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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 condition 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 condition. 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 Mikelić (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(e2){\mathcal{O}(\varepsilon^2)} whereas it is O(e){\mathcal{O}(\varepsilon)} in the BJ condition, where ε is the separation of scales parameter. KeywordsBeavers and Joseph condition-Domain of validity-Separation of scales
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(uU), (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=ε1,(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, ε1 (to be more precise, it is generally admitted
that the domain is ε0.1). 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, εO(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
H1,l
L1,
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(uU)|
|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(uU)|
|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
h1
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%xvF
iQFxipF=0,xivF
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)
vF(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 vF(x2=h)=0.
The flow of the free fluid is a plane Poiseuil flow between two parallel plates with an
adherence condition on x2=hand on x2=0. Therefore, we have classically
pF=pF(x1), dpF
dx1=cst,v
F
1=1
2µ
dpF
dx1
x2(hx2), (8)
or in dimensional form
vF
1=1
2µ
dpF
dX1
X2(hX2). (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.BothvPand pPare functions of xand y.The
dimensionless Stokes equation and incompressibility condition are in the form
µ%xvP
iQPxipP=0,xivP
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 '
vP=0on '.
We hav e now
pP
c
h=O!µcvP
c
l2",QP=hp
P
c
vP
cµc=O(ε2).
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
vP=v(0)P(y,x)+εv(1)P(y,x)+ε2v(2)P(y,x)+...
pP=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)=pF(x)=p(x),
p(1)P=τ
i(y)p
xip(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(ε2)
µ%xvBL
iε2xipBL =0,xivBL
i=0,
Then we look for vBL and pBL in the form of asymptotic expansions like (12).
vBL =v(0)BL(y,x)+εv(1)BL(y,x)+ε2v(2)BL(y,x)+...
pBL =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=
kBL
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=
KBL
ij
µ
p
xj.
By taking into account the perturbation, the dimensionless boundary condition becomes
vF
i(x2=0)=ε2KBL
ij
µ
p
xj,
123
About the Beavers and Joseph Boundary Condition 265
which in our particular boundary value problem reduces to
vF
1(x2=0)=ε2KBL
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
hK
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/h1, 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.
References
Auriault, J.-L.: Heterogeneous medium. Is an equivalent description possible? Int. J. Eng. Sci. 29(7), 785–795
(1991)
Auriault, J.-L.: On the domain of validity of Brinkman’s equation. Transp. Porous Media. (2009) (to appear)
123
266 J.-L. Auriault
Auriault, J.-L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transp.
Porous Media 60, 89–108 (2005)
Beavers, G.S., Joseph, D.L.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1),
197–207 (1967)
Cieszko, M., Kubik, J.: Derivation of matching conditions at the contact surface between fluid-saturated porous
solid and bulk fluid. Transp. Porous Media 34, 319–336 (1999)
Jäger, W., and Mikeli´c, A.: Modelling interface laws for transport phenomena between an unconfined fluid
and a porous medium using homogenization. Transp. Porous Media. (2009) (to appear)
Larson, R.E., Higdon, J.J.L.: Microscopic flow near the surface of two-dimensional porous media. Part 1.
Axial flow. J. Fluid Mech. 166, 449–472 (1986)
Larson, R.E., Higdon, J.J.L.: Microscopic flow near the surface of two-dimensional porous media. Part 2.
Transver se flow. J. Fluid Mech. 178, 119–136 (1987)
Levy, T., Sanchez-Palencia, E.: On boundary conditions for fluid flow in porous media. Int. J. Eng. Sci. 13,
923–940 (1975)
Nield, D.A.: The Beavers–Joseph boundary condition and related matters: a historical and critical note. Transp.
Porous Media. (2009) (to appear)
Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homo-
geneous fluid. I. Theoretical development. Int. J. Heat Mass Transf. 38(14), 2635–2646 (1995a)
Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homo-
geneous fluid. II. Comparison with experiment. Int. J. Heat Mass Transf. 38(14), 2647–2655 (1995b)
Richardson, S.: A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49(part 2),
327–336 (1971)
Saffman, P.G.: On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50(9),
959–964 (1971)
Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127.
Springer-Verlag, Berlin (1980)
Taylor,G.I.: A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49(part2), 319–326
(1971)
Val d é s - P a r a d a , F. J. , G o y e a u , B . , Oc h oa - Ta p i a, J . A . : J u m p m o me n t u m b o u n d a ry c o n d i t i o n a t a fl ui d - p o r o u s
dividing surface: derivation of the closure problem. Chem. Eng. Sci. 62, 4025–4039 (2007)
123
... An approximation that we can use to treat the equation (11) is (Discacciati and Quarteroni, 2009;Auriault, 2010) ...
... Remark 1. We note that the Beavers-Joseph-Saffman boundary conditions were found experimentally in (Beavers and Joseph, 1967;Saffman, 1971) and demonstrated in Mikelić, 2000, 2009), but only in a 2D laminar case (as mentioned in (Auriault, 2010)), and the extension to a generic geometry is non-trivial (Eggenweiler and Rybak, 2021). Moreover, in (Auriault, 2010) they found that the corrector of the simplified boundary condition (12) is ϵ 2 . ...
... We note that the Beavers-Joseph-Saffman boundary conditions were found experimentally in (Beavers and Joseph, 1967;Saffman, 1971) and demonstrated in Mikelić, 2000, 2009), but only in a 2D laminar case (as mentioned in (Auriault, 2010)), and the extension to a generic geometry is non-trivial (Eggenweiler and Rybak, 2021). Moreover, in (Auriault, 2010) they found that the corrector of the simplified boundary condition (12) is ϵ 2 . For these reasons, for most of the paper, we will consider the simplified boundary condition (12), nevertheless some comparisons with the BJS boundary condition are made for the sake of completeness. ...
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... The boundary of porous medium with a free viscous fluid is the continuity of pressure and the normal mass flux which give rise to wipe up the tangential velocity of the synovial fluid [17][18][19][20] . Such boundary condition was proposed by Beavers and Joseph (BJ) in 1967 as a empirical formula 18,19,21 . The analysis showed that BJ condition is macroscopic law which describe a phenomena at a macroscopic scale whereas characteristic length is much larger than the pore characteristic size 18,19,21 . ...
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Stokes’s equation in the fluid domain and Brinkman’s equation in the porous media are combined in the current study which is designated by the Stokes-Brinkman coupling. The current paper gives a theoretical analysis of the Stokes-Brinkman coupling. It has been shown that such a model is a good match for the knee joint. A flow model has been investigated in order to get a better understanding of the convective diffusion of the viscous flow along the articular surfaces between the joints. The Beavers and Joseph slip conditions which are a specific boundary condition for the synovial fluid are used to solve the governing system of partial differential equations for the synovial fluid and the results are provided here. Through the development of an analytical solution and numerical simulation (using the finite volume approach) it is hoped that the mechanisms of nutritional transport into the synovial joint will be better understood. According to the data the average concentration has a negative connection with both the axial distance and the duration spent in the experiment. Many graphs have been utilized to gain understanding into the problem’s various characteristics including velocity and concentration, among others. Hyaluronate (HA) is considered to be present in porous cartilage surfaces and the viscosity of synovial fluid fluctuates in response to the amount of HA present.
... One example, which is also the focus of the present work, is the interface between an overlying flow and a rigid porous surface. Recent work (Ochoa-Tapia & Whitaker 1995;Mikelić & Jäger 2000;Auriault 2010a;Minale 2014) have treated the inhomogeneous interface problem theoretically with upscaling techniques. Ochoa-Tapia 2 U. Lācis, S. Bagheri & Whitaker (1995) used a volume-averaging technique to derive a shear-stress jump condition. ...
... Mikelić & Jäger (2000) used homogenization and method of matched asymptotic expansions to show that the Saffman (1971) version of the empirical boundary condition by Beavers & Joseph (1967) (called BJ condition hereafter) is mathematically justified and its slip parameter can be computed by solving microscale problems in an interface unit cell. Auriault (2010a) also used a homogenization technique to derive a BJ-type of boundary condition valid for pressure-driven flows; he obtained however the condition at different order compared to Mikelić & Jäger (2000), as seen in discussion by Jäger & Mikelić (2010) and Auriault (2010b). More recently, Carraro et al. (2015) repeated the procedure of Mikelić & Jäger (2000) to determine the boundary condition of penetration (wall-normal) velocity component. ...
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Interfacial boundary conditions determined from empirical or ad-hoc models remain the standard approach to model fluid flows over porous media, even in situations where the topology of the porous medium is known. We propose a non-empirical and accurate method to compute the effective boundary conditions at the interface between a porous surface and an overlying flow. Using multiscale expansion (homogenization) approach, we derive a tensorial generalized version of the empirical condition suggested by Beavers & Joseph (1967). The components of the tensors determining the effective slip velocity at the interface are obtained by solving a set of Stokes equations in a small computational domain near the interface containing both free flow and porous medium. Using the lid-driven cavity flow with a porous bed, we demonstrate that the derived boundary condition is accurate and robust by comparing an effective model to direct numerical simulations. Finally, we provide an open source code that solves the microscale problems and computes the velocity boundary condition without free parameters over any porous bed.
... Most works treating the boundary conditions are empirical [35][36][37][38]. Those contributions which have recently treated the interface problems from first-principles have focused on rigid porous media and one-dimensional problems, such as the laminar channel flow [39,40] only (for which there is no transfer of mass or momentum between the material and the free fluid), or infiltration flow only [41]. ...
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Despite the ubiquity of fluid flows interacting with porous and elastic materials, we lack a validated non-empirical macroscale method for characterizing the flow over and through a poroelastic medium. We propose a computational tool to describe such configurations by deriving and validating a continuum model for the poroelastic bed and its interface with the above free fluid. We show that, using stress continuity condition and slip velocity condition at the interface, the effective model captures the effects of small changes in the microstructure anisotropy correctly and predicts the overall behaviour in a physically consistent and controllable manner. Moreover, we show that the performance of the effective model is accurate by validating with fully microscopic resolved simulations. The proposed computational tool can be used in investigations in a wide range of fields, including mechanical engineering, bio-engineering and geophysics.
... Beavers and Joseph conducted experimental research on the coupled problem within both the pipe flow region and porous medium region. 9 Their findings revealed a jump in slip velocity component perpendicular to the interface between these regions, which is proportional to the tangential stress of the liquid at that interface. This led to the development of the Beavers-Joseph slip velocity condition (BJ condition) for describing this coupled boundary condition, widely used in treating coupled flows. ...
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Based on the flow characteristics of fluids in various reservoir media, fractured-vuggy oil reservoirs can be classified into seepage zones and conduit flow zones. An interface exists between these two regions, where the movement of formation fluid near this interface is characterized by a coupling or transitional phenomenon between seepage and conduit flow. However, the complexity of this coupling interface poses challenges for traditional numerical simulations in accurately representing the intricate fluid dynamics within fractured-vuggy oil reservoirs. This limitation impacts the development planning and production adjustment strategies for fractured-vuggy oil reservoirs. Consequently, achieving accurate characterization and numerical simulation of these systems remains a critical challenge that requires urgent attention. A new mathematical model for oil-water two-phase flow in fractured-vuggy oil reservoirs is presented, which developed based on a novel coupling method. The model introduces the concept of the proportion coefficient of porous media within unit grids and defines a coupling region. It employs an enhanced Stokes–Brinkman equation to address the coupling issue by incorporating the proportion coefficient of porous media, thereby facilitating a more accurate description of the coupling interface through the use of the coupling region. Additionally, this proportion coefficient characterizes the unfilled cave boundary, simplifying the representation of model boundary conditions. The secondary development on the open-source fluid dynamics software is conducted by using matrix & laboratory (MATLAB). The governing equations of the mathematical model are discretized utilizing finite volume methods and applying staggered grid techniques along with a semi-implicit calculation format for pressure coupling—the Semi-Implicit Method for Pressure Linked Equations algorithm—to solve for both pressure and velocity fields. Under identical mechanism models, comparisons between simulation results from this two-phase flow program and those obtained from Eclipse reveal that our program demonstrates superior performance in accurately depicting flow states within unfilled caves, thus validating its numerical simulation outcomes for two-phase flow in fractured cave reservoirs. Utilizing the S48 fault-dipole unit as a case study, this research conducted numerical simulations to investigate the water-in-place (WIP) behavior in fractured-vuggy oil reservoirs. The primary focus was on analyzing the upward trend of WIP and its influencing factors during production across various combinations of fractures and dipoles, thereby validating the feasibility of the numerical modeling approach in real-world reservoirs. The simulation results indicated that when multiple dissolution cavities at different locations communicated with the well bottom sequentially, the WIP in the production well exhibited a staircase-like increase. Furthermore, as the distance between bottom water and well bottom increased, its effect on water intrusion into the well diminished, leading to a slower variation in the WIP curve. These characteristics manifested as sudden influxes of water flooding, rapid increases in water levels, and gradual rises—all consistent with actual field production observations. The newly established numerical simulation method for fractured-vuggy oil reservoirs quantitatively describes two-phase flow dynamics within these systems, thus effectively predicting their production behaviors and providing guidance aimed at enhancing recovery rates typically observed in fractured-vuggy oil reservoirs.
... Equation (11) can be simplified letting M → ∞ , which gives (Discacciati and Quarteroni 2009;Auriault 2010) Using (12) in place of (11) we get a difference of about with respect to the whole BJS (Discacciati and Quarteroni 2009). ...
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Lymph Nodes (LNs) are crucial to the immune and lymphatic systems, filtering harmful substances and regulating lymph transport. LNs consist of a lymphoid compartment (LC) that forms a porous bulk region, and a subcapsular sinus (SCS), which is a free-fluid region. Mathematical and mechanical challenges arise in understanding lymph flow dynamics. The highly vascularized lymph node connects the lymphatic and blood systems, emphasizing its essential role in maintaining the fluid balance in the body. In this work, we describe a mathematical model in a steady setting to describe the lymph transport in a lymph node. We couple the fluid flow in the SCS governed by an incompressible Stokes equation with the fluid flow in LC, described by a model obtained by means of asymptotic homogenisation technique, taking into account the multiscale nature of the node and the fluid exchange with the blood vessels inside it. We solve this model using numerical simulations and we analyze the lymph transport inside the node to elucidate its regulatory mechanisms and significance. Our results highlight the crucial role of the microstructure of the lymph node in regularising its fluid balance. These results can pave the way to a better understanding of the mechanisms underlying the lymph node’s multiscale functionalities which can be significantly affected by specific physiological and pathological conditions, such as those characterising malignant tissues.
... The Beavers-Joseph-Saffman interface condition (2.4) used in this work is initially established in a two-dimensional setting, and expanding it to three dimensions presents a significant and complex challenge [2,[84][85][86][87][88]. Moreover, it would be interesting to study how the physico-chemical properties of the interface affect the solution [67]. ...
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We study the coupling between time-dependent Darcy–Brinkman and the Darcy equations at the microscale subjected to inhomogeneous body forces and initial conditions to describe a double porosity problem. We derive the homogenized governing equations for this problem using the asymptotic homogenization technique, and as macroscopic results, we obtain a coupling between two Darcy equations, one of which with memory effects, with mass exchange between phases. The memory effects are a consequence of considering the time dependence in the Darcy–Brinkman equation, and they allow us to study in more detail the role of time in the problem under consideration. After the formulation of the model, we solve it in a simplified setting and we use it to describe the movement of fluid within a vascularized lymph node.
... The superscript "0" corresponds to the parameters values on the interface Γ. Saffman (1971) simplified the Beavers-Joseph interface condition by neglecting the flow velocity v d The Beavers-Joseph interface condition was extended by Neale and Nader (1974) for the case when the fluid flow in the porous domain instead of Darcy's law is described by the Brinkman model with the parameter = b ∕ , where and b are the fluid dynamic and effective viscosities. The Beavers-Joseph condition and its modified variants were also studied in other numerous works (Cieszko and Kubik 1999;Jamet 2007, 2009;Jäger and Mikelić 2009;Auriault 2010;Hou and Qin 2019). ...
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A modified formulation of the interfacial boundary conditions for the coupling of the Stokes and Brinkman models that describe the incompressible fluid flow in the free space and porous medium domains is proposed using the dimension analysis procedure. The two conditions of continuity of the normal velocity and pressure are supplemented by the interfacial vorticity and tangential velocity jumps conditions. The condition of the vorticity jump establishes the proportionality of the vorticities in a free fluid and porous medium domains. The condition of velocity jump that is similar to the classical Beavers–Joseph condition defines the vorticity as a linear combination of the tangential velocity components in the free space and porous medium. The derived conditions are directly applicable for a class of 2D problems with an arbitrary shaped boundary. The test problems are solved numerically using the Stokes flow microscopic model and analytically for the Stokes–Brinkman flow model to determine the coefficients in the introduced linear dependencies of the interfacial velocity and vorticity. The case is considered for the porous media formed by circular or square cylinders located in the centers of rectangular cells. As a result, the corresponding coefficients in the boundary conditions are found as a porosity function for two types of the porous medium configuration. The approximate analytical estimation of the coefficients confirms the obtained numerical dependencies. The verifications of the calculated coefficients were made by analytical and numerical studying 2D fluid flow problems. It is shown that the fluid flow calculated on the basis of the Stokes–Brinkman flow model with modified boundary conditions agrees well with the results of the microscopic Stokes flow model. The advantages of the proposed boundary conditions are discussed.
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In Part I of this paper a stress jump condition was developed based on the non-local form of the volume averaged Stokes' equations. The excess stress terms that appeared in the jump condition were represented in a manner that led to a tangential stress boundary condition containing a single adjustable coefficient of order one. In this paper we compare the theory with the experimental studies of Beavers and Joseph [J. Fluid Mech. 30, 197–207 (1967)], and we explore the use of a variable porosity model as a substitute for the jump condition. The latter approach does not lead to a successful representation of all the experimental data, but it does provide some insight into the complexities of the boundary region between a porous medium and a homogenous fluid.
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The compatibility conditions matching macroscopic mechanical fields at the contact surface between fluid-saturated porous solid and adjacent bulk fluid are considered. Special attention is paid to the derivation of conditions for tangential components of the fluid flow velocities and to the verification of validity of the condition postulated by Beavers and Joseph. It has been shown that at the contact surface between two media, a dissipation of mechanical energy due to the fluid viscosity does exist and thus the form of a dissipation function has been proposed. It has been proven that this relation determines the form of two linear compatibility conditions derived for the tangential components of the relative fluid velocities and that these conditions describe the experimental results more precisely than the condition postulated by Beavers and Joseph.
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We investigate the validity of Darcy's law when the separation of scales is poor. We use the method of multi-scale asymptotic expansions which gives the macroscopic behaviour from the pore scale description. The first order approximation is the Darcy's law. When the separation of scales is poor, eventual correctors to Darcy's law are obtained by investigating the following orders of approximation, thus enabling us to study its robustness. We investigate the two first correctors. Thus, the accuracy of the macroscopic flow law is improved from O(ε) to O(ε3), where ε is the separation of scale parameter. The second corrector shows a Brinkman's term. For macroscopically homogeneous porous media, the first corrector cancels out, thatpoints out the robustness of Darcy's law in this case.
Article
Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.
Article
A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.
Article
The present paper contains an analysis of the model of a porous material proposed in part 1, and carries out calculations which allow comparison between theory and the experiments described therein. The relevant boundary conditions to be applied at an interface between a fluid and such a material are considered.
Article
A model problem is analysed to study the microscopic flow near the surface of two-dimensional porous media. In the idealized problem we consider axial flow through infinite and semi-infinite lattices of cylindrical inclusions. The effect of lattice geometry and inclusion shape on the permeability and surface flow are examined. Calculations show that the definition of a slip coefficient for a porous medium is meaningful only for extremely dilute systems. Brinkman's equation gives reasonable predictions for the rate of decay of the mean velocity for certain simple geometries, but fails for to predict the correct behaviour for media anisotropic in the plane normal to the flow direction.
Article
In problems where a viscous fluid flows past a porous solid it has frequently been assumed that the tangential component of surface velocity is zero. When the porous solid has an open structure with large pores the external surface stress may produce a tangential flow below the surface. Recently, Beavers & Joseph (1967) have assumed that the surface velocity UB depends on the mean tangential stress [μ(du/dy)]y=0[\mu(d\overline{u}/dy)]_{y=0} in the fluid outside the porous solid through the relation \[ \left[\mu\frac{d\overline{u}}{dy}\right]_{y=0} = \frac{\mu\alpha}{k^{\frac{1}{2}}}(U_B-Q), \] where Q is the volume flow rate per unit cross-section within the porous material due to the pressure gradient, k is the Darcy constant and α is a constant which depends only on the nature of the porosity. An artificial mathematical model of a porous medium is proposed for which the flow can be calculated both inside and outside the surface. This conceptual model was materialized and the experimental results agree with the calculations. The calculated values of α so found are not quite independent of the external means of producing the external tangential stress.
Article
A model problem is analysed to study the microscopic flow near the surface of porous media. In the idealized system, we consider two-dimensional media consisting of infinite and semi-infinite periodic lattices of cylindrical inclusions. In Part 1, results for axial flow were presented. In this work results for transverse flow are presented and discussed in the context of macroscopic approaches such as slip coefficients and Brinkman's equation.