FLUID FLOWS IN DIELECTRIC POROUS MEDIA. Doina CIORANESCU, Patrizia DONATO and Horia 1. ENE. 1. INTRODUCTION AND FORMULATION OF THE ...
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International Series of Numerical Mathematics, Vol. 107, © 1992 Birkhliuser Verlag Basel
FLUID FLOWS IN DIELECTRIC POROUS MEDIA
Doina CIORANESCU, Patrizia DONATO and Horia 1. ENE
1. INTRODUCTION AND FORMULATION OF THE PROBLEM.
Let
n be an open set in IRn (n
2) and Y = [0, II [x .. x [0, In [ the representative cell.
~
Denote by T an open subset of Y with smooth boundary aT, such that by r(c:T) the set of all translated images of c:T of the form c:(kl
+ T),
Tc
Y. Denote
k E zn, kl
=
(kIll, .. , knln ).
We make the following assumption: The holes r(c:T) do not intersect the boundary Denote now by T. the set of the holes contained in
By this construction
an.
n and set
n. is periodically perforated by holes of size of the same order as the
period (see Figure 1).
o o o o
0 0 0 0
0 0 0 0
000 0 0 0 0 0 0 0 0 0
Figure 1 In the sequel we use the notations
- Y' = y\T,
- Iwl =
the Lebesgue mesure of
w (n-dimensional
if
w is
a n-dimensional open set,
(n - 1 )-dimensional if w is a curve), - X..,
= the characteristic function
of w,
- v = the zero extension to the whole n for any fonction v defined on n., - M..,(4)) = I~I 4>(x)dx, the mean value of 4> on w.
L
D. Cioranescu, P. Donato and H. I. Ene
4
Consider in 11e the Stokes system with non homogeneous slip boundary condition on {}T.
-~u'
+ "ilp' = f
divu e
(Ll)
=0
in f!.
in f!,
=0 au' + ac:"Yu· = _pe . n + ona11
ue
an
where u e
= (ui, .. , u~)
exterior body forces,
Q
g'
on {}T"
stands for the velocity field, p. for the pression,
f is the field of
and 'Y are positive constants, n is the exterior unit normal to 11 •.
The boundary condition on (}Te means that the stress vector gives rice firstly, to a braking phenomenon due to the presence of the term ac:"Yu' and secondly, to a proportionality with the exterior surface forces due to the presence of g•. Assume that the data
f and g. satisfy
i) f E (L2(f!)t (1.2)
{ ii) g. =
lex) + g(::'),
with
l
E
(Hl(11)t and
9 E (L2(aT)t :'periodic such that MaT(gi) =I
o.
Let introduce the space
The variational formulation of system (1.1) is then the following: Find u· E V., p' E L 2 (f!.) such that
f
10,
"ilu· "ilep dx + ac:"Y
f
u· "ilep dx
u·ep du -
f
fep dx+
= 0,
Yep E Ve.
=
10,
f
laT,
10,
f
laT,
f
10,
p. divep dx =
g''P du,
Yep E V.,
Classical results give the existence of a unique solution of this problem. We are interested to give the asymptotic behaviour of (u',p') when c: -+ O. We are here in the classical homogenization framework. In [7] H.I. Ene and E. Sanchez-Palencia studied the Stokes How in a periodic porous medium with Dirichlet conditions on the boundary of the holes. The limit law describing
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Fluid flows in dielectric porous media
the homogenized medium is a Darcy's law. In [3) D. Cioranescu and P. Donato consider the Laplace equation with non homogeneous Fourier conditions on the boundary of ,the holes containing a term of type O!c"Yu·. Following the values of 7, several a priori estimates are obtained which lead to different limit laws. In the case we present here (i.e. Stokes system with non homogeneous Fourier boundary conditions ) we obtain at the limit, following the values of 7, a Darcy's law (-y a Brinkmann equation (7
= 1) or the
Stokes equation (-y
< 1),
> 1). This phenomenon was
already observed by C. Conca [6) when studying the Stokes equation with homogeneous Fourier bondary conditions. It was also noticed by G. Allaire who considers the Stokes
c. The boundary conditions on the holes are either of Dirichlet type [1) or of slip type [2). In this situation it is
equation in a perforated domain with holes of size r. with r.
¢:
the geometry of the domain, more precisely the size r. which determines the type of the limit law. In this paper we give the main results we obtained for system (1.1) and their physical interpretation. We refer the reader to [4) for complete proofs and for furher results and comments. 2. RESULTS. I. Case 7
< 1 (Darcy's law).
From system (1.1) we have the following estimates: IIc"Yu·II(L2«(J.»n ~ c
IIc~V'u·II(L2«(J.»n ~ c where c is a constant independent of c. Hence, up to a subsequence
(2.1) Following along the lines of [3), let us introduce the linear form 11-;' defined by
where h E L 2 (8T).
D. Cioranescu, P. Donato and H. I. Ene
6
From lemma 3.1 of [3] we can easily prove the following proposition: PROPOSITION
2.1. Let {v'}
c
H1(O,) be a sequence satisfying
and suppose there exists C E]O, 1[ such that
with c a constant independent of c. Then
where Jlk
= 1~IIaT hey) ds.
Applying lemma 5.1 of [6] one has an extension pe of the pressure pe such that
Consequently, up to a subsequence
(2.2) We can now state the homogenization result: THEOREM
2.2. The limit fonction u given by (2.1) satisfies u
.
wlth Jll =
laTI lYT
and Jlg =
1 0 = -(Jllg + Jlg aJll
\l P)
1 r TYi JeT g(y)ds.
II. Case 1=1 ( Brinkmann's equation). When I
= 1 we derive from
(1.1) the estimate
with c independent of c. Then there exist (see D. Cioranescu-J. Saint Jean Paulin [5]) extension operators Qe E £(Hl(Oe); Hl(O» such that
(2.3)
Q'(w') ~ u
in (Ht(O)t weakly.
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Fluid flows in dielectric porous media
Moreover, convergence (2.2) still holds. Let XA , system given on the reference cell Y:
+ VqA =
-.6.X A
divX A = 0
(2.4)
8(X A
Ay)
an -
0
qA
be solution of the following Stokes
in Y*
in y*
+ qA • n = 0
on 87',
XA Y-periodic •
2
for any matrIX A E IRR . Define
Let Q E £(Hl(Y*)j Hl(y*)) be an extension operator defined over Y as constructed in (5) and set x wHx) = eQ(wA - Ay)( -) + Ax, e
x E
Then the following convergences hold:
WA ~ Ax VWA ~ A
in (Hloc(IRR)t weakly in (L 2 (n)t 2 weakly.
Analogously, if we define
we have
where 'PAis the mean value of q;. over Y. Introduce finally
which, thanks to (2.4), satisfies the equation -div
'7A + V qA = 0
Due to the periodicity, one has the convergence
inn.
n.
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D. Cioranescu, P. Donato and H. I. Ene
Moreover, AA and PA being linear in A one has
(2.5)
We can give a more precise form to the coefficients system (2.4) defines
n2
{X kh , qkh }k,
solutions
aijkh.
h = 1, ... , n of
To do that, remark that
+ 'Vqkh = 0 in Y· div Xkh = 0 in Y·
_tJ.X kh
(2.6)
O(Xkh _
II kh)
--"-'-..".-----'- + qkh . n = 0
on
IIkh
aT
Y-periodic
Xkh
where
on
= (II~h)i with II~h = OkiYh.
Then,(2.5) and (2.6) yield the formulae
aijkh
=
1 -a (x
kh
aYI
p
kh
- II ) -
a (x').. - II'}) .. dx.
OYI
Let now give the homogenization result. THEOREM
2.3. The Eonction u defined by (2.3) satisfies the equation
{
- ;,0 [( aijkh VXj
U
=0
on
- PijOkh)
an.
~Uk 1+ J-laUi =
VXh
J-lIg?
+ J-lg,
in
n
III. Case -y> 1 (Equation de Stokes). In this case convergences (2.2) and (2.3) still hold and we can prove the following result: THEOREME
2.4. The Eonction
{
U
defined by (2.3) is solution oE the equation
- ,,0 [( aijkh VXj
U
=0
on
- PijO kh )
on.
~Uk 1 = PIg? + J-lg,
VXh
in
n
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Fluid flows in dielectric porous media
IV. Variant of system (1.1) (Darcy's law). We can also consider the Stokes equation with a slightly different slip condition -~u~
+ \lp~ = f
divu e u~
=0
=0 =0
ue • n
where
T
is the unit tangent to
ne.
ne
ne sur an in
on aTe
au ane . + €Xc ., T
in
U
~
•T
= 9e . T
on
!l'1"
V.L e ,
In this case, for any value of /' we obtain always at the
limit a Darcy's type law (see [4) for details).
3. RELATED MECHANICAL MODELS. Problem (1.1) describes the flow of an incompressible viscous fluid through a porous medium under the action of an exterior electric field. One generally knows (cf. J. R. Melcher [8)), that the electric surface charges act on the boundary between the solid and fluid part of the medium. These charges give rice to a double layer which permits the slip of the fluid. The boundary condition on aTe can be rewritten under the form
This means that the stress vector Ufj • n j induces a slowing effect on the motion of the fluid, expressed by the coefficient ae'Y. Moreover, if there are exterior forces like, for instance, an
electric field, then the non homogeneity of the boundary condition on the holes is expressed in terms of surface charges contained in g~. We know that in a periodic heterogeneous medium the electric field can be obtained by standard homogenization and, consequently, ge has the form (1.2)ii (cf. E. Sanchez-Palencia [9)). In all the cases studied here, at the limit appear additional terms issued from ge. Let us point out that these terms, usually
introduced by the physicists as a result of observations, are obtained rigorously by the homogenization method. In the case / < 1 we obtain a Darcy's law with the additinal terms J.lg and J.ll. The braking of the motion beeing important, one has a slow flow. It is interesting to note that 1
the term - - shows that the viscosity of the fluid increases. QJ.ll
The case /
= 1 is
a critical one. We have at the limit a Brinkmann type law: the
slowing effect is not too important.
D. Cioranescu, P. Donato and H. I. Ene
10
In the third case, when '"1
> 1, we get at the limit the Stokes equation with supple-
mentary terms. The fluid behaves like a free fluid. The slip beeing quite important the behaviour of the fluid is not affected by the presence of solid inclusions. We have also to mention that in all the cases, at the limit, the fluid is not any more incompressible. To conclude, a last remark deals with the differences between the boundary condition in system (1.1) and the boundary conditions used by C. Conca [6) and G. Allaire [2). They studied purely mechanical slip conditions without exterior contributions whereas we are in the presence of an applied electric field which is at the origin of the slip of the fluid. REFERENCES [1) G. ALLAIRE, Homogenization of Navier-Stokes equations in open sets perforated with tiny holes. Arch. Rat. Mech. Anal.,6 (1989), 497-537. [2) G. ALLAIRE, Homogenization of the Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math., XLIV, 6 (1991), 605-642. [3) D. CIORANESCU and P. DONATO, Homogeneisation du probleme du Neumann non homogene dans des ouverts perfores, Asymptotic Analysis, 1 (1988), 115-138. [4) D. CIORANESCU, P. DONATO and H. I. ENE, to appear. [5) D. CIORANESCU and J. SAINT JEAN PAULIN,Homogenization in open sets with holes, J. Math. Anal. Appl.,71 (1979),590-607. [6) C. CONCA, On the application of the homogenization theory to a class of problems arising in fluid mechanics, Journal Math. pures et Appl., 64 (1985), 31-75 [7) H. I. ENE and E. SANCHEZ - PALENCIA, Equation et phenomenes de surface pour l'ecoulement dans un modele de milieu poreux, Journal Mecanique, 14 (1975), 73-
108. [8) J. R. MELCHER, Continuum Electromechanics, MIT Press, Cambridge, Massachusetts and London, England (1981). [9) E. SANCHEZ - PALENCIA, Non homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer Verlag (1980). Doina CIORANESCU Laboratoire Analyse Numerique Universite Pierre et Marie Curie
Tour 55-65, 5eme etage " place Jussieu 75252 Paris Cedex 05, France.
Patri.ia DONATO Istituto eli Matematica Facolta di Sclenze M.F.N. Universita eli Salerno 84081 Baronissi (Salerno), Italy.
Horia I.ENE InstitutuI de Matematica al Academiei Str. Academiei 14 P. O. Box 1-764 70700 Bucuresti, Romania.
