Newtonian and Non-Newtonian Fluids through Permeable Boundaries

We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface Γ of a domain Ω, where Ω ⊂ R. Γ is defined as a smooth two-dimensional (at least class C) manifold in Ω. The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface Γ. We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normaln, where the tangential components are assumed to be zero.These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of Ω; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed.


Introduction
The flow of incompressible Navier-Stokes fluids and fluids of second grade through permeable boundaries and past porous walls has been studied under various conditions.The equation of motion for incompressible flows in Newtonian fluids (Navier-Stokes equations) under no-slip boundary conditions has been studied extensively from many perspectives.Since the pioneering papers of Leray [1][2][3] and Hopf [4] questions of the existence, stability [5,6], and uniqueness of both classical and weak solutions have received more than their fair share of attention.
Unlike Newtonian fluids, fluids of second grade (and other non-Newtonian species) have the property of developing "normal stresses differences" at boundaries.It was shown, for example, by Berker [16] that if an incompressible flow of a fluid of grade two satisfies the homogeneous Dirichlet boundary condition.The stress at the boundary is given by t = (− + || 2 )n + [ + 2  ] ∧ n, where n is the unit exterior normal to the boundary and  = ∇∧k is the vorticity.The wedge denotes a vector product.Thus there is a normal component of stress at the boundary in addition to the pressure.The question becomes what governs the flow across the boundary?Possible ways of circumventing this question may be to "prescribe" the normal component of the velocity field at the boundary or to prescribe mass or momentum flux.The prescription of shear stress has also been suggested.( [16,17]).Nonlinear or non-Newtonian fluids are fluids like molten metals, multigrade oils, printing inks, paints, suspensions, polymer solutions, molten plastics, blood, protein solutions, and ice [18].These fluids cannot be described by the above model.The study of these interesting substances has proved to be very important with the growth of the polymer and plastics industry over the last four decades.Consequently, an interest has arisen to study the flow of these nonlinear fluids and, in the case of this model, second-grade fluids, through permeable boundaries.The boundary conditions alone in such circumstances are an interesting topic for study.Works by Berker [16] and Rajagopal and Gupta [19] can be mentioned in this regard.

Mathematical Problems in Engineering
In this study we shall provide an alternative approach through the formulation of "dynamics at the boundary, " the idea being that the normal component of velocity at the boundary is viewed as an unknown function which satisfies a differential equation intricately coupled to the flow in the region "enclosed" by the boundary.
A glimpse of the history of the research on non-Newtonian and Newtonian fluids around porous boundaries is given in Section 2. Notation and definitions precede Sections 4 and 5 which deal with the constitutive equations and the modelling of permeability.In Section 6 the expressions on the interface Γ are given.The alternative model is studied and the stability and uniqueness are proved in Section 7. Section 8 concludes the study and further explorations are discussed.

Backgroud
Berker [16] studied the two-dimensional creeping flow of a second-order fluid with nonparallel porous walls.An additional velocity boundary condition was needed.The other conditions they used were due to the usual no-slip conditions.This additional velocity boundary condition was to prescribe the rate of shear at the wall.The problem was then solved numerically by a standard routine.
In 1989 Rajagopal and Kaloni [20] wrote remarks on boundary conditions for flows of fluids of the differential type.Rajagopal [21] discusses a lot of related issues.Rajagopal and Gupta [19] studied the flow of an incompressible fluid of second grade past an infinite porous plate subjected to either suction or blowing at the plate.They studied fluids modelled by No assumptions were made about the material moduli  1 and  2 .For the boundary value problem they considered, it was found that the velocity distributions do not depend on the normal stress modulus  2 , but the pressure does.They found that it was possible to produce an exact solution which is asymptotic in nature for both "suction" and "blowing" at the plate if the material modulus  1 > 0. For  1 < 0, they found that such solutions could not exist in the case of blowing, a result which was in keeping with the classical incompressible fluid.Fosdick and Rajagopal [22] have shown that the model (1) whose material modulus  1 < 0 exhibits anomalous behaviour was not to be expected of any fluid of rheological interest (also see [23]).Proudman studied an example of steady laminar flow at a large Reynolds number [24].
Beavers and Joseph [25] studied the flow of a Newtonian fluid over a porous surface in 1967.They found that if the governing differential system was not to be underdetermined, it was necessary to specify some condition on the tangential component of the velocity of the free fluid at the porous interface.It is usual in these analyses to approximate the fluid motion near the true boundary with an adherence condition for the tangential component of velocity of the free fluid at some boundary.Because of a certain ambiguity which is implied by the notion of a "true" boundary for a permeable material, it was found useful to define a nominal boundary.They fixed a nominal boundary by first defining a smooth geometric surface and then assuming that the outermost perimeters of all the surface pores of the permeable material are in this surface.Thus, if the surface pores were filled with solid material to the level of their respective perimeters, a smooth impermeable boundary of the assumed shape would result.This definition is precise when the geometry is simple (planes, spheres, cylinders, etc.) but may not be fully adequate in more complex situations.Beavers and Joseph's [25] experiment was designed to examine the tangential flow in the boundary region of a permeable interface.The results of this experiment indicate that the effects of viscous shear appear to penetrate into the permeable material in a boundary layer region, producing a velocity distribution similar to that depicted in the following figure.The tangential component of the velocity of the free fluid at the porous boundary can be considerably greater than the mean filter velocity within the body of the porous material.
In Figure 1 the plane  = 0 defines a nominal surface for the permeable material.The flow through the body of the permeable material, which is homogeneous and isotropic, is assumed to be governed by Darcy's Law.Read more of the status on Darcy's Law in [26].In the absence of body forces Darcy's Law may be written as  = −(/)(/), where  is the "permeability" of the material and  is the volume flow rate per unit of the cross-sectional area.As such,  represents the filter velocity rather than the true velocity of the fluid in the pores.The measured pressure gradient is denoted by /.

Basic Notation
We work in Euclidean 3 space.The following notation will be used throughout: denotes the Euclidean norm.
k ∧ u := denotes the usual vector product of the vectors k and u If A and B are second order tensors we shall use the notations A : B = ∑ 3 ,=1     and |A| 2 = A : A. Let Ω ⊂ R 3 be a bounded domain with a smooth (at least C 2 ) boundary Γ.Let n = n() denote the unit exterior normal to Γ at .We shall be concerned with smooth vector fields k = k() defined in Ω such that on Γ it has the form   k() = −()n(), where   is the trace operator denoting boundary values and  is a smooth scalar field defined on Γ. Associated with ∇k we define the symmetric and skew-symmetric tensors A and W as A = A(k) = ∇k+(∇k)  and W = W(k) = ∇k−(∇k)  , where (∇k)  denotes the transpose of the gradient of k.The rate of deformation tensor is related to A by D(k) = (1/2)A(k).We note that if k is solenoidal (∇⋅k = 0) then trace A(k) = 2∇⋅k = 0 and, for any vector a, W(k)a = ∧a, where  = ∇∧k denotes the vorticity associated with k.

The Constitutive Equations
The stress tensor for the linear viscous Newtonian model is T = −I + (∇k + (∇k)  ), with  as the pressure,  as the coefficient of viscosity, and k as the velocity of the fluid.This model describes the flow of fluids like water and other similar fluids.Lamb [27] and Ladyzhenskaya [28] wrote mathematical theories on viscous incompressible flow.
Fluids of a differential type [29][30][31], of which Rivlin-Ericksen fluids are a subclass, are depicted by a popular nonlinear model.Fluids of complexity n form an important subclass of the fluids of a differential type.For incompressible fluids of complexity  the Cauchy stress tensor is of the form T = −I + F(A 1 , . . ., A  ).The pressure  is not a thermodynamic variable and the term −I reflects Pascal's law, which is inherent to all fluids.A 1 , . . ., A  are the first  Rivlin-Ericksen tensors [21] defined recursively by (3) Fluids of grade n are examples of fluids of complexity .The stress tensors for fluids of grades 1 and 2 respectively, are assumed to be of the form T [1] = −I + A 1 , T [2] = T [1] where  and   are material coefficients (possibly temperature-dependent).
For incompressible fluids of second grade, the stressdeformation relation then becomes T = T [2] = −I + A +  1   A where  and k are the pressure and the velocity fields.Here  is the coefficient of viscosity and  1 and  2 are material coefficients or "normal stress moduli." In this case A = A 1 .
To use the relation (5) for the modelling of a fluid, the fluid has to be compatible with thermodynamics in the sense that all flows of the fluid must satisfy the Clausius-Duhem inequality, and the assumption must be made that the specific Helmholtz free energy is at a minimum when the fluid is in equilibrium.Under these assumptions,  1 and  2 [32] must satisfy Considerations of stability of the rest state require the assumptions  and  1 to be nonnegative; that is,  > 0,  1 > 0. See [32].Under assumption (6), which we shall follow throughout, the form of the stress tensor T given in (5) reduces to a more compact expression.To obtain this we note that ∇k = (1/2)(A + W) and (∇k Therefore, by ( 5) and ( 7) where we have set  1 = .

Modelling of Permeability
We study the motion of fluids around and through a fixed porous container filled with the same fluid.The interior of the porous container is an open bounded set Ω ⊂ R 3 and the porous boundary, Γ, is smooth.The surrounding fluid domain, Ω  , is bounded and its outer boundary is denoted by Γ  .The exterior normal to Ω on Γ is denoted by n. Figure 2 illustrates the situation where the curvature of the boundary Γ of Ω is nonnegative.Permeability of the walls of the container is described by assuming that at the boundary Γ the flow k has the direction of the normal: The velocity component  is treated as an unknown and an evolution equation has to be found for it.We model the surface Γ as having an effective area measure  which has a density function () with respect to the area measure .
Thus  = ().The effective area through which fluid can permeate is not more than the surface area and therefore 0 ≤ () ≤ 1 for any  ∈ Γ.If () ≡ 0, the wall is impermeable and if () ≡ 1, there is no wall.In order to obtain expressions for mass and momentum in a boundary patch Γ  , we let the patch be heuristically represented by a volume  built from copies of Γ  (Figure 3).This is in line with the Beavers-Joseph thinking which was discussed before.For this volume we set up a coordinate system consisting of a "radial part" , which has the direction of the normal vector n, and a "surface part" made up by vectors tangential to Γ  .For the mass of  we obtain where  is some measure of thickness.With the aid of these concepts we introduce the surface density of the fluid at  ∈ Γ as where  is the volume density of the fluid.
To obtain the equation of motion for fluid in the boundary, we assume that the rate of change of linear momentum in the boundary is explained by stress forces at both sides of the boundary.
Let T and T  denote the stress tensors at the sides of the boundary facing Ω and Ω  , respectively, and let P and P  denote the transfer-of-momentum tensors on the two sides.On an arbitrary boundary patch Γ  ⊂ Γ the law of conservation of linear momentum is stated in the following way: with  as defined in (11), and it follows that From ( 13) we have ()  (, ) + n ⋅ [P − P  ]n = n ⋅ Tn − n ⋅ T  n.In the domain Ω the momentum flux tensor is given by P = k ⊗ k.In accordance with this, we shall take P =  2 n ⊗ n at the boundary.The tensor P  will be taken as zero.
We take T  = ℓI to obtain from ( 13) From the incompressibility of the flow in Ω it follows that

Expressions at the Interface
In order to obtain expressions for the stress tensors T and T  as well as the acceleration at the boundary through which only normal flow occurs, we obtain a formal expression for the symmetric tensor A on a surface which is immersed in fluid.We shall eventually use these expressions in postulating the form of T and T  and in formulating a boundary condition which expresses zero tangential acceleration at a wall.We consider a smooth vector field k(x) defined on a domain Ω ⊂ R 3 and a smooth two-dimensional (at least class  2 ) manifold Γ ⊂ Ω so that k and ∇k are defined on Γ.Let n(x) be the unit normal to Γ at the point x ∈ Γ.
At any point x on Γ we consider two orthogonal curves  1 and  2 in a neighbourhood of  parametrised by arc lengths  1 and  2 , respectively.Let  1 and  2 be the unit tangents to the principal normal curves at a point on the surface.For local coordinates we use the orthogonal system formed by  1 ,  2 , and n.Under the convention that  1 ∧ 2 = n we have n∧ 1 =  2 and n ∧  2 = − 1 .Let  1 and  2 represent the principal curvatures at a point on the surface and let  =  1 + 2 denote the mean curvature.

Assumptions
(1) We shall assume throughout that the surface density is bounded and bounded away from zero; that is, there exist constants  and  such that Also, we assume  ∈  ∞ (Γ).
(2) Apart from the smoothness of Γ we make two additional assumptions regarding the shape of Ω; namely, that the curvatures  1 ,  2 , and  are constrained in the following way: (a) There exist constants  and  such that (b) There exists a constant  such that Note that these assumptions allow cases where  1 and  2 can be of opposite signs.The Frenet-Serret [34][35][36] formulae in this case, providing that there is no torsion, are then The surface gradient ∇  of a scalar function  may be written as where the trace operator  1 denotes the normal derivative.Also consider If f is a vector field defined on Γ, the surface gradient ∇  is defined as the tensor Surface divergence and surface curl are defined as The relationship between the surface operators and the volume operators for a function defined in Ω is given by We use ( 20)-( 25) to prove more important results to make the calculations easier.

Lemma 2.
Let  1 and  2 be two orthogonal unit tangential vectors and let n be the exterior unit normal vector to Γ. Let , , and  be scalar functions; then Proof.Consider the following: We shall apply the expressions above to k.By the Frenet-Serret formulae (torsion is zero) (/ 1 )(  k) = −(/ 1 )n − (n/ 1 ) = −(/ 1 )n +  1  1 , and, similarly, The transpose is given by To find an expression for A at Γ, we need an expression for ∇k on the boundary: Although we know that the divergence of k will be zero, it is helpful to observe that  =   (∇⋅k Hence We proceed to find expressions for   (k ⋅ ∇)k,   (∇k), and   (∇k)  .
We know that  ∧ n = W(k)n = (n ⋅ ∇)k − (∇k)  n, and Therefore, Multiply (34) with − to obtain From (31) we now obtain The transpose is Thus we have Let us define the symmetrical tensors M and N by with a tangential vector.Then, for a vector field of the form k = −n on Γ, from (38) we have In local coordinates we have the representations If ∇ ⋅ k = 0, it follows that trA = 0, which is in line with incompressibility.
We would further like to obtain expressions for the terms n ⋅ Δk, n ⋅ [(k ⋅ ∇)A]n, and n ⋅ [AW − WA]n on the boundary Γ.

Lemma 3.
Let n be the exterior normal to the boundary Γ, k ∈ D, and A = −2M + N with M and N as defined in (39).We assume that ∇ ⋅ k = 0 and  o  ∧ n = 2∇  , which implies that where   denotes the Gauss-curvature.
Proof.(a) We have chosen  1 ,  2 , and n so that  1 ∧  2 = n.In view of the incompressibility and the fact that there is zero tangential velocity (b) Consider the tensor ∇⋅A built from "row vectors" with (e 1 , e 2 , e 3 ) a basis for R 3 .Then Here we used the fact that n = −n.Determine n ⋅ [∇  ⋅ ] term by term to obtain denotes the Gauss-curvature and is bounded by assumptions ( 17) and ( 18).Hence The term we use in the proof of (57) is therefore Here we make use of the additional boundary conditions (52) and (55) and the fact that Wn =  o  ∧ n to obtain that (51)

Explicit Form of the Dynamic Boundary Condition.
It is shown that for a smooth two-dimensional manifold Γ contained in a domain Ω ⊂ R 3 the following is true for a vector field k which is of the form k = −n on Γ: where M and N are defined in (39).If k is solenoidal, as in the case under consideration,  = 0.A straightforward application of Stokes' theorem shows that  is tangential to the boundary, which implies that  is tangential to the boundary.Indeed, let Γ  be any patch of the surface Γ; then where  is a vector tangential to the boundary.Now if f =   k = −n, then ∫ Γ  f ⋅ = 0, and that implies that ∫ Γ  (∇∧k)⋅ n = 0 for all Γ  ⊂ Γ, which in turn implies that (∇∧k)⋅n = 0.
In the problem under consideration we shall assume that the "rate of deformation" tensor A has precisely the form (52) on the boundary Γ with n the unit exterior normal (the traditional rate of deformation is defined as D = (1/2)A).
We shall consider a kinematic boundary condition, which has a physical meaning in that there are no tangential components of deformation at the interface boundary.This concerns the form of the tensor N.
Towards this, we observe from (52) that It follows from (54) that there are no tangential components of deformation at Γ if and only if  = 0; that is, This is the kinematic boundary condition.The various terms in n ⋅ Tn, with T on a surface Γ, given by ( 8), may be expressed as follows (see Lemma 3): Guided by these expressions and ( 8), we assume that, at Γ, For the stress tensor T  in the fluid exterior to Ω we assume that n ⋅ T  n = ℓ().This amounts to the situation where the fluid in Ω  is at rest.As a result we have From ( 13), (14), and (60) we obtain and  = .Equation ( 61) is the explicit form of the dynamic boundary condition.

An Alternative Model: Problem A
Although it was possible to prove stability and uniqueness for the original model (see [33,37]), we could not find a way to a possible proof of existence for a classical solution.In this chapter we describe an alternative model which displays all the properties of the original problem with respect to stability and uniqueness.
In the alternative model the dynamics at the boundary are formulated by assuming a "shear flow" of the form with  1 and  2 as the surface parameters (like arc length).It is assumed that the "body force" acting on the shearing fluid at the boundary is proportional to the difference between the pressures    and ℓ().Under these assumptions the equation governing the evolution of  is where   k = −n, and  is the resulting pressure through the boundary with thickness .Δ  is the Laplace-Beltrami operator (Δ  = ∇  ⋅ ∇  ) and ∇  denotes the surface gradient.
The parameter  has the physical dimension of length and may be thought of as the "thickness" of the "shear layer" (see [38], Sect 123, p. 506).Equation ( 63) is derived by calculating the stress tensor for a shear flow and noticing that terms of the form k * ⋅ ∇  vanish.The term  −1 ℓ() may be left out since, as before, it disappears when projections are taken.The kinematic boundary condition is still imposed.
7.1.Definitions.All spaces of vector fields are denoted by boldface letters.
(5) For the deformation we use the following notation for the norm and scalar products: D is a closed subspace of H 2 (Ω).Elements of D satisfy the kinematical boundary conditions (55).
The associated scalar product is According to assumption (16) this is equivalent to the usual L 2 metric.It is assumed that the function  ∈  ∞ (Γ).
(9) For the purpose of stability and uniqueness we define the following norm: (10) We shall deal extensively with the energy associated with fluids of second grade defined for the purpose of this study by with  1 = ( − 2) and  2 = .Ẽ1/2 V is evidently a norm on H 1  (Ω).We shall refer to the quantity Ẽ1/2 V as the energy norm of k.
(11) The constant , which appears in inequalities, denotes a generic positive constant.Sometimes it is necessary to indicate the quantities on which a constant depends in brackets or by a subscript.

Important Identities
Identity 1.For any symmetric tensor A and any arbitrary tensor B, we have A : Proof.Consider the following: Expressions are necessary for inner products of the form (  F, F) Ω , where F is either a vector or a second order tensor.  =   + k ⋅ ∇ is the material derivative.In order to obtain simple expressions for the scalar product, we notice that if ∘ denotes either the usual scalar product or the "colon" product, then provided ∇ ⋅ k = 0. Hence the following identity.
Identity 2. For any smooth vector or tensor quantity F(, ) and any v ∈ D, we have Proof.By the divergence theorem Later in this study we shall employ the energy method.It will become necessary to use the various boundary conditions in order to prove stability.The following is important to obtain the required results.
Proof.Integrating by parts and using the fact that k is solenoidal We note that, in particular for k ∈ D, the imbedding of H 2 (Ω) in the space of bounded continuous functions makes the choice  = |k| 2 possible, and it follows from Identity 3 that

Mathematical Problems in Engineering
For k ∈ D we may also choose  = |A(k)|2 , and it follows that since N = 0 on D. The following will be of immediate importance.
Thus, if the curvature  is positive everywhere on Γ, it becomes apparent that A(k) = 0 if and only if k = 0. Identity 5.For any bilinear form  on a Hilbert space , we have, for any v, w ∈  and with u = v − w that (v, v) − (w, w) = (u, v) + (w, u).Identity 6.Let f and g be tensor fields of the same order and let ∘ denote the "scalar product" in such tensor fields.For v ∈ D it is true that Proof.Consider the following: (82) and the result follows.
The following two lemmas are important in establishing a Poincaré inequality.

Lemma 5. The bilinear forms
.  1 and  2 are positive constants.
Proof.For u and k ∈ H 1  (Ω) and by ( 16), (17), and the Schwartz inequality Proof.From the smoothness of Γ (which is always assumed), it follows that the embedding  : [40].From the boundedness and coerciveness proved above it follows that there exists a smallest eigenvalue  and associated eigenfunction u ∈ H 1  (Ω) for which (u, u) = 1 (see [34]): > 0, for if it is zero, it follows that u = 0, which cannot be.It follows from (90) that for any k ∈ H 1  (Ω) the inequality holds and that  is the largest such constant.Finally, we set  = 2/.
Remark 8.It is now easy to see that ‖A(⋅)‖ Ω is a norm on H 1  (Ω).
In fact, we have the following lemma.

Lemma 9. For all
to both sides of the inequality (89): From the definition of the energy norm it is clear that and the result follows.
From Lemma 7 it is clear that these are the best estimates of this form.
Lemma 10.The norms ‖A(k)‖ Ω and Ẽ1/2 V are equivalent to the norm in the Sobolev space H 1 (Ω).
Proof.From (83) and (89) it follows that The addition of the two inequalities above yields Let  = min (1, /); then Equation (83) yields and from the Trace theorem it follows that From (92) it is evident that the energy norm is equivalent to the norm ‖A(k)‖ Ω .
Remark 11.From the above lemma we may claim from the embedding H 1 (Ω) → L 3 (Γ), [40], that there exists a constant  > 0 such that

Stability and Uniqueness for the Model Problem
We now derive an energy identity for the solutions of (101).Take the L 2 (Ω), scalar product with k on both sides of (101) 1 .This produces According to the formulation of the original problem on the boundary where () = n ⋅ Tn, we obtain From (101) 2 we obtain Note that here we have to make the assumption that  − 2 > 0, which gives us a restriction on .We define a parameter It is now clear that stability can only be proved under the assumption that  2 ∈ (0, 1/2).
The Poincaré inequality (see [39]) states that there exists a smallest constant  such that ‖‖ 2 0,Γ ≥ ‖∇  ‖ 2 0,Γ .Applying the Schwartz inequality and the above Poincaré inequality, we obtain   ẼV () ≤ −‖A(k)‖  The uniqueness of the solution of Problem A is treated in the same way as the uniqueness of the solution of the original problem (see [33]).

Conclusion
An extensive study was conducted to find expressions for the stress tensors of Newtonian and non-Newtonian fluids at a permeable surface.We employed the Serret-Frenet formulae exactly for this reason.Stability of the rest state and uniqueness were proven for a special case where a shear flow was taken into account.
These results proved to be valuable in applications for the study of blood flow, where they were applied to model the permeability of special capillaries in the formation of cerebrospinal fluid [41,42].Here the authors have presented a mathematical model of the flow of blood through the permeable boundary of a blocked choroidal capillary in which the parameters could be controlled.The blood plasma was modelled as a Newtonian fluid and the nonlinear Stokes equations were supplemented with a boundary condition at the permeable interface of the specialized capillary.The existence of a unique weak solution, which depends on the viscosity and the nature of the curvature of the capillary, was proved.By incorporating in this model all the ultrafiltration parameters, which are presented in [41,42], the authors have attempted (within the prescribed morphological and physiological properties of the microvascular environment) to adapt the model used by Maritz and Sauer [33] to reallife situations.Further applications could be found in the modelling of other permeable systems in the human body like the lymphatic glands and the urinary system.
With this research the authors have tried to prepare the ground for the applications of these results in the exploring of permeable surfaces in biosciences, engineering, and the natural sciences.The open question regarding the existence

Figure 1 :
Figure 1: Velocity profile for the rectilinear flow in a horizontal channel formed by a permeable lower wall ( = 0) and an impermeable upper wall ( = ℎ).

Figure 2 :
Figure 2: Profile for normal flow through the permeable wall Γ.