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Transp Porous Med (2010) 83:257–266
DOI 10.1007/s11242-009-9435-9
About the Beavers and Joseph Boundary Condition
Jean-Louis Auriault
Received: 12 May 2009 / Accepted: 17 June 2009 / Published online: 9 July 2009
©SpringerScience+BusinessMediaB.V.2009
Abstract A large number of papers are adopting the Beavers and Joseph (BJ) condition
(Beavers and Joseph, J Fluid Mech 30(1):197–207, 1967) for describing the boundary condi-
tion between a saturated porous medium and a free fluid, in place of the adherence condition.
The aim of the paper is to bring some insight into the domain of validity of the BJ condi-
tion. After a short review of some papers on the subject, we point out that the experimental
conditions of BJ do not show a good separation of scales. That makes the BJ condition not
transposable to different macroscopic situations. When the separation of scales is good, an
intrinsic boundary condition is obtained by using the homogenization technique of multiple
scale asymptotic expansions. As in the BJ condition and other theoretical works, e.g., Jäger
and Mikeli´c (Transp Porous media, 2009, to appear) we obtain the adherence condition of the
free fluid at the first order approximation. However, the corrector to the adherence condition
is O(ε2)whereas it is O(ε)in the BJ condition, where εis the separation of scales parameter.
Keywords Beavers and Joseph condition ·Domain of validity ·Separation of scales
1 Introduction
The usual macroscopic conditions at the boundary of a porous medium with a free viscous
fluid are the continuity of the pressure and the normal mass flux whereas the tangential veloc-
ity of the free fluid cancels out. This latter condition was questioned by (Beavers and Joseph
1967) who proposed the empirical formula
∂u
∂n=α
√K(u−U), (1)
J.-L. Auriault (B
)
Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire,
BP 53 , 38041 Grenoble Cedex, France
e-mail: jean-louis.auriault@hmg.inpg.fr
123
258 J.-L. Auriault
where uis the tangential velocity of the free fluid, nis the exterior normal to the porous
medium, Kis the permeability, Uis the tangential velocity of the Darcy’s flux and αis a
dimensionless parameter which “depends only on the properties of the fluid and the permeable
material”.
Attempts to theoretically support the Beavers and Joseph (BJ) condition or to propose
modified BJ condition are numerous, see Nield (2009), for relevant papers prior to 1975.
Recently the subject received much attention. Among all papers, let us cite Saffman (1971),
Ochoa-Tapia and Whitaker (1995a,b), Cieszko and Kubik (1999),Val d é s- P ar a da e t al . (2007),
Jäger and Mikeli´c (2009). Experimental investigations are fewer, e.g., Beavers and Joseph
(1967), Taylor (1971). Of great interest are also numerical investigations of the pore scale
velocity in the vicinity of a macroscopic porous block surface, Larson and Higdon (1986,
1987).
This note is concerned with the domain of validity of the BJ condition or its modified
versions. For this purpose, we successively revisit rapidly the experiments of Beavers and
Joseph (1967)andTaylor (1971) and the theoretical approaches in Saffman (1971)andJäger
and Mikeli´c(2009). This is done in the light of the homogenization theory which makes a
systematic use of the notion of separation of scales.
The analysis is based on the obvious that the BJ condition is a macroscopic law: it describes
a phenomenon at a macroscopic scale which characteristic length is (very) much larger than
the pore characteristic size. Thus, a separation of scales should be implied. Macroscopic
equivalent descriptions of finely heterogeneous phenomena are possible in the presence of
such a separation of scales, only
l
L=ε≪1,(2)
where land Lare characteristic lengths of the heterogeneity and of the macroscopic boundary
value problem (phenomenon and/or geometry), respectively. Following Auriault (1991), the
alternative is
– Either a separation of scales is present, ε≪1 (to be more precise, it is generally admitted
that the domain is ε≤0.1). A macroscopic description is possible which is intrinsic to
the investigated phenomenon and to the heterogeneous medium: the macroscopic laws
are independent of the macroscopic boundary value problem in consideration.
– Or a separation of scales is absent, ε≥O(1). A macroscopic equivalent description is
not possible. Of course, it is possible after some averaging process to obtain a pseudo-
macroscopic description. However, this pseudo-description depends on the macroscopic
boundary value problem at stake: in particular the coefficients entering the laws are not
effective coefficients.
We show that the BJ condition belongs to this second category. This is confirmed by the
different above cited authors, from the founder paper by Beavers and Joseph (1967). BJ
condition is sometimes seen as a consequence of the Brinkman’s equation. Since the domain
of validity of this equation is drastically restricted to very particular porous media such as
swarms of particles or fibrous materials with extremely large porosities Auriault (2009), we
will not address this subject in the paper.
In Sect. 2, we present a critical review of selected previous studies on BJ condition from
the founding paper Beavers and Joseph (1967)totherecentpaperbyJäger and Mikeli´c
(2009). Then in Sect.3, we study the boundary condition between a free fluid and a porous
medium in the particular case of the BJ experiment, when such a boundary macroscopic
condition is possible, i.e., when a separation of scales is present.
123
About the Beavers and Joseph Boundary Condition 259
2 The BJ Condition: A Critical Review
2.1 Experimental Investigations
Experimental setup in Beavers and Joseph (1967), is schematized in Fig. 1.Aporousblock
of length Land height His placed in a rectangular channel, with a gap of thickness hwhich
enables the free fluid to flow over the porous medium. A good estimate of the characteristic
size of the pores is given by
l=√K
We immediately see that the porous medium itself shows a good separation of scales
l
H≪1,l
L≪1,
which enables the use of Darcy’s law to describe the fluid flow through the porous block.
Contrarily, Fig. 6 in Beavers and Joseph (1967), shows that there is no separation of scales
between the free fluid channel thickness and the pore size for data which support BJ condition
σ=h
l<10.
Therefore, a macroscopic description of the free fluid flow cannot be addressed independently
of the macroscopic flow in the porous medium: there exist a macroscopic Darcy’s law for the
porous medium +free fluid system, only. The free fluid channel plays the role of a particular
pore of this system. Therefore, the BJ condition is a pseudo-macroscopic description. Its
structure as well as its coefficients depend on the macroscopic boundary value problem in
consideration. “The assumption of rectilinear flow in the channel breaks down” for values
of σ=O(1),(Beavers and Joseph 1967). Numerical investigations of the pore scale fluid
velocity show that when fitting numerical data to BJ condition, “the slip velocity is likely
to be extremely sensitive to the position of the interface” which is chosen to perform the
fitting (Larson and Higdon 1986,1987). In the same way Saffman (1971), pointed out that
“in applying the value of αdetermined in one experiment to the predictions of a second
experiment, care must be taken to ensure that the surface is drawn in the same way”.
Fig. 1 Experimental setup
123
260 J.-L. Auriault
In Taylor (1971), experiments are conducted with an ideal porous medium consisting of
parallel plates perpendicular to the free fluid channel, itself being limited by a moving plate.
This very particular pore geometry induces a one-dimensional velocity field at the pore scale
and in the free fluid, which enables analytical investigations (Richardson 1971). For this type
of porous media, αremains a constant when h/l>0.5. BJ condition is satisfactorily verified.
2.2 Theoretical Approaches
Apart from the Richarson work (Richardson 1971), one of the first tentative to theoretically
justify the BJ condition is due to Saffman (1971). Saffman applies a statistical approach
and ensemble averaging to strongly non-homogeneous porous media to obtain the following
macroscopic boundary condition
u=√K
α
∂u
∂n+O(K). (3)
Implicitly the free fluid characteristic length his much larger than the pore size lsince the
demonstration requires in the free fluid (see Fig. 1for the notations)
X2
l→∞.
This is coherent with the fact that the fluid velocity Uin the porous medium is absent in (3):
the flows in both the porous medium and the free fluid are provoked by the same macroscopic
gradient of pressure and velocities Uand uare in the ratio, see relations (7)below
U
u=O!l2
h2"≪1.
The paper by Jäger and Mikeli´c (2009) is a tentative to demonstrate the BJ condition by using
the rigorous technique of asymptotic expansions which itself is based on the separation of
scales small parameter ε. At the first order of approximation, the classical adherence con-
dition is recovered on the porous medium surface. However, the corrector to this first order
condition is obtained by assuming the shear stress on the interface in the free fluid to be equal
to the surface average of the shear stress in the pore fluid. This is not physically admissible,
because of the presence of the porous skeleton: on the interface, the free fluid is in contact
with both the pore fluid and the solid matrix. The authors recover the BJ condition, which
is surprising. The perturbation to the adherence condition is due to the pore fluid velocity
which should yield an ε2order corrector, see (7), whereas the corrector in (1)isO(ε)
|α
√K(u−U)|
|∂u
∂n|=O!l
h"=O(ε).
2.3 Conclusion
The above review shows that the Beavers and Joseph boundary condition and the Saffman
boundary condition call for the following remarks.
– The Beavers and Joseph condition was experimentally checked for boundary value prob-
lems with poor separation of scales: ε=l
h>0.1. In such cases, an intrinsic macroscopic
boundary condition does not exist; an eventual boundary condition, whatever the way it
is obtained, is strongly dependent on the macroscopic boundary value problem at stake.
123
About the Beavers and Joseph Boundary Condition 261
– We have from the BJ condition (1)
|α
√K(u−U)|
|∂u
∂n|=O!l
h"=O(ε)≪1,(4)
when the separation of scales is good. In such a case, the BJ condition reduces to the
adherence condition at the first order of approximation.
–Thedifferenttheoreticalinvestigationsdonotmakefulluseofthenotionofseparation
of scales or are questionable.
3 Boundary Macroscopic Condition Between a Free Fluid and a Porous Medium
For the sake of simplicity, we consider the Beavers and Joseph experiment, see Fig. 1,where
L=O(H)=O(h),whenamacroscopicboundaryconditioncanbeinvestigatedbetween
the free fluid and the porous medium , i.e., when a separation of scales is present
ε=l
h≪1
We also assume that the porous medium is periodic, macroscopically homogeneous and iso-
tropic. As in (1) inertia is neglected. Due to the particular boundary value problem at stake,
both macroscopic velocities in the free fluid and in the porous medium are in the X1direction.
3.1 Estimates
The flows in the free fluid and in the porous medium are described by the Navier equation
and the incompressibility condition
µ%Xvi−∇Xip=0,∇Xivi=0.(5)
Both flows are forced by a macroscopic gradient of pressure
|∇Xp|=O#pc
L$.
Throughout the paper, an index cshows a characteristic value. The viscous terms in the free
fluid Fand in the porous medium Pare of very different orders of magnitude since the shear
stresses are related to lengths hand l, respectively,
|µ%XvF
i|=O!µcvF
c
h2",|µ%XvP
i|=O!µcvP
c
l2".
From (5), we deduce that
|∇Xp|=O#pc
L$=O!µcvF
c
h2"=O!µcvP
c
l2",(6)
which gives
vP
c
vF
c=U
u=O!l2
h2"≪1.(7)
We successively address the flows in the free fluid, in the porous medium and in the boundary
layer to be introduced to match the velocity and the pressure fields in these two regions. It is
123
262 J.-L. Auriault
convenient to make dimensionless the equations that describe the three flows. Here, we use
has the characteristic length
x=X
h,
where xis the macroscopic dimensionless space variable.
3.2 Flow in the Free Fluid (F)
In the free fluid, the dimensionless Stokes equation and incompressibility condition are in
the form (an asterisk shows a dimensionless quantity, and obviously, µ∗=1)
µ∗%xv∗F
i−QF∇xip∗F=0,∇xiv∗F
i=0,
where from (6)
QF=hp
F
c
vF
cµc=O(1).
On the boundary x2=0, the free fluid velocity vFcancels out on the porous matrix surface
and equals the porous medium velocity vPon the rest of the surface. To the first order of
magnitude, we have with (7)
v∗F(x2=0)=0.
The above adherence condition is fully coherent with estimate (4). We remark that the above
condition is just as if the porous medium was replaced by a non-porous solid. This is the
direct consequence of the separation of scales (2). The boundary value problem for the free
fluid is completed by v∗F(x2=h∗)=0.
The flow of the free fluid is a plane Poiseuil flow between two parallel plates with an
adherence condition on x2=h∗and on x2=0. Therefore, we have classically
p∗F=p∗F(x1), dp∗F
dx1=cst,v
∗F
1=−1
2µ∗
dp∗F
dx1
x2(h∗−x2), (8)
or in dimensional form
vF
1=−1
2µ
dpF
dX1
X2(h−X2). (9)
To the first order of approximation, the flow of the free fluid ignores the porous medium.
3.3 Flow in the Pores (P)
Let us recall the main results of this classical problem. The porous structure is periodic of
period &. The pores introduce a second characteristic length lto which corresponds a sec-
ond dimensionless space variable y=X
l.Bothv∗Pand p∗Pare functions of xand y.The
dimensionless Stokes equation and incompressibility condition are in the form
µ∗%xv∗P
i−QP∇xip∗P=0,∇xiv∗P
i=0,(10)
where
QP=hp
P
c
vP
cµc,
123
About the Beavers and Joseph Boundary Condition 263
to which we add the boundary condition on the pore surface '
v∗P=0on '.
We hav e now
pP
c
h=O!µcvP
c
l2",QP=hp
P
c
vP
cµc=O(ε−2).
Remark that
pP
c=pF
c,v
P
c=O#ε2vF
c$(11)
By following the homogenization method, we look for the velocity and the pressure in the
pores in the form
v∗P=v∗(0)P(y,x)+εv∗(1)P(y,x)+ε2v∗(2)P(y,x)+...
p∗P=p∗(0)P(y,x)+εp∗(1)P(y,x)+ε2p∗(2)P(y,x)+...,
(12)
with y=x/εand where the different terms in the expansions are y-periodic. We need the
Darcy’s pore fields, only, i.e., the pore scale fields of the velocity v∗(0)Pand the pressures
p∗(0)Pand p∗(1)P.Thisisaclassicalproblemwhichinvestigationcanbefound,e.g.,in
Auriault (1991). v∗(0)Pand p∗(1)Pare periodic and verify
µ∗%yv∗(0)P
i−∇xip∗(0)P−∇yip∗(1)P=0,∇yiv∗(0)P
i=0(13)
The solution is
p∗(0)P=p∗(0)P(x)=p∗F(x)=p∗(x),
p∗(1)P=−τ∗
i(y)∂p∗
∂xi+¯p∗(1)P(x), (14)
v∗(0)P
i=−
k∗
ij(y)
µ∗
∂p∗
∂xj(15)
The above solutions are valid in the bulk porous medium, away from the macroscopic bound-
aries of the porous sample where the periodicity is broken. After averaging on the porous
medium period &
⟨.⟩=1
|&|%
&
.d&,&v∗(0)P
i'=−K∗
µ∗
∂p∗(0)P
∂xi,(16)
where the isotropy of the porous medium at the macroscopic scale has been taken into
account. Relation (16) stands for the dimensionless first order flow law, i.e., the Darcy’s law.
The dimensionless form of (16)is
&v(0)P
i'=−K
µ
∂p(0)P
∂Xi,K=O(l2). (17)
123
264 J.-L. Auriault
3.4 Boundary Layer (BL)
At the first order of approximation the flow of the free fluid ignores the presence of the porous
medium. This one causes a small perturbation of the free fluid flow, in the vicinity of the
boundary X2=0. Near this boundary, the flow in the porous medium is also modified. To
investigate this perturbation, we classically introduce a y1-periodic boundary layer (Levy and
Sanchez-Palencia 1975;Sanchez-Palencia 1980) of period &BL(0<y1<1,−∞ <y2<
∞). Let vBL and pBL be the perturbation in the free fluid y2>0 and in the pores y2<0.
Due to the presence of the pore structure these two fields are functions of both dimensionless
space variables y=X/land x=X/h
vBL =vBL(y,x), pBL =pP(y,x).
Then, the characteristic length to be used for the shear stress is l, as in the porous medium flow
investigated above. Obviously vBL =O(vP), pBL =O(εpF)for y2>0andpBL =O(pP)
for y2<0. We take advantage of the linearity of Eq. 5and we use vP
cand pcto make dimen-
sionless the equations. They are similar to (10)withagainQ=O(ε−2)
µ∗%xv∗BL
i−ε−2∇xip∗BL =0,∇xiv∗BL
i=0,
Then we look for vBL and pBL in the form of asymptotic expansions like (12).
v∗BL =v∗(0)BL(y,x)+εv∗(1)BL(y,x)+ε2v∗(2)BL(y,x)+...
p∗BL =p∗(0)BL(y,x)+εp∗(1)BL(y,x)+ε2p∗(2)BL(y,x)+...,
(18)
with p∗(0)BL(x2>0)=0. As above we have p∗(0)BL(y,x)=p∗(x)for x2<0. Then
v(0)BL and p(1)BL verify
µ%yv∗(0)BL
i−∇yip∗(1)BL =0,∇yiv∗(0)BL
i=0,y2>0,(19)
µ%yv∗(0)BL
i−∇xip∗−∇yip∗(1)BL =0,∇yiv∗(0)BL
i=0,y2<0.(20)
The boundary conditions are
–v∗(0)BL and p∗(1)BL are &BL-periodic,
–lim
y2→∞ v∗(0)BL =0and lim
y2→∞ p∗(1)BL =0
–lim
y2→−∞ v∗(0)BL =v∗(0)P(y,x)and lim
y2→−∞ p∗(1)BL =p∗(1)P(y,x)
The solution v∗(0)BL is in the form
v∗(0)BL
i=−
k∗BL
ij (y)
µ∗
∂p∗
∂xj.(21)
With a view to obtain a macroscopic boundary condition, let us take the surface average of
v∗(0)BL
i
⟨.⟩x2=0=
1
%
0
.dS,&v∗(0)BL
i'x2=0=−
K∗BL
ij
µ∗
∂p∗
∂xj.
By taking into account the perturbation, the dimensionless boundary condition becomes
v∗F
i(x2=0)=−ε2K∗BL
ij
µ∗
∂p∗
∂xj,
123
About the Beavers and Joseph Boundary Condition 265
which in our particular boundary value problem reduces to
v∗F
1(x2=0)=−ε2K∗BL
µ∗
∂p∗
∂x1
.
In dimensional form we obtain
vF
1(x2=0)=−KBL
µ
∂p
∂X1
,KBL =O(l2). (22)
Equation (22) represents the corrected boundary condition for the velocity at the interface
free fluid–porous medium for the boundary value problem shown in Fig. 1, when a separation
of scales is present. It can be easily generalized to different situations. Equation (22) calls
for some remarks
– The corrector in (22)isoftheorderO(ε2), whereas in the BJ condition (1)itisofthe
order O(ε).
– The structures of relations (22)and(1)aredifferent.
– In the case of the particular boundary value problem shown in Fig. 1, it is possible to put
(22)intheformof(1). From (9)wehave
dvF
1
dX2
(X2=0)=−h
2µ
dp
dX1
.
Therefore, the boundary condition (22) can be put in the form
vF
1=2KBL
h
dvF
1
dX2
.(23)
However,
2KBL
h≪√K
α,
and relation (23) is a priori valid for the boundary value problem shown in Fig. 1, only.
4 Conclusion
The Beavers and Joseph condition was obtained from experimental data for flows through a
canal and a porous medium which do not show a separation of scales, ε=√K/h>0.1.
Therefore, the corresponding separated flow laws and the boundary condition between both
flows (the BJ condition) cannot be used for different macroscopic boundary value problems.
They are dedicated to the particular setup at stake.
When a separation of scales is present, ε=√K/h≪1, an intrinsic boundary condition
was demonstrated. However, it was shown that the corrector to the first order condition—the
adherence condition—is O(ε2), whereas in the BJ condition it is O(ε).
Finally, the adherence condition reveals itself a very good approximation.
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