flow in the porous part. We recall that theeffective porous media pressure p0 is the L− periodic in x1function satisfying

div(

K(f(x)−∇p0)

)

= 0 in Ω2 (57)

p0 = peff + Cblω

∂ueff1

∂x2(x1, 0) on Σ; (58)

K(f(x)−∇p0)|x2=−H · e2 = 0, (59)

with ueff , peff being the solution to the problem (48)-(51)and Cbl

ω being the pressure stabilization constant. K is thepermeability tensor.



Proposition 4

1

ε2(

vε + βbl,ε ∂v01

∂x2(x1, 0)

)

−K(f(x)−∇p0) 0

weakly in L2(Ω2)2, as ε → 0; (60)

pε − p0 → 0 strongly in L2(Ω2), as ε → 0; (61)

||pε − peff ||H−1/2(Σ) ≤ C√ε, (62)



4. STEP Is it possible that the physical pressure pε isdiscontinuous? In the case of the experiment by Beavers

and Joseph it is easy to see that p0 = H(−x2)Cblω

∂v01∂x2

(x1, 0)

and we have

pε = (Cblω − ωbl,ε(x))

∂v01∂x2

(x1, 0) + O(ε).

The leading part of pε is a continuous function but changingvery rapidly in the neighborhood of Σ from Cbl

ω to 0.Averaging leads to a discontinuous function.


Open problems

OPEN PROBLEMS :a) 3D non-tangential flows.b) Comparison with the results of Ochoa-Tapia andWhitaker ? They proposed the interface law involvingtangential derivatives of the normal stress and continuity ofvelocities. Then they argued that in most cases it reducesto i) pressure continuity and ii) law by Beavers and Joseph.c) Comparison with the models used by Discacciati, Miglioand Quarteroni (Applied Numerical Mathematics (2002)):

σeffnτ = 0 and σeffnn = −p0 + gravity.

d) Comparison with Payne and Straughan (J. Math. PuresAppl. (1998)):


Open problems

α√3

trKueff1 =

∂ueff1

∂x2+

∂ueff2

∂x1and σeffnn = −p0.

e) Interface laws for Biot’s consolidation theory? Recentarticle by A. Mikelic and M. F. Wheeler"On the interface law between a deformable porous medium containing aviscous fluid and an elastic body" , accepted for publication inM3AS: Mathematical Models and Methods in AppliedSciences, 2011.

f) Recent simulations of the coupled flow + reaction-diffusion equationsin the presence of the interface porous medium/unconfined fluid by T.Carraro, S. Goll, A. Marciniak-Czochra . . . with confirmation of thepressure jump through direct simulation of the ε-problem. Interface lawsfor the heat conducting fluid and the heat conducting solid structure (P.Bastian, W. Jager, A. Marciniak-Czochra, A. Mikelic, . . . ).


Les lois d’interface pour la pression et pour la vitesse ... · He used a statistical approach to...

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