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Abstract
We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent . By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the
-Laplace equation,
, with a coefficient called ‘flow factor’, which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle.
1. Introduction
In this paper we consider the stationary flow of an incompressible non-Newtonian fluid in a thin domain in
, where ε is a positive small parameter related to the thickness of the domain. More precisely, we consider a Hele–Shaw cell, i.e. the domain is confined between two curved surfaces in close proximity and contains a cylindrical obstacle. The fluid is modeled as a power-law fluid, i.e. the viscosity depends on the rate of strain via a power-law. The boundary
consists of two disjoint part
(Dirichlet boundary) and
(Neumann boundary). The Dirichlet part corresponds to the upper and lower surfaces which are separated by a non-uniform thickness, as well as the lateral surface of the obstacle. The Neumann part, which corresponds to the lateral boundary of the cell, is regarded as a penetrable inlet/outlet zone where an external pressure gradient is applied as a surface force along
. Moreover, the flow is assumed to be governed by the Stokes equation, therefore inertial effects are neglected. It is also assumed that the flow is isothermal and that gravity can be neglected.
The main aim of the present work is to derive a lower-dimensional model for the flow by studying the asymptotic behavior of the system described above, as . This dimensional reduction will be achieved by using an adaptation of a technique called two-scale convergence for thin domains which was developed by Marušić and Marušić-Paloka [Citation1]. The present results generalize [Citation2] where the Newtonian case was considered without any obstacle in the geometry. Moreover, we generalize the classical Poiseuille law, see e.g. [Citation3, p. 222], for Newtonian flow. In particular the Newtonian Hele–Shaw flow is suitable to visualize streamlines around an obstacle [Citation4], as well as, connecting the velocity to the gradient of the pressure by linear dependence. The main result of the present paper is a nonlinear Poiseuille law, in which the limit velocity and the limit pressure gradient follows a power-law [Citation5, Sec. 7.3.1]. A novelty is that we consider pressure-driven flow that can be found in many realistic applications, see e.g. [Citation6–9].
Non-Newtonian flow in Hele–Shaw like domains appears in many industrial applications, e.g. polymer processing, transportation of oil in pipelines, hydrology, as well as in food processing. In particular, we focus on some works where analysis and mathematical modeling of Poiseuille flow of power-law fluids are considered. Aronsson and Janfalk [Citation10] derived the equations of motion and proved some exact solutions and a representation theorem. A more rigorous study was performed by Mikelić and Tapiéro [Citation11] who derived a nonlinear Poiseuille law as a limit case of the Navier–Stokes equations when the thickness of the cell tends to zero. However, to avoid technical difficulties the authors chose to employ a no-slip condition on the whole boundary. The typical situation in a Hele–Shaw cell is that some part of the boundary should be penetrable, thus allowing fluid particles to enter and leave the domain. At such boundaries it is natural to prescribe the mass flux or impose a boundary condition for the pressure, i.e. by prescribing the momentum flux. Note also that, most examples in Hele–Shaw's original paper showed domains where a solid obstacle was placed in the interior of the domain, which is another characteristic of Hele–Shaw flow. Thus, the present work complements [Citation11] in the sense that a mixed boundary condition is employed, both surfaces may be curved and the domain may contain an interior obstacle. Related models that take thermal effects into account can be found in [Citation12,Citation13]. Non-Newtonian Hele–shaw flow of other type than power-law relation has been studied in [Citation14,Citation15].
Poiseuille's law also plays an important role in lubrication theory, formulated in the Newtonian case by Reynolds [Citation16], see also [Citation17, Ch. 22]. Recall that the fundamental problem in lubrication theory is to describe fluid flow in the gap between adjacent surfaces which are in relative motion to each other and in general there are no obstacles in the domain. Bayada and Chambat [Citation18], gave the first rigorous mathematical derivation of Reynolds equation, considering the Stokes problem in a thin domain and assuming that the velocity field satisfies a Dirichlet condition on the whole boundary. The authors proved that the limit pressure satisfies the classical Reynolds equation with a Neumann condition. In contrast, our limit equation has a Dirichlet condition for the pressure. The connection between Hele–Shaw theory and lubrication theory has been further explored in [Citation19]. A lubrication problem with power-law fluids was considered in [Citation20], under the assumption that the lower surface is flat and moving whereas the upper surface is rough and stationary. Flow of power-law fluids through a thin porous medium, i.e. periodic array of vertical cylinders confined between two parallel plates with no-slip boundary condition on the whole boundary was studied in [Citation21].
The paper is organized as follows: In Section 2 we give a precise description of the Hele–shaw cell and present some notation and preliminary results. In Section 3 we set the problem and formulate our main result. To prove the main result we need several a priori estimates and some results related to two-scale convergence for thin domains. These results are proved in Section 4– 5. Finally, the main result is proved in Section 5–6. In addition, several technical results are proved in Appendices 1–4.
2. Preliminaries, notation and some technical results
2.1. Euclidean structure
Let denote the set of real
matrices
equipped with the Euclidean scalar product
where
in
denotes the transpose of X, the symmetrical and skew-symmetric parts of X are defined respectively
The identity element in
is denoted as
. We identify
with
(column vectors) and denote the scalar product in
as
2.2. Geometry of the domain
Let ω denote a perforated Lipschitz domain in say
where L>0 and D is the closed disc
of radius 0<R<L. The boundary of ω is decomposed into two disjoint parts
Thus, both
and
have positive measure (arc length). Note that the main result of this paper is valid for more general domains ω that share the essential properties of the definition given above.
The (unscaled) fluid domain is defined as
where
are Lipschitz functions defined on the closure of ω, i.e.
and satisfy the conditions
(1)
(1) where
with
is the thickness of the fluid domain and
is the center surface of the fluid domain.
A point in the domain Ω will be denoted as where
and
Similarly, the components of a vector function
are denoted as
, where
.
The boundary of Ω is divided into two disjoint parts
where
is referred to as the ‘lateral’ surface.
The thin domain is defined by anisotropic scaling of the unscaled domain Ω of unit thickness, using the parameter
, i.e.
In other words
is a Hele–Shaw cell of thickness
. Also, the boundary of
is divided into two disjoint parts
2.3. Function spaces
We shall work mainly with the following function spaces
where
with conjugate exponent
and the divergence operator
is defined by
(2)
(2)
are closed subspace of
. Since
has positive surface measure we can equip
with the norm
which is equivalent to the usual
-norm by Korn's inequality [Citation22]. Moreover,
denotes the quotient space of
by
defined as the set of all equivalence classes
equipped with the norm
For more details concerning the characterization of this space, see e.g. [Citation23,Citation24].
We introduce the operator defined by
(3)
(3) where
is the symmetrical part of X. Then a has the following properties
a is continuous
, with
;
a is
-homogeneous, i.e.
(4)
(4)
a is monotone, i.e.
(5)
(5) holds for all
For more details, see e.g. [Citation25].
In the following subsections, we give some variants of Korn's inequality, Bogovski and de Rham's operators for our thin domain.
2.4. Korn's inequality
These estimates show how the constants depend on the parameter in the thin domain .
Theorem 2.1
Korn's inequality
There exist constants and
depending only on Ω and
such that
(6)
(6)
(7)
(7) for all
. These constants are such that
where
and
denote the best constants in the inequalities (Equation6
(6)
(6) ) and (Equation7
(7)
(7) ).
Proof.
See Appendix 1.
2.5. The Bogovski operator
The divergence operator is onto, see [Citation24, Theorem 5.4] where the case p = 2 was considered. The generalization to is straight-forward since the vector field
in [Citation24, Lemma 5.1] is of class
Related results can also be found in [Citation26–28]. However, the divergence operator is not one-to-one since
is always nontrivial. By collapsing the nullspace to zero we can construct an invertible operator
defined by
, for all
, which is one-to-one, onto and therefore A is an isomorphism. From Jensen's inequality it follows that
for all
, hence A is continuous. The Open Mapping theorem asserts that the inverse
of A is continuous. The operator B is called the Bogovski operator.
2.6. De Rham's operator
Since B is a continuous isomorphism, we can identify
where
is the annihilator of
defined as
and by the Riesz Representation theorem, we have
. The dual operator
of B is usually called De Rham's operator or the ‘pressure operator’. Note that
is also continuous with
. To emphasize that B depends on the domain
, we write
. Thus, to obtain a
-bound for the pressure we need to investigate how the operator norm
depends on ε.
Theorem 2.2
Let
denote the Bogovski operator. Then there exist positive constants
and
depending only on Ω and
such that
(8)
(8)
Proof.
See Appendix 2.
Remark 2.3
We would like to point out that the lower bound in the Theorem 2.2 for the norm of the Bogovski operator is not needed to prove the main result, it is included only to show that the norm as
and that an upper bound of the form
is the best possible estimate that can be obtained.
Theorem 2.4
De Rham's Theorem
Let be such that
Then, there exists a unique
, such that
Proof.
Since satisfying
for all
this implies that
, therefore, we can choose q in
such that
. Furthermore, by duality we have
Then, we define
, which implies that
, consequently,
i.e.
as desired.
3. Setting the problem and the main result
3.1. Power-law fluid
Let us suppose that is the velocity field,
is the pressure and
is the gradient of velocity. The rate of strain tensor
is defined by
The stress tensor
is defined by the following non-Newtonian constitutive law
(9)
(9) When the function μ (dynamic viscosity) is constant, we say that the fluid is Newtonian. In the present paper we study the flow of a power-law fluid, where the viscosity
is a nonlinear function. More precisely, we assume that
(10)
(10) which defines the rheology of the fluid, where
and
are constant parameters called power-law index and the consistency respectively. For more details see [Citation29, pp. 16–22] or [Citation30, p. 55]. The parameter p is dimensionless while
has units which depend on the value of p. When p = 2 we recover the case of a Newtonian fluid studied in [Citation2].
3.2. Boundary value problem
Let us consider stationary pressure-driven isothermal fluid flow in the Hele–Shaw cell . We assume that inertial and body forces can be neglected. Moreover, we assume that the fluid is incompressible and obeys the non-Newtonian constitutive relation (Equation9
(9)
(9) ). We model the flow by the following boundary value problem in
(11)
(11) where
and
are the unknowns,
is the outward unit normal of
and
, the external pressure, is a given function. We assume that
is defined on the whole domain, more precisely
, i.e.
depends only on the variable
.
3.3. Existence and uniqueness
In order to prove existence and uniqueness of solutions, we consider the problem (Equation11(11)
(11) ) in the general form
(12)
(12) where
and
are given vector functions, which are interpreted as body force and surface force respectively.
We use the standard methods of the calculus of variations to interpret the problem (Equation12(12)
(12) ) as a problem of minimization.
Lemma 3.1
Let J be the functional defined as
Then, there exists a unique minimizer
in
, i.e.
for all v in the admissible class
.
Proof.
See Appendix 3.
3.4. Weak formulation
Take a test function and apply the Divergence theorem to
, we have
Since v = 0 on
and
, from the definition of the stress tensor
we see that,
Thus, the weak formulation of problem (Equation12
(12)
(12) ) is
(13)
(13) for all v in
Theorem 3.2
Given and
, there exists a unique weak solution
of the problem (Equation12
(12)
(12) ) satisfying the weak formulation (Equation13
(13)
(13) ).
Proof.
By Lemma 3.1, there exists a unique minimizer which satisfies the Euler–Lagrange equation,
(14)
(14) for all
. Define
by
In this notation (Equation14
(14)
(14) ) becomes
From De Rham's theorem we deduce the existence and uniqueness of a pressure function
such that
This is the weak formulation (Equation13
(13)
(13) ) and therefore
is the unique weak solution of (Equation12
(12)
(12) ).
3.5. Main result
Taking f = 0 and in the weak formulation (Equation13
(13)
(13) ) we have that the weak solution
of (Equation11
(11)
(11) ) satisfies
(15)
(15) for all
Applying the Divergence theorem to the surface integral and using that v = 0 on
, yields
(16)
(16) for all
where
denotes the normalized pressure defined by
(17)
(17) Recall that, by assumption,
depends only of
. This implies that the third component in
is zero, i.e.
To give a precise asymptotic description of the system (Equation11(11)
(11) ), we need the definition of two-scale convergence for thin domains introduced by Marušić and Marušić-Paloka in [Citation1].
Definition 3.3
We say that a sequence , where
, in
two-scale converges to u in
provided that
(18)
(18) for all v in
and we write
Moreover, we say that
two-scale converges strongly to u if
(19)
(19) and we write
(strongly).
To characterize the limit velocity, we introduce the so called permeability function and the flow factor for a thin domain, both depending on the rheology of the fluid and the geometry of the domain.
Definition 3.4
The solution ψ of the boundary value problem
(20)
(20) where
is called the permeability function of Ω. That is, for each
,
is the solution of
where
is the p-laplacian in the variable y.
Definition 3.5
The flow factor of Ω is defined as
(21)
(21) where ψ is the permeability function, i.e. the solution of boundary value problem (Equation20
(20)
(20) ).
Lemma 3.6
For our geometry the permeability function ψ and the flow factor ϱ for Ω, are given by
(22)
(22) for all
and
(23)
(23) for all
where
and
are defined in (Equation1
(1)
(1) ).
Proof.
The proof of this lemma will be given in Section 6.
Remark 3.7
The smoothness of ψ depends on the smoothness of . From (Equation22
(22)
(22) ) we see that ψ is of class
, since h is a Lipschitz function on
. From (Equation23
(23)
(23) ) and the fact that h is bounded from below by the positive constant by (Equation1
(1)
(1) ), we deduce also that the flow factor ϱ and
belong to
.
Before we formulate the main result, let us introduce the following definitions that will be used throughout this paper: Let ϕ be a scalar function and be a vector function, then we define
,
,
and
as
and
Theorem 3.8
Main result
For each the boundary value problem (Equation11
(11)
(11) ) has a unique solution
such that
(24)
(24) where the constants
and
depends only on Ω and
. Let ψ be the permeability function defined by (Equation22
(22)
(22) ). Then, the following two-scale convergence holds,
where
is the limit velocity defined by
and
is the unique solution of the boundary value problem
(25)
(25) where ϱ is the flow factor defined by (Equation23
(23)
(23) ) and
is the outward unit normal to the surface of the obstacle.
We observe that the Dirichlet condition on for the velocity field in the original problem becomes a Neumann condition on
for the limit pressure and the Neumann condition on
for the stress tensor in the original problem becomes a Dirichlet condition on
for the limit pressure.
Furthermore, by contrast, if one imposes a non-homogeneous Dirichlet condition for the velocity field on one ends up with a Neumann condition on
(this is shown in [Citation18, equation (5.7)]). A major difference between the two boundary conditions is that the limit pressure is uniquely determined by the Dirichlet condition, whereas it is only determined up to an additive constant in the case of a Neumann condition. Moreover, under the Neumann condition the obstacle becomes a streamline of the y-averaged flow (see Lemma 5.4 (iv)). The limit velocity u is uniquely determined in both formulations and the equation
tells us that pressure variation in the thickness of Ω can be neglected. In the Newtonian case, p = 2, the flow factor is given by
and therefore, the pressure equation (Equation25
(25)
(25) ) becomes the classical coefficient in the Reynolds equation formulated in [Citation16]. This shows that our results can be extended to the context of lubrication theory, see e.g. [Citation19]. In addition, for future work, it would be interesting to allow the upper and lower surfaces to be in contact, i.e.
in some points
of ω, thus creating more obstacles in the domain. Then, the problem (Equation12
(12)
(12) ) becomes a singular Hele–Shaw flow problem and therefore, the limit problem (Equation25
(25)
(25) ) will be degenerate. For some results in the Newtonian case and
on
see [Citation1, Sec. 3.3.].
4. Estimates
To prove the main result (Theorem 3.8) we need uniform a priori estimates for the velocity, the monotone operator and the pressure with respect to the parameter ε.
Theorem 4.1
Velocity estimates
For the velocity field the following estimates hold
(26)
(26) where
and
are the constants in Theorem 2.1.
Proof.
Taking in (Equation16
(16)
(16) ) and using the fact that
we have,
(27)
(27) Applying the Hölder inequality in (Equation27
(27)
(27) ) and the Korn inequality (Equation6
(6)
(6) ) we find
Thus,
(28)
(28) This together with the first and second inequalities in the Korn inequality again in the left hand side, we obtain
Theorem 4.2
Monotone operator estimate
The monotone operator (Equation3(3)
(3) ) satisfies the following estimate
(29)
(29) where
is the constants in Theorem 2.1.
Proof.
Follows from the estimate obtained in the proof of Theorem 4.1.
Theorem 4.3
Pressure estimate
The normalized pressure satisfies the estimate
(30)
(30) where
and
are the constants defined in Theorems 2.1 and 2.2 respectively.
Proof.
Let be defined by
From (Equation16
(16)
(16) ) we see that
belongs to
, and by the Hölder inequality and the Korn inequalities (Equation6
(6)
(6) ) and (Equation7
(7)
(7) ) we obtain,
This together with inequality (Equation28
(28)
(28) ) it follows
whence, we deduce that
Let
in
be defined by the pressure operator, i.e.
This means that the unknown pressure function in (Equation11
(11)
(11) ) can be recovered via (Equation17
(17)
(17) ), i.e.
. Note that
depends only on the boundary values of
on
. From (Equation8
(8)
(8) ) in Theorem 2.2 we infer,
5. Two-scale convergence results
Here we prove some compactness results for two-scale convergence
5.1. Compactness results
Since we have
So, raising both sides to power
and using the Theorems 4.1, 4.2 and 4.3 yield
and
Thus, we can choose some constants C and
, which are independent of ε as
such that
(31)
(31)
Lemma 5.1
For any sequence with
in
satisfying the bounds (Equation31
(31)
(31) ), there exist a
with
, and a
such that, up to a subsequence
,
Proof.
For the sake of completeness we provide the proof which follows the same ideas as in [Citation1, Theorem 1]. Since (Equation31(31)
(31) ) holds, there exist a subsequence which is still denoted by
and
such that
for any
. Similarly, there exists
such that, up to a subsequence
for any
.
Since vanishes on
we have
(32)
(32) for all
such that
on
. Multiplying (Equation32
(32)
(32) ) by
and passing to the limit as
we see that
(33)
(33) Let us characterize the subsequence limit d, by considering it block by block, i.e.
Similarly the blocks of the matrix Φ are denoted as
Choosing
and
in (Equation33
(33)
(33) ) gives
We conclude that
and
. Next, choosing
and
in (Equation33
(33)
(33) ) gives
By definition of weak derivative we deduce that
and
. Thus,
Hence the trace of u on
is well defined by [Citation1, Lemma 4 (i)]. Applying the Divergence theorem to (Equation33
(33)
(33) ) we obtain
for all
such that
on
. We conclude that u satisfies the Dirichlet condition u = 0 on
. The convergence of the symmetrical part of gradient u follows by linearity. The two-scale convergence for the normalized pressure follows directly from the definition.
5.2. Two-scale convergence and the monotone operator
We introduce the monotone operator (Equation3(3)
(3) ) in the weak formulation (Equation16
(16)
(16) ), i.e.
(34)
(34)
Lemma 5.2
Given , let
be defined as
Then, we have the strong two-scale convergences
,
.
In particular, is an admissible test function in the two-scale sense.
Proof.
The gradient of is defined as
for all
. Therefore, the weak two-scale convergence
follows from Lemma 5.1 (ii). The weak two-scale convergence
is deduced from the homogeneity property of a defined in (Equation4
(4)
(4) ). Thus, it is enough to prove
(35)
(35) Then, by the Change of Variable theorem, and noting that
, we obtain
Multiplying by
and homogeneity property, we get
Passing to the limit as
, using the Dominated Convergence theorem and the continuity of a, we obtain (Equation35
(35)
(35) ).
Lemma 5.3
The sequence in
satisfies the bound (Equation31
(31)
(31) ). There exists
such that up to a subsequence
Moreover,
(36)
(36) and if (Equation36
(36)
(36) ) holds as an equality, then
(37)
(37) for all test functions v in
.
Proof.
In view of the monotonicity condition (Equation5(5)
(5) ), we have
(38)
(38) where
is a test function of the same form as in Lemma 5.2. This inequality yields
Multiplying this inequality by the factor
we find
(39)
(39) Let us analyze the integrals on the right hand side in (Equation39
(39)
(39) ) in terms of limits as
. The limit of the first integral exists by hypothesis and is given by
(40)
(40) The second integral is convergent by Lemma 5.1 (ii) and Lemma 5.2, as the integrand is a product of strongly two-scale convergent and two-scale convergent sequences and [Citation1, Lemma 2].
(41)
(41) Thus, we can estimate the inferior limit of the integral on the left hand side in (Equation39
(39)
(39) ), by means of limits (Equation40
(40)
(40) ) and (Equation41
(41)
(41) ), as
, obtaining
(42)
(42) and the inequality (Equation36
(36)
(36) ) is established by taking u = v.
Assuming that equality holds in (Equation36(36)
(36) ), we see from (Equation42
(42)
(42) ) that
After some rearranging, it follows that
(43)
(43) Fix
and set
with
in (Equation43
(43)
(43) ), we obtain
Dividing by t, using the continuity of a, and letting
give
Finally, replacing w by
, we deduce that
for all
.
5.3. Conservation of volume
In this subsection we investigate the conservation of the incompressibility of the flow when we let . To this end it is convenient to introduce the following space of functions.
Definition 5.4
Let be defined as the closure of
in the norm
Lemma 5.5
Let be a sequence as in Lemma 5.1 with the additional condition that
in
. Then any subsequential two-scale limit
satisfies
(44)
(44) and
(45)
(45)
Proof.
As in [Citation1, Proposition 4], the divergence free condition for can be stated as
for all scalar test functions
. It follows that
(46)
(46) for all
such that v = 0 on
. Multiplying (Equation46
(46)
(46) ) by
and passing to the limit, as
yield
hence
. From this and the no-slip boundary conditions on
we deduce (Equation44
(44)
(44) ).
By choosing v in (Equation46(46)
(46) ) which does not depend on y, i.e.
, such that v = 0 on
and passing to the limit, as
, we obtain
for all
such that v = 0 on
. This implies that the line integral on
depends only on
, whence by the Divergence theorem, we have
Since
we see that
It follows from [Citation22, Proposition 2] that the trace in the dual sense is the restriction of
on
, which is well defined as,
(47)
(47) Thus, (Equation45
(45)
(45) ) is proved.
Lemma 5.6
All functions satisfy
,
v = 0 on
,
,
on
Proof.
The proofs of –
follow by the same arguments used in the proof of Lemma 5.1 and (iii)–(iv) can be deduced by following the ideas used in the proof of Lemma 5.5.
5.4. Momentum equation
We end this Section, by proving the main result of this paper. Indeed, we will derive a lower-dimensional model for the flow from the Stokes Equation (Equation11(11)
(11) ).
Lemma 5.7
The thin film equation
Let and
be as in Lemma 5.1 and let ζ denote a subsequential limit of
as in Lemma 5.3. Then the first component
of u and
satisfy
(48)
(48)
Proof.
By choosing test functions in (Equation34(34)
(34) ), of the form
we have
Passing to the limit as
, we obtain
(49)
(49) for all
. Choosing
, we see that
or in other words,
This shows that
does not depend of variable y, i.e.
.
Furthermore, multiplying (Equation34(34)
(34) ) with
, choosing
and using that
in
, we can assert that
(50)
(50) Using
, we can rewrite the first term in (Equation49
(49)
(49) ) as
we deduce
(51)
(51) which holds for all
(see Definition 5.4), since
is dense in
. Then taking v = u in (Equation51
(51)
(51) ) and using Lemma 5.5, i.e. (Equation44
(44)
(44) ) and (Equation45
(45)
(45) ), we have
(52)
(52) From (Equation50
(50)
(50) ) and (Equation52
(52)
(52) ) we deduce that equality holds in (Equation36
(36)
(36) ). Thus (Equation37
(37)
(37) ) implies that (Equation49
(49)
(49) ) can be written as
(53)
(53) From the definition of the operator a, i.e. equality (Equation3
(3)
(3) ) and (Equation44
(44)
(44) ), we obtain
Inserting this into (Equation53
(53)
(53) ) yields
(54)
(54) for all
. This is the weak formulation of the boundary value problem (Equation48
(48)
(48) ).
6. Regularity of pressure and pressure equation
In this section, we will prove some properties of solutions to the lower-dimensional problem. Indeed, we will prove a regularity result for the pressure
and a nonlinear relation between the velocity u and the gradient of q. However, to be able to do this we first present the proof of Lemma 3.6.
Proof
Proof of Lemma 3.6
We note that the boundary value problem (Equation20(20)
(20) ) is formally an ordinary differential equation in variable y, with the parameter
, i.e.
By formal integration, we see that there exist a scalar function
such that
From the observation,
it follows that
Whence by integration once again the last equality becomes
(55)
(55) From the boundary condition
and
, we see that
where
is defined in (Equation1
(1)
(1) ) and therefore we get (Equation22
(22)
(22) ) from (Equation55
(55)
(55) ). To obtain the flow factor
, we integrate the permeability function on
, i.e.
Observe that the last integral can be written as
Lemma 6.1
Let be the permeability function defined by (Equation22
(22)
(22) ) and
be any solution of the boundary value problem (Equation48
(48)
(48) ). Then
belongs to
, with
on
.
Proof.
To show the -regularity for
, in view of the Lemma 5.4, we take test functions
of the form
where Θ is any vector field in
such that
on
, in (Equation54
(54)
(54) ), we obtain
where ϱ is the flow factor defined in (Equation23
(23)
(23) ). Since
, we can define a linear functional with
determined by
i.e.
Thus,
(56)
(56) where C is the a positive constant given by
Furthermore, since the functions ϱ and
belong to
, (see Remark 3.7) we can replace Θ with
in (Equation56
(56)
(56) ) which gives
for all
such that
on
, and therefore, in view of Lemma A.5 in Appendix 4, we conclude that
belongs to
with
on
.
Remark 6.2
Note that with
on
.
Lemma 6.3
Let be any solution of (Equation48
(48)
(48) ). Then u is uniquely determined by
, more precisely
(57)
(57) where
and ψ must satisfy the scalar ordinary differential Equation (Equation20
(20)
(20) ).
Proof.
In view of Lemma 6.1, the problem (Equation48(48)
(48) ) can be written as an ordinary differential equation in the variable y, i.e.
(58)
(58) where
is a parameter and
can be regarded as a constant vector in
.
We want to show that (Equation57(57)
(57) ) is a solution of the form
Indeed,
As in [Citation11, Propostion 3.4], by formal integration, we see that there exist a vector function
such that
Thus,
whence by integration once again and taking into account the no-slip condition at
we obtain the velocity field
(59)
(59) Since
on
, as well, we find that
Observe that
is the gradient with respect to C of a the integral functional
with convex Lagrangian
Consequently using analogue of Lemma 3.1 we see that C is uniquely determined and the unique solution to the Euler equation is
where
is defined as in (Equation1
(1)
(1) ) and so
is uniquely determined, and finally (Equation57
(57)
(57) ) is deduced from (Equation59
(59)
(59) ), using (Equation55
(55)
(55) ).
To summarize this section, from Lemma 5.5, we deduce that q satisfies the boundary value problem (Equation25(25)
(25) ), hence q in
is uniquely determined by
. Thereafter it follows from Lemma 6.3 that u in
is also unique. Since all subsequential limits u and q are the same, we conclude that the whole sequence of solutions
to the problem (Equation11
(11)
(11) ) two-scale converges to
.
Acknowledgements
We thank the anonymous referees for many valuable comments and suggestions which improved the final version of this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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Appendices
Appendix 1. Proof of Theorem 2.1
In this appendix, we generalize the Korn inequalities proved in [Citation2], where the case p = 2 was considered. Let □ be the cube in defined by
and let
be the cube defined by
. Moreover we define the planes
and
Lemma A.1
Let and
denote the best constants in the inequalities
(A1)
(A1)
(A2)
(A2) for all
such that v = 0 on
. Then
(A3)
(A3)
(A4)
(A4) for all
such that v = 0 on
.
Proof.
Suppose and define
in
as the function which satisfies
Making the change of variable
and applying (EquationA1
(A1)
(A1) ) to
we obtain
which implies (EquationA3
(A3)
(A3) ). Similarly, but using (EquationA2
(A2)
(A2) ) instead of (EquationA1
(A1)
(A1) ), we obtain (EquationA4
(A4)
(A4) ).
To estimate the constant in the Korn inequality under anisotropic scaling of the domain, we use an extension operator in combination with a covering argument. In view of assumptions in (Equation1(1)
(1) ) we can deduce that
is also thin domain, so that we can consider the unscaled domain Ω defined as
Indeed, due to the Dirichlet condition on
, we can extend any
by zero to
, which is confined between the hyperplane
.
Finally, we can assume a sufficiently large domain of Ω given by
such that
A.1. Extension operator
By consideration above, we prove the result
Lemma A.2
There exists a bounded linear operator
such that:
Ev = v, for all v in
for some positive constant
which depends only on ω, it holds
for all v in
.
Proof.
Since Ω is a bounded open subset of with Lipschitz boundary, there exists a continuous linear extension operator E from
into
, such that
See for instance, [Citation31, p. 25. Theorem 1.4.3.4], whence
follows. To show
, first we note that
Applying this inequality to Ev yields
By the continuity of the extension operator, there exists a constant
, which depends only on Ω, such that
From the Korn inequality for
(see [Citation24, Theorem 4.8]) there exists a positive constant
depends only in Ω,
Hence,
where
, and thus, we conclude the proof of the lemma.
A.2. Covering argument
For , let
denote the square in
of length ε centered in z, i.e.
For each
, there exist a finite collection of points
in
such that
Define the cube
Then
for all
. Since each cube
is a translation of the cube
we can use the inequality (EquationA3
(A3)
(A3) ) locally in Lemma A.1.
Proof
Proof of Theorem 2.1
Suppose . To use the extension operator in Lemma A.2, let
in
be defined by extension
From Lemma A.1
, the covering argument and inequality (EquationA3
(A3)
(A3) ) of Lemma A.2 it follows that
or
This together with Lemma A.2
yields
This proves the first inequality in Theorem 2.1. The second inequality is proved in a similar way.
Appendix 2. Proof of Theorem 2.2
We will need a simple scaling argument to prove the upper bound of norm of the Bogovski operator. To obtain a lower bound we consider an eigenvalue problem.
Simple scaling argument
For any , let
denote the linear scaling operator defined by
where
and similarly for
. We note that
is invertible with
. Using a simple change of variables we show that
maps
onto
with
Moreover,
maps
onto
such that
In particular, we define
acting on
, as
It is readily checked that
is well defined in
.
Lemma A.3
Let
denote the Bogovski operators for Ω and
respectively. Then
(A5)
(A5)
(A6)
(A6)
Proof.
The statements (EquationA5(A5)
(A5) ) and (EquationA6
(A6)
(A6) ) are equivalent, so it suffices to prove the first one. Given
, suppose that u in
satisfies
Then
satisfies
Inverting, the last relation we see that
Lemma A.4
For all v in and all
we have
where
is the constant in Theorem 2.1.
Proof.
From
we obtain
Integrating this inequality over
gives
By the Korn inequality (Equation7
(7)
(7) ) and some rearrangement, we take infimum on both sides, the lemma is proved.
Proof
Proof of Theorem 2.2
Assume . From Lemma A.3 and Lemma A.4 we deduce,
which implies the upper bound in (Equation8
(8)
(8) )
To prove the lower bound, we consider the eigenfunction ψ in
corresponding to the eigenvalue problem
where
, and λ is the smallest eigenvalue of
on ω. It is well know that λ can be characterized as the minimum of the Rayleigh quotient
see [Citation32] for more details. Now suppose u in
satisfies
Since u = 0 on
and
on
we have
(A7)
(A7) Using
and applying the Korn inequality (Equation6
(6)
(6) ) in the last integral on right hand side of (EquationA7
(A7)
(A7) ) we deduce that
for all
such that
. Consequently, taking the infimum on the last integral we obtain
Estimating the right hand side with the norm of the Bogovski operator, we see that
whence, we deduce,
This proves the lower bound in (Equation8
(8)
(8) ) with
Appendix 3. Proof of Lemma 3.1
Proof
Proof of Lemma 3.1
For fixed , consider the minimization problem
(1) Let us begin to show that is weakly coercive on
. By the Young inequality with
for appropriate r>0, the Trace theorem (see [Citation23, p. 153, Sec. 2.5]) and the Korn inequality (Equation6
(6)
(6) ), we obtain
where C is a constant from the Trace inequality in the Trace theorem. We choose
, to obtain that
where K is a positive constant which depends on
. Hence,
if
, this proves the weak coercivity of J.
(2) Choose any minimizing sequence such that
Then the weak coercivity condition implies that
is bounded in
,
, and since,
is reflexive there exist a subsequence
and a function
in
such that
Since
is a closed subspace of
, then
is weakly closed i.e.
and so
(3) The weak lower semi continuity of J implies that
therefore
is a minimizer of J over
.
(4) We turn to deriving the corresponding Euler–Lagrange equation. Let in
be a minimizer of J and pick
. Then
belongs the admissible class
for all
. Let
, then
(A8)
(A8) where,
Note that,
Consequently, by Dominated Convergence theorem, the limit (EquationA8
(A8)
(A8) ) exist as
, and thus,
Since
for all t it must hold that
, hence the minimizer
satisfies the weak form of the Euler–Lagrange equation
( 5) Finally, let us prove the uniqueness of the minimizer
. Let
and
be two different minimizers, then by Korn's inequality (Equation6
(6)
(6) ) we have
on a set of positive measure. Since J is strictly convex it follows that
which contradicts that
is a minimizer, hence
.
Appendix 4. Characterization of the normalized limit pressure ![](//:0)
![](//:0)
Following [Citation33, p. 153. Proposition IX.3], we present a characterization of the subspace of to which the normalized limit pressure
belongs.
Lemma A.5
The following statements are equivalent
and there exists a constant
such that
for all
such that
on
;
with
on
.
Proof.
If with
on
, then by the Divergence theorem we have that
for all
such that
on
. And therefore
Hence,
where
. By density [Citation22, Lemma 2] this holds for all
such that
and
on
.
Conversely, assuming that , we can define the gradient of
as the linear functional
, given by
defined on the subspace
. By assumption
so,
is bounded. According to the Hahn–Banach theorem the
can be extended to a bounded linear functional on
. Thus by the Riesz representation theorem, there exists
such that
Therefore,
for all
such that
on
. This implies that
and therefore
. Now by Divergence theorem we have
for all
such that
on
. We conclude that
with
on
.