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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 152743, 47 pages
http://dx.doi.org/10.1155/2012/152743
Research Article

A Viscous Fluid Flow through a Thin Channel with Mixed Rigid-Elastic Boundary: Variational and Asymptotic Analysis

1Institut Camille Jordan UMR, CNRS 5208, PRES University of Lyon/University of Saint-Etienne, 23 Rue Paul Michelon, 42023 Saint-Etienne, France
2Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014 700 Bucharest, Romania

Received 23 January 2012; Accepted 29 February 2012

Academic Editor: D. O'Regan

Copyright © 2012 R. Fares et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

1. Introduction

A few years ago we began to publish several papers dealing with an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure. Problems involving the interaction between a fluid and a deformable structure have been studied extensively in the last years due to their applications in many areas such as: engineering, biomechanics, biology, and hydroelasticity.

In order to model the blood flow through a vessel we considered in [1] a non steady viscous flow in a thin rectangle with elastic walls, when at the ends of the flow domain periodicity conditions are imposed. The asymptotic approach was then extended for the nonperiodic case in [2], when the inflow and outflow velocities are given functions. Generalizations of the previous cited papers were obtained in [35]. In the first two papers we extended the results of [1] to a-three dimensional case, by considering a periodic, axisymmetric flow in a thin cylinder with elastic lateral boundary, while in [4] we studied two different cases: when the inertial term of the equation for the wall displacement is much smaller than the stress term and when these two terms are of the same order. Moreover, [5] deals with the asymptotic analysis for a fluid with variable viscosity.

In all these papers, the fluid flow was described by the Stokes equations and the behavior of the elastic structure was simulated either by the Sophie Germain’s or by the Koiter’s equation. The fluid-structure interaction was mathematically expressed by the equality of the fluid velocity at the boundary and the time derivative of the wall displacement. We constructed an asymptotic solution and we proved that it represents a good approximation for the exact solution, by obtaining a small error between them.

In this paper we consider the nonsteady Stokes flow in a thin tube structure. In two-dimensional case, a tube structure is some connected union of thin rectangles, having a dimension much smaller than the other one. The main difference between this paper and our previous works [2, 5] is that now the boundary of the flow domain contains elastic parts and rigid parts, as well.

The flow domain consists of two thin rectangles with elastic boundary. The junction between the rectangles is realized by means of rigid boundaries. The interaction between the viscous fluid and the elastic boundaries produces normal displacements. The elastic boundaries behavior is described by the Sophie Germain’s equation.

This domain models a vessel structure where a stent was placed. In fact, for treating arterial stenoses or occlusions, percutaneous angioplasty is indicated. It uses small inflatable balloons, single-use, that help dilate the artery at the site of narrowing. During angioplasty, a small wire mesh tube called a stent may be permanently placed in the newly opened artery or vein to help it remain open.

There are two types of stents: bare stents (wire mesh) and covered stents (also commonly called stent grafts). The first are in use as part of the expansion of retrecissements vessels, the second impermeable and can be used to prevent or treat hemorrhage. Stents are used for arteries in the heart, the kidney, the arm, or the leg. They are also used for the aorta in the abdomen or chest (see Figures 1 and 2).

152743.fig.001
Figure 1: Bifurcation stenting.
152743.fig.002
Figure 2: Stenting.

We assimilate then the part of the vessel where the stent is placed to a domain with rigid boundaries.

We suppose that the viscous fluid has a variable viscosity depending on a longitudinal variable for each rectangle. This situation models a blood flow in a vessel structure where the viscosity depends on the concentration of some substances diluted in blood or some blood cells. Indeed, the asymptotic analysis of the convection-diffusion equation set in such domains [6, 7] shows that in the case of the Neumann (impermeability) condition at the lateral boundary and small Reynolds numbers, the concentration is asymptotically close to the one-dimensional description, that is, the convection-diffusion equation set on the graph. The solution of the problem on the graph is the leading term of the asymptotic expansion, and it evidently depends on the longitudinal variable. On the other hand, the viscosity often depends on the concentration of the diluted substances or distributed cells, and so, it depends on the longitudinal variable. Of course, the fluid motion equation is coupled with the diffusion-convection equation in this case. However, if the velocity is small (in our case, it is of order 𝜀2), then neglecting the convection, in comparison with the diffusion term or iterating with respect to the small term,we get the steady state diffusion equation; in absence of the source term in the right-hand side, it has a piecewise-linear asymptotic solution on the graph for the concentration. So, in this simplified situation, the diffusion equation can be solved before the fluid motion equation, and we obtain for the flow, the Stokes or Navier-Stokes equation with a variable viscosity depending (via concentration) on the longitudinal variable.

There are, of course, many other practical problems involving fluids with variable viscosity. For example, the presence of bacteria in suspension (see [8]) may change locally the viscosity.

The outline of the paper is as follows. In Section 2 we give a description of the flow domain and of the coupled system which models our problem. The next section presents the variational formulation of the problem. The literature contains an important number of papers dealing with a variational approach of fluid-structure interaction problems. For instance, results concerning the existence of weak or strong solutions when the fluid domain is either fixed or depending on time can be found in [912]. We establish in this section results such as: existence, uniqueness, regularity and a priori estimates. In Section 4 we construct the asymptotic solution. As in our previous papers [15], the problem depends on two small parameters. The first small parameter, 𝜀, is defined as the ratio of the dimensions of the thin rectangles; the second one, 𝛿, corresponds to the softness of the wall. For various values of the small parameters 𝜀 and 𝛿, an asymptotic expansion of the solution is constructed; the parameter 𝛿 is taken of the form 𝛿=𝜀𝛾, with 𝛾, 𝛾3. The asymptotic expansion is different for the cases: 𝛾>3 and 𝛾=3. The asymptotic solution contains three types of terms: the regular part, defined as in [1], the boundary layer correctors corresponding to the boundary conditions and the boundary layer correctors which realize the junction between the motion in the two rectangles. The first two types of correctors have already been introduced in [1, 2]. The third type is characteristic for structures with junction regions. The asymptotic solution is more complicated in this case since it contains also some truncation functions introduced in order to restrict the influence of the boundary layer correctors to the regions to which they correspond. We present and solve the problems for all the components of the asymptotic solution. For the two cases 𝛾>3 and 𝛾=3 the order of solving the problems is presented and the leading term of the asymptotic expansion is described. In the last section, we establish the error between the exact solution and the asymptotic one, by means of the a priori estimates obtained in Section 3. The small error between the two solutions justifies our asymptotic expansion.

2. The Physical Problem

We consider an incompressible, viscous fluid, with variable viscosity, occupying a thin domain, 𝜀. The flow domain is a thin tube structure, composed by two thin rectangles with lateral elastic boundaries, connected by a region with rigid boundaries. We introduce the first small parameter of our problem, 𝜀, 𝜀=1/𝑞, 𝑞 in connection with the ratio of the two dimensions of the rectangles, as below. The thin rectangles are given by:𝐷1𝜀=𝑥1,𝑥220<𝑥1<1,𝜀<𝑥2,𝐷<𝜀2𝜀=𝑥1,𝑥22𝜀<𝑥1<𝜀,0<𝑥2,<1(2.1) and the junction region is𝐷𝑟𝜀=𝑥1,𝑥22𝜀<𝑥1<2𝜀,𝜀<𝑥2𝑥<2𝜀1,𝑥22𝜀<𝑥1<0,𝜀<𝑥2<0,𝑥21+𝑥22𝜀2𝑥1,𝑥22𝜀<𝑥1<2𝜀,𝜀<𝑥2𝑥<2𝜀,12𝜀2+𝑥22𝜀2𝜀2.(2.2) The flow domain 𝜀2 is given by (𝐷1𝜀{𝑥12𝜀})(𝐷2𝜀{𝑥22𝜀})𝐷𝑟𝜀, as shown in Figure 3.

152743.fig.003
Figure 3: The domain 𝜀.

The main difference between this paper and our previous works [2, 5] is that now the boundary of the flow domain 𝜀 contains elastic parts and rigid parts as well. Let 𝐴𝜀+𝐵𝜀+ and 𝐴𝜀𝐵𝜀 be the rigid parts of the boundary of 𝜀 defined as follows:𝐴𝜀+𝐵𝜀+=𝑥1,𝑥22𝜀<𝑥1<2𝜀,𝜀<𝑥2𝑥<2𝜀,12𝜀2+𝑥22𝜀2=𝜀2,𝐴𝜀𝐵𝜀=𝑥1,𝑥22𝑥1=𝜀,0𝑥2𝑥<2𝜀1,𝑥220𝑥1<2𝜀,𝑥2𝑥=𝜀1,𝑥22𝜀<𝑥1<0,𝜀<𝑥2<0,𝑥21+𝑥22=𝜀2;(2.3) the elastic parts of 𝜕𝜀 are given by:Γ𝜀±=±𝜀,𝑥22𝜀<𝑥2𝑥<11,±𝜀2𝜀<𝑥1<1.(2.4) Let 𝐹𝜀1={(1,𝑥2)𝜀<𝑥2<𝜀} and 𝐹𝜀2={(𝑥1,1)𝜀<𝑥1<𝜀} be the inflow and outflow parts of the boundary of 𝜀. Denoting 𝐹𝜀±=Γ𝜀±𝐴𝜀±𝐵𝜀±, we can write 𝜕𝜀=𝐹𝜀+𝐹𝜀𝐹𝜀1𝐹𝜀2.

We study the nonsteady, slow flow of the viscous fluid in the domain 𝜀 previously described, when the fluid interacts with the elastic boundaries, Γ𝜀±. The interaction between the fluid and the elastic boundaries produces the normal displacements 𝑑±=𝑑±(𝑥1,𝑥2,𝑡). We neglect the tangential displacements and we consider that the elastic boundaries are clamped. We study the problem for 𝑡(0,𝑇), with 𝑇 an arbitrary positive constant independent on 𝜀 and we assume that the membranes are not very elastic so that the displacement of the boundaries is small enough. Consequently, at each time 𝑡, we approximate the position of the elastic membranes by its initial position and, hence, the fluid flow equations are considered in the initial configuration. We suppose that the displacements 𝑑±𝐹𝜀± have the following form:𝑑±𝑥1,𝑥2=𝑑,𝑡±𝑥1,𝑡on𝐷1𝜀Γ𝜀±,𝑑±𝑥2,𝑡on𝐷2𝜀Γ𝜀±,0on𝐹𝜀±Γ𝜀±.(2.5)

The problem described above, with nonhomogeneous boundary conditions for the velocity, is modeled by the following coupled system:𝜌𝑓𝜕𝐮𝜕𝑡2div𝑥𝜈(𝑥)𝐷𝑥𝐮+𝑥𝑝=𝐟in𝜀×(0,𝑇),div𝑥𝐮=0in𝜀𝜕×(0,𝑇),𝜌2𝑑±𝜕𝑡2+3𝐸𝜕124𝑑±𝜕𝑥4𝑖𝜕+𝜇5𝑑±𝜕𝑥4𝑖𝜕𝑡=𝑔±±𝑝|𝐷𝑖𝜀𝜀±Γon𝐷𝑖𝜀Γ𝜀±𝑑×(0,𝑇),𝑖=1,2,+=0on𝐴𝜀+𝐵𝜀+×𝑑(0,𝑇),=0on𝐴𝜀𝐵𝜀×(0,𝑇),𝐮=𝝍𝜀on𝜕𝜀Γ𝜀+Γ𝜀𝑑×(0,𝑇),±(1,±𝜀,𝑡)=𝜕𝑑±𝜕𝑥1𝑑(1,±𝜀,𝑡)=0in(0,𝑇),±(2𝜀,±𝜀,𝑡)=𝜕𝑑±𝜕𝑥1(𝑑2𝜀,±𝜀,𝑡)=0in(0,𝑇),±(±𝜀,1,𝑡)=𝜕𝑑±𝜕𝑥2𝑑(±𝜀,1,𝑡)=0in(0,𝑇),±(±𝜀,2𝜀,𝑡)=𝜕𝑑±𝜕𝑥2(±𝜀,2𝜀,𝑡)=0in(0,𝑇),𝐮𝐧=±𝜕𝑑±𝜕𝑡,𝐮𝝉=0onΓ𝜀±×(0,𝑇),𝐮(𝑥,0)=𝟎in𝜀,𝑑±(𝑥,0)=𝜕𝑑±𝜕𝑡(𝑥,0)=0onΓ𝜀±,(2.6) with 𝐧 the outer unit normal on the boundary of 𝜀 and 𝝉 the unit tangent vector to 𝜕𝜀. The given data contained by the previous system are: some material constants and some given functions. 𝜌𝑓,𝜌,𝜇,𝐸 are positive given constants in connection with the properties of materials representing the density of the fluid, the density of the elastic walls, a viscosity coefficient, and the two-dimensional Young’s modulus (The two dimensional Young’s modulus 𝐸 is defined as 𝐸(3)/(1̂𝜈2) where 𝐸(3) is the common three dimensional Young modulus and ̂𝜈 is the Poisson ratio.), respectively, and the positive constant stands for the thickness of the elastic walls.

The given functions are: 𝜈, the variable viscosity of the fluid, which satisfies 𝜈𝐶1(𝜀),𝜈(𝑥)𝛼>0 for all 𝑥𝜀, 𝐟, the exterior force applied to the fluid, 𝑔±, the exterior forces applied on the elastic walls, with 𝑔±𝐹𝜀±,𝑔±𝑥1,𝑥2=𝑔,𝑡±𝑥1,𝑡on𝐷1𝜀Γ𝜀±,𝑔±𝑥2,𝑡on𝐷2𝜀Γ𝜀±,0on𝐹𝜀±Γ𝜀±(2.7) and a small inflow-outflow velocity defined by means of the function 𝜓𝜀 defined as follows:𝝍𝜀𝑥1,𝑥2,𝑡=𝜀2𝝍𝜉1,𝜉2,𝑡,𝑥𝜀,(2.8) with (𝜉1,𝜉2)=(𝑥1/𝜀,𝑥2/𝜀). The function 𝝍 is the trace of a function denoted also by 𝝍 with the following properties:𝝍𝜉1,𝜉2𝜉,𝑡=𝜓2𝐞,𝑡1𝜉,1,𝜉2𝝍𝜉(2,)×(1,1),1,𝜉2𝜉,𝑡=𝜓1𝐞,𝑡2𝜉,1,𝜉2(1,1)×(2,),div𝜉𝝍=0,in𝐷𝑟,𝝍=𝟎on𝐹+𝐹,𝝍𝜉1,𝜉2,0=𝟎,11𝜓𝜉2,𝑡d𝜉2+11𝜓𝜉1,𝑡d𝜉1=0,(2.9) where 𝐷𝑟=(1/𝜀)𝐷𝑟𝜀,𝐹±=𝐹𝜀±. The unknowns of the system (2.6) are the velocity of the fluid, 𝐮, the pressure of the fluid, 𝑝, and the normal displacements of the elastic walls, 𝑑±.

The fluid flow is described by the nonsteady Stokes equations. For the normal displacements we consider the Sophie-Germain’s equation. A “viscous” type term, 𝜇(𝜕5𝑑𝑖±/𝜕𝑥4𝑖𝜕𝑡), is added to the usual forth-order equation for the normal displacements. It corresponds to the viscoelastic behavior of the wall (the so called Kelvin-Voigt model). Usually, the Young’s modulus, 𝐸, has a value of 104106 Pa. On the other hand, we assume that the characteristic longitudinal space scale for vessels is of order of cm and the time scale is of order of seconds. Let us use the S.I. system of units. This leads us to the necessity of scaling of every derivative is 𝑥𝑖 by the factor 102; that is, the fourth derivative will contain the additional factor 108. If is of order 103 m or 102 m, then the coefficient 𝜌 can be taken in the further analysis as a value of order of 1. The coefficient 3𝐸/12 in (2.6)3 will be replaced (after scaling in 𝑥𝑖) by a great coefficient 𝛿1 with the value 𝛿 of order of 107 to 104. If the ratio of thickness and the length of the vessel 𝜀 are of order 102, then 𝛿 is of order from 𝜀2 to 𝜀4. We assume that the “viscous” term is much smaller than the term with the coefficient 𝛿1 and hence the new coefficient denoted also by 𝜇, obtained after scaling in 𝑥𝑖, is 𝑂(1). More details concerning (2.6) can be found, for instance in [1].

Due to the properties of the function 𝝍𝜀, the compatibility condition for the coupled system which describes the physical problem is0=𝜕𝜀d𝐮(𝑥,𝑡)𝐧d𝑠=d𝑡12𝜀𝑑+𝑥1,𝑡𝑑𝑥1,𝑡d𝑥1+12𝜀𝑑+𝑥2,𝑡𝑑𝑥2,𝑡d𝑥2.(2.10) Using next the initial condition for the displacements, condition for the above coupled system becomes12𝜀𝑑+𝑥1,𝑡𝑑𝑥1,𝑡d𝑥1+12𝜀𝑑+𝑥2,𝑡𝑑𝑥2,𝑡d𝑥2=0.(2.11) This condition states that the global area of the flow domain is preserved.

3. Variational Formulation: Existence, Uniqueness, Regularity, and A Priori Estimates

In order to obtain the above properties for the solution of the physical problem, we introduce the variational framework.

To simplify the computations, we consider first (2.6) with homogeneous boundary conditions on 𝐹𝜀1 and 𝐹𝜀2, that is, the problem for 𝝍𝜀=𝟎. Then, the same properties for the solution of (2.6) follow with the usual technique for nonhomogeneous problems.

Taking into account the conditions for the velocity and for the displacements and the condition (2.11) we introduce the following spaces:𝑉𝜀=𝐻𝐯1𝜀2div𝐯=0in𝜀,𝐯=𝟎on𝜕𝜀Γ𝜀+Γ𝜀,𝐯𝝉=0onΓ𝜀±,𝑊𝜀=𝛽+,𝛽𝐻1𝐹𝜀+×𝐻1𝐹𝜀𝛽+,𝛽𝐻20Γ𝜀+×𝐻20Γ𝜀,𝛽+,𝛽=(0,0)on𝐴𝜀+𝐵𝜀+×𝐴𝜀𝐵𝜀,12𝜀𝛽1+𝑥1𝛽1𝑥1d𝑥1+12𝜀𝛽2+𝑥2𝛽2𝑥2d𝑥2.=0(3.1) Choosing for the data the regularity 𝐟𝐿2(0,𝑇;(𝐿2(𝜀))2) and 𝐠=(𝑔+,𝑔)𝐿2(0,𝑇;𝐹𝜀+)×𝐿2(0,𝑇;𝐹𝜀) we consider the following variational problem:Find(𝐮,𝐝)𝐿2(0,𝑇;𝑉𝜀)×𝐻1(0,𝑇;𝑊𝜀̇̈𝐝),with𝐮,𝐿20,𝑇;(𝑉𝜀)×𝐿20,𝑇;(𝑊𝜀)𝜌whichsatisesa.e.in(0,𝑇)𝑓dd𝑡𝜀𝐮𝝋d𝑥+2𝜀+𝜈D𝐮D𝝋d𝑥2𝑖=1d𝜌d𝑡12𝜀𝜕𝐝𝜕𝑡𝜷d𝑥𝑖+3𝐸1212𝜀𝜕2𝐝𝜕𝑥2𝑖𝜕2𝜷𝜕𝑥2𝑖d𝑥𝑖+𝜇12𝜀𝜕3𝐝𝜕𝑥2𝑖𝜕𝜕𝑡2𝜷𝜕𝑥2𝑖d𝑥𝑖=𝜀𝐟𝝋d𝑥+2𝑖=112𝜀𝐠𝜷d𝑥𝑖𝝋𝑉𝜀,𝜷𝑊𝜀,with𝝋𝐧=±𝛽±onΓ𝜀±,𝐮𝐧=±𝜕𝑑±𝜕𝑡onΓ𝜀±,̇𝐮(0)=𝟎,𝐝(0)=𝐝(0)=𝟎.(3.2) Here and below 𝐝=(𝑑+,𝑑) and 𝜷=(𝛽+,𝛽).

For the nonhomogeneous boundary conditions we still obtain the variational formulation (3.2) with 𝐮 replaced by 𝐮𝝍𝜀 and 𝐟 replaced by 𝐟𝜌𝑓(𝜕𝝍𝜀/𝜕𝑡)+2div(𝜈𝐷𝝍𝜀).

Theorem 3.1. The variational problem (3.2) has the unique solution (𝐮,𝐝) with (̇̈𝐮,𝐝)𝐿2(0,𝑇;(𝐿2(𝜀))2)×𝐿2(0,𝑇;𝐿2(𝐹𝜀+)×𝐿2(𝐹𝜀)).

Proof. Let us start with the proof of the uniqueness for the solution of (3.2). Consider (𝐮1,𝐝1) and (𝐮2,𝐝2) two solutions of (3.2) and define (𝐮,𝐝)=(𝐮1𝐮2,𝐝1𝐝2). Subtracting the two relations (3.2)1 and taking as test function (𝝋,𝜷)=(𝐮,𝐝) we get: 𝜌𝑓2dd𝑡𝜀𝐮2d𝑥+2𝜀+𝜈D𝐮D𝐮d𝑥2𝑖=1𝜌2dd𝑡12𝜀𝜕𝐝𝜕𝑡2d𝑥𝑖+3𝐸d24d𝑡12𝜀𝜕2𝐝𝜕𝑥2𝑖2d𝑥𝑖+𝜇12𝜀𝜕3𝐝𝜕𝑥2𝑖𝜕𝑡2d𝑥𝑖=0.(3.3) Integrating from 0 to 𝑡 this equality and taking into account the initial conditions, we obtain: 𝐮=𝟎 a.e. in (0,𝑇) and 𝐝=𝟎 a.e. in (0,𝑇). Hence the problem (3.2) has a unique solution.
For proving the existence and the regularity of the functions 𝐮 and 𝐝, we will use the Galerkin’s method.
We begin with the construction of a basis for the space 𝑊𝜀. Let {𝜁𝑗}𝑗 be a basis of 𝐻20(2𝜀,1) chosen by considering the eigenfunctions of the following problem: 𝜁𝑗(𝑖𝑣)=𝛼𝑗𝜁𝑗𝜁in(2𝜀,1),𝑗(2𝜀)=𝜁𝑗𝜁(1)=0,𝑗(2𝜀)=𝜁𝑗(1)=0,(3.4) where 𝜁𝑗(𝑖𝑣) is the fourth derivative of 𝜁𝑗 and 𝛼𝑗>0, for all 𝑗. We define {𝜷𝑗}𝑗 as follows: 𝜷𝑗=(𝛽+𝑗,𝛽𝑗) where 𝛽±𝑗𝑥1,𝑥2=𝜁𝑗𝑥1𝑥for1,𝑥2𝑥(2𝜀,1)×{±𝜀},0for1,𝑥2𝐴𝜀±𝐵𝜀±,𝜁𝑗𝑥2𝑥for1,𝑥2{±𝜀}×(2𝜀,1).(3.5) It is easy to check that {𝜷𝑗}𝑗 is a basis for 𝑊𝜀. We choose the functions of the basis {𝜁𝑗}𝑗 such that 𝜀𝜷𝑖𝜷𝑘d𝑥=𝛿𝑖𝑘.(3.6) As a consequence of the previous relation we also get: 𝜀𝜷𝑖𝜷𝑘=𝛼𝑘𝛿𝑖𝑘.(3.7) We consider now {𝝍𝑖}𝑖 a basis of the space 𝑉𝜀0={𝐮(𝐻10(𝜀))2div𝐮=0in𝜀}, constructed with the eigenfunctions of the following Stokes problem: 2div𝜈D𝝍𝑖+𝑞𝑖=𝜆𝑖𝝍𝑖in𝜀,div𝝍𝑖=0in𝜀,𝝍𝑖=0on𝜕𝜀,(3.8) with 𝜆𝑖>0 for all 𝑖. The functions 𝝍𝑖,𝑖 are uniquely determined from the condition 𝜀𝝍𝑗𝝍𝑘d𝑥=𝛿𝑗𝑘,(3.9) which implies 2𝜀𝜈D𝝍𝑖D𝝍𝑘d𝑥=𝜆𝑘𝛿𝑗𝑘.(3.10) Next, for any 𝜷𝑗 we consider the following problem: 𝝋Find𝑗,𝑝𝑗𝑉𝜀×𝐿2𝜀𝝋suchthat2div𝜈D𝑗+𝑝𝑗𝐻=0in1𝜀2,𝝋𝑗𝐧=±𝛽±𝑗onΓ𝜀±.(3.11) Following the classical results of [13] for nonhomogeneous Stokes problems, we obtain a unique 𝝋𝑗 and a function 𝑝𝑗 unique up to an additive constant. Moreover, for any 𝑗,𝑘𝜀𝜈D𝝋𝑗D𝝍𝑘d𝑥=0.(3.12) By means of the functions {𝜷𝑗}𝑗,{𝝍𝑖}𝑖,{𝝋𝑗}𝑗, we are now in a position to define, for each 𝑛,𝑚, an approximate solution (𝐮𝑚𝑛,𝐝𝑛) of (3.2) as follows: 𝐮𝑚𝑛(𝑥,𝑡)=𝑚𝑖=1𝑎𝑖(𝑡)𝝍𝑖(𝑥)+𝑛𝑗=1̇𝑏𝑗(𝑡)𝝋𝑗𝐝(𝑥),𝑛(𝑥,𝑡)=𝑛𝑗=1𝑏𝑗(𝑡)𝜷𝑗(𝑥),(3.13) with 𝑎𝑖,𝑏𝑗[0,𝑇],𝑖=1,,𝑚,𝑗=1,,𝑛 scalar unknown functions. These functions are determined below from the condition that (𝐮𝑚𝑛,𝐝𝑛) is the solution for the problem: 𝜌𝑓𝜀𝜕𝐮𝑚𝑛𝜕𝑡𝝍𝑖d𝑥+2𝜀𝜈D𝐮𝑚𝑛D𝝍𝑖d𝑥=𝜀𝐟𝝍𝑖𝜌d𝑥,for𝑖{1,,𝑚},𝑓𝜀𝜕𝐮𝑚𝑛𝜕𝑡𝝋𝑗d𝑥+2𝜀𝜈D𝐮𝑚𝑛D𝝋𝑗+d𝑥2𝑙=1𝜌12𝜀𝜕2𝐝𝑛𝜕𝑡2𝜷𝑗d𝑥𝑙+3𝐸1212𝜀𝜕2𝐝𝑛𝜕𝑥2𝑙𝜕2𝜷𝑗𝜕𝑥2𝑙d𝑥𝑙+𝜇12𝜀𝜕3𝐝𝑛𝜕𝑥2𝑙𝜕𝜕𝑡2𝜷𝐣𝜕𝑥2𝑙d𝑥𝑙=𝜀𝐟𝝋𝑗d𝑥+2𝑙=112𝜀𝐠𝜷𝑗d𝑥𝑙𝐮,for𝑗{1,,𝑛},𝑚𝑛𝐧=±𝜕𝑑±𝑛𝜕𝑡onΓ𝜀±,𝐮𝑚𝑛(0)=𝟎,𝐝𝑛̇𝐝(0)=𝑛(0)=𝟎.(3.14) We introduce the notations: 𝑝𝑖𝑘=𝜀𝝋𝑘𝝍𝑖d𝑥,𝑞𝑘𝑗=𝜀𝝋𝑘𝝋𝑗d𝑥,𝑟𝑖𝑘=2𝜀𝜈D𝝋𝑖D𝝋𝑘d𝑥.(3.15) Taking into account the previous notations and the relations (3.6), (3.7), (3.9), (3.10), (3.12) we get from (3.14) the following 𝑚+𝑛 linear differential system for the unknown functions 𝑎𝑖,𝑏𝑗,𝑖=1,,𝑚,𝑗=1,,𝑛: 𝜌𝑓̇𝑎𝑖(𝑡)+𝜆𝑖𝑎𝑖(𝑡)+𝑛𝑘=1𝑝𝑖𝑘̈𝑏𝑘(𝑡)=𝜀𝐟𝝍𝑖𝜌d𝑥,𝑓𝑚𝑘=1𝑝𝑘𝑗̇𝑎𝑘(𝑡)+𝜌𝑓𝑚𝑘=1𝑞𝑘𝑗̈𝑏𝑘(𝑡)+𝑛𝑘=1𝑟𝑘𝑗̇𝑏𝑘̈𝑏(𝑡)+𝜌𝑗(𝑡)+3𝐸𝛼12𝑗𝑏𝑗(𝑡)+𝜇𝛼𝑗̇𝑏𝑗=(𝑡)𝜀𝐟𝝋𝑗d𝑥+2𝑙=112𝜀𝐠𝜷𝑗d𝑥𝑙,𝑎𝑖(0)=𝑏𝑗̇𝑏(0)=𝑗(0)=0,𝑖=1,,𝑚,𝑗=1,,𝑛.(3.16) The previous system uniquely determines the unknown functions 𝑎𝑖,𝑏𝑗,𝑖=1,,𝑚,𝑗=1,,𝑛. For more details see [1, Section  3].
In the sequel we obtain some a priori estimates which give the regularity of the solution for (3.2). Computing 𝑚𝑖=1𝑎𝑖(𝑡)(3.14)1+𝑛𝑗=1̇𝑏𝑗(𝑡)(3.14)2 and using (3.13) we get:𝜌𝑓2dd𝑡𝜀𝐮𝑚𝑛2d𝑥+𝜀𝜈D𝐮𝑚𝑛2+2𝑖=1𝜌2dd𝑡12𝜀𝜕𝐝𝑛𝜕𝑡2d𝑥𝑖+3𝐸d24d𝑡12𝜀𝜕2𝐝𝑛𝜕𝑥2𝑖2d𝑥𝑖+𝜇12𝜀𝜕3𝐝𝑛𝜕𝑥2𝑖𝜕𝑡2d𝑥𝑖=𝜀𝐟𝐮𝑚𝑛d𝑥+2𝑖=112𝜀𝐠𝜕𝐝𝑛𝜕𝑡d𝑥𝑖.(3.17) Integrating from 0 to 𝑡, using the property of 𝜈 and the initial conditions we obtain, as in [1], the first estimates: 𝐮𝑛𝑚𝐿(0,𝑇;(𝐿2(𝜀))2)𝐶(𝐟,𝐠),D𝐮𝑛𝑚𝐿2(0,𝑇;(𝐿2(𝜀))4)𝐶(𝐟,𝐠),𝜕𝑑±𝑛𝜕𝑡𝐿(0,𝑇;𝐿2(Γ𝜀±))𝜕𝐶(𝐟,𝐠),2𝑑±𝑛𝜕𝑠2𝐿(0,𝑇;𝐿2(Γ𝜀±))𝜕𝐶(𝐟,𝐠),3𝑑±𝑛𝜕𝑠2𝜕𝑡𝐿2(Γ𝜀±×(0,𝑇))𝐶(𝐟,𝐠),(3.18) with 𝐶(𝐟,𝐠)=𝐶(𝑇,𝜇,𝜌𝑓,𝜌,𝐸,)(𝐟𝐿2(0,𝑇;(𝐿2(𝜀))2)+𝐠𝐿2((0,𝑇);𝐿2(𝐹𝜀+×𝐹𝜀))) and 𝑠 is the variable of the parametrisation for Γ𝜀+ or on Γ𝜀.
The second estimates are obtained computing 𝑚𝑖=1̇𝑎𝑖(𝑡)(3.14)1+𝑛𝑗=1̈𝑏𝑗(𝑡)(3.14)2:𝜕𝐮𝑛𝑚𝜕𝑡𝐿2(0,𝑇;(𝐿2(𝜀))2)3𝐸121/2𝐶(𝐟,𝐠),D𝐮𝑛𝑚𝐿((0,𝑇);(𝐿2(𝜀))4)3𝐸121/2𝜕𝐶(𝐟,𝐠),2𝑑±𝑛𝜕𝑡2𝐿2(Γ𝜀±×(0,𝑇))3𝐸121/2𝜕𝐶(𝐟,𝐠),3𝑑±𝑛𝜕𝑠2𝜕𝑡𝐿(0,𝑇;𝐿2(Γ𝜀±))3𝐸121/2𝐶(𝐟,𝐠).(3.19) From (3.18)1,2 and (3.19)1,2 we get the boundedness of {𝐮𝑚𝑛}𝑚,𝑛 in 𝐿(0,𝑇;𝑉𝜀)𝐻1(0,𝑇;(𝐿2(𝜀))2). Consequently, we can pass to the limit in (3.14) which yields the existence result of the theorem. To achieve the proof, we notice that the regularity stated in (3.2) follows from the estimates (3.18) and (3.19).

As a consequence of Theorem 3.1 we introduce the pressure which appears in (2.6)1,3.

Corollary 3.2. There exists a unique function 𝑝𝐿2(0,𝑇;𝐻1(𝜀)) such that (𝐮,𝑝,𝐝) satisfies (2.6)1,3 a.e. in 𝜀×(0,𝑇) and on (𝐷𝑖𝜀Γ𝜀±)×(0,𝑇), respectively.

To obtain this result we follow the ideas of [1, Section  3].

The last result of this section presents the estimates for (𝐮,𝑝,𝐝), the unique solution of (2.6).

Corollary 3.3. Let (𝐮,𝑝,𝐝) be the solution of the problem (2.6) corresponding to the data 𝐟,𝐠, with 𝐟𝐿2(0,𝑇;(𝐿2(𝜀))2) and 𝐠𝐿2(0,𝑇;𝐹𝜀+)×𝐿2(0,𝑇;𝐹𝜀). Then the following estimates hold: 𝐮𝐿(0,𝑇;(𝐿2(𝜀))2)𝐶(𝐟,𝐠),D𝐮𝐿2(0,𝑇;(𝐿2(𝜀))4)𝐶(𝐟,𝐠),𝜕𝐮𝜕𝑡𝐿2(0,𝑇;(𝐿2(𝜀))2)3𝐸121/2𝐶(𝐟,𝐠),𝜕𝑑±𝜕𝑡𝐿(0,𝑇;𝐿2(Γ𝜀±))𝜕𝐶(𝐟,𝐠),2𝑑±𝜕𝑠2𝐿(0,𝑇;𝐿2(Γ𝜀±))𝜕𝐶(𝐟,𝐠),2𝑑±𝜕𝑡2𝐿2(Γ𝜀±×(0,𝑇))3𝐸121/2𝜕𝐶(𝐟,𝐠),3𝑑±𝜕𝑠2𝜕𝑡𝐿(0,𝑇;𝐿2(Γ𝜀±))3𝐸121/2𝐶(𝐟,𝐠),𝑝𝐿2(0,𝑇;(𝐿2(𝜀))2)3𝐸121/2𝐶(𝐟,𝐠).(3.20)

Remark 3.4. If we consider the nonhomogeneous problem (which corresponds to 𝝍𝜀) we obtain for the corresponding solution the same estimates (3.20) with a different 𝐟, but which depends in the same way on 𝜀.

4. Asymptotic Analysis

In order to approximate the solution of (2.6), we consider more regular data than in Section 3. We suppose that:𝝍𝜀𝒞𝜀×(0,𝑇)2,𝒞𝐟𝜀×(0,𝑇)2,𝑔±𝒞𝐹𝜀±,×(0,𝑇)𝜈𝒞𝜀,𝑡<𝑇suchthat𝐟(𝑥,𝑡)=𝑔±(𝑥,𝑡)=𝝍𝜀(𝑥,𝑡)=0(𝑥,𝑡)𝜀×0,𝑡,(4.1) and 𝐟, 𝑔± and 𝜈 are chosen as follows:𝐟𝑥1,𝑥2=𝑓𝑥,𝑡1𝐞,𝑡1in𝜀𝑥1>13𝑓𝑥×(0,𝑇),2𝐞,𝑡2in𝜀𝑥2>13×(0,𝑇),𝟎in𝜀𝑥113,𝑥213𝑔×(0,𝑇),±𝑥1,𝑥2=𝑔,𝑡±𝑥1𝐹,𝑡on𝜀±𝑥1>13𝑔×(0,𝑇),±𝑥2𝐹,𝑡on𝜀±𝑥2>130𝐹×(0,𝑇),on𝜀±𝑥113,𝑥213×𝜈𝑥(0,𝑇),1,𝑥2=𝜈𝑥1in𝜀13<𝑥1<23,𝜈𝑥2in𝜀13<𝑥2<23,𝜈0in𝜀𝑥113,𝑥213𝑥123𝑥223,(4.2) with 𝜈0 constant.

4.1. Construction of the Asymptotic Solution

In the sequel we introduce the second small parameter 𝛿=12/3𝐸 and we take 𝛿=𝜀𝛾, with 𝛾, 𝛾3. The asymptotic solution approximating the periodic flow in an infinite rectangle (see [1, Section  5]) is modified by using two types of correctors: the first type corresponds to the boundary conditions on 𝑥1=1 and 𝑥2=1 and the second one represents the boundary layer functions in a neighborhood of (𝑥1,𝑥2)=𝟎. So, the asymptotic solution contains the regular part of the solution (which has two terms, due to the form of the flow domain), two boundary layer functions corresponding to 𝑥1=1 and to 𝑥2=1, respectively, and the correctors in (𝑥1,𝑥2)=𝟎. Since the terms of the asymptotic solution should vanish in different parts of the flow domain, we multiply them with the truncation functions defined as follows: 𝜒, 𝜂, 𝜒,𝜂𝒞(),3𝜒(𝜏)=0,|𝜏|<2,1,|𝜏|>3,𝜂(𝜏)=0,|𝜏|<4,71,|𝜏|>8.(4.3)

We construct the asymptotic solution as below:̂𝐮𝑎(𝑘)𝑥1,𝑥2,𝑡=𝐮1(𝑘)𝑥1,𝑥2𝜀𝜒𝑥,𝑡1𝜀+𝐮2(𝑘)𝑥1𝜀,𝑥2𝜒𝑥,𝑡2𝜀+𝐮(𝑘)1𝑏𝑙𝑥11𝜀,𝑥2𝜀𝜂𝑥,𝑡1+𝐮(𝑘)2𝑏𝑙𝑥1𝜀,𝑥21𝜀𝜂𝑥,𝑡2+𝐮(𝑘)0𝑏𝑙𝑥1𝜀,𝑥2𝜀𝜂,𝑡1𝑥1𝜂1𝑥2,̂𝑝𝑎(𝑘)𝑥1,𝑥2,𝑡=𝑝1(𝑘)𝑥1,𝑥2𝜀𝜒𝑥,𝑡1𝜀+𝑝2(𝑘)𝑥1𝜀,𝑥2𝜒𝑥,𝑡2𝜀+𝑝(𝑘)1𝑏𝑙𝑥11𝜀,𝑥2𝜀𝜂𝑥,𝑡1+𝑝(𝑘)2𝑏𝑙𝑥1𝜀,𝑥21𝜀𝜂𝑥,𝑡2+𝑝(𝑘)0𝑏𝑙𝑥1𝜀,𝑥2𝜀𝜂,𝑡1𝑥1𝜂1𝑥2,𝑑(𝑘)±𝑎𝑥1,𝑥2,𝑡=𝑑(𝑘)±1𝑥1𝜒𝑥,𝑡1𝜀+𝑑(𝑘)±2𝑥2𝜒𝑥,𝑡2𝜀+𝑑(𝑘)1±𝑏𝑙𝑥11𝜀𝜂𝑥,𝑡1+𝑑(𝑘)2±𝑏𝑙𝑥21𝜀𝜂𝑥,𝑡2+𝑑(𝑘)0±𝑏𝑙𝑥1𝜀,𝑥2𝜀𝜂,𝑡1𝑥1𝜂1𝑥2.(4.4)

Due to the definition of the truncation functions, we notice that in 𝜀{1/4𝑥𝑖3/4}, that is, in a neighborhood of the region of variable viscosity, the asymptotic solution reduces to its regular part, (𝐮𝑖(𝑘),𝑝𝑖(𝑘),𝑑(𝑘)±𝑖), in a neighborhood of {𝑥𝑖=1} it reduces to (𝐮𝑖(𝑘)+𝐮(𝑘)𝑖𝑏𝑙,𝑝𝑖(𝑘)+𝑝(𝑘)𝑖𝑏𝑙,𝑑(𝑘)±𝑖+𝑑(𝑘)𝑖±𝑏𝑙), 𝑖=1,2, while in 𝐷𝑟𝜀 the asymptotic solution is equal to the corrector in 𝑥=𝟎,(𝐮(𝑘)0𝑏𝑙,𝑝(𝑘)0𝑏𝑙,𝑑(𝑘)0±𝑏𝑙). This means that the regular part of the asymptotic solution (which can be computed and has a very simple expression) represents an approximation for the exact solution in all the flow domain except some neighborhoods of 𝑥1=1, 𝑥2=1 and 𝑥=𝟎.

We give next the expressions of the three components of the asymptotic solution and we explain their role in this construction.

4.1.1. The Regular Part of the Asymptotic Solution

For each rectangle 𝐷1𝜀 and 𝐷2𝜀 we define the corresponding regular part of the asymptotic expansion as in [1]. The regular part corresponding to 𝐷1𝜀 has the expression:𝐮1(𝑘)𝑥1,𝑥2𝜀=,𝑡𝑘𝑗=0𝜀𝑗+2𝑢11,𝑗𝑥1,𝑥2𝜀𝐞,𝑡1+𝑘𝑗=0𝜀𝑗+3𝑢12,𝑗𝑥1,𝑥2𝜀𝐞,𝑡2,𝑝1(𝑘)𝑥1,𝑥2𝜀=,𝑡𝑘𝑗=0𝜀𝑗+1𝑝1𝑗𝑥1,𝑥2𝜀+,𝑡𝑘𝑗=0𝜀𝑗𝑞1𝑗𝑥1,𝑑,𝑡(𝑘)±1𝑥1=,𝑡𝑘𝑗=0𝜀𝑗+𝛾𝑑1(±)𝑗𝑥1.,𝑡(4.5) It represents the solution of the problem (2.6)1,2,3,11 set in the infinite rectangle in 𝑂𝑥1 direction, (,)×(𝜀,𝜀); the functions which appear in (4.5) can be computed explicitly and represent a good approximation of the exact solution of (2.6) in a neighborhood of the region with variable viscosity of 𝐷1𝜀 (as we will prove in the last section).

In a similar way we introduce the regular part of the asymptotic solution corresponding to 𝐷2𝜀.

Hence, the sum of the first two terms of the asymptotic solution represents a good approximation of the exact solution of (2.6) in some neighborhood of the region of variable viscosity of the flow domain, but it is not close to it on the inflow/outflow boundaries and in 𝐷𝑟𝜀. Since the purpose of the asymptotic construction is to approximate the exact solution with a small error between the exact and asymptotic solution, we modify and complete the regular part by two types of boundary layer correctors.

4.1.2. The Boundary Layer Correctors for 𝑥1=1 and 𝑥2=1

These boundary layer functions are introduced in order to repair the traces of the regular part of the asymptotic solution on 𝑥1=1 and 𝑥2=1. They are given by:𝐮(𝑘)𝑖𝑏𝑙𝑥𝜀=,𝑡𝑘𝑗=0𝜀𝑗+2𝐮𝑗(𝑖)𝑥𝜀,𝑝,𝑡(𝑘)𝑖𝑏𝑙𝑥𝜀=,𝑡𝑘𝑗=0𝜀𝑗+1𝑝𝑗(𝑖)𝑥𝜀,𝑑,𝑡(𝑘)𝑖±𝑏𝑙𝑥𝑖𝜀=,𝑡𝑘𝑗=0𝜀𝑗+𝛾𝑑𝑗(𝑖)𝑥𝑖𝜀,𝑡,𝑖{1,2}.(4.6) The corrector with 𝑖=1 corresponds to the end 𝑥1=1 and that with 𝑖=2 corresponds to 𝑥2=1.

From the definition of the truncation functions and of the asymptotic solution we notice that the influence of each corrector defined above is significant only near the corresponding end of the flow domain.

4.1.3. The Boundary Layer Corrector in 𝑥=0

This corrector is necessary in order to realize the junction between the two parts of the asymptotic solution, corresponding to the two branches of the flow domain and to obtain the conditions on the rigid boundaries of the domain. The expressions of the correctors in 𝑥=𝟎 are given by:𝐮(𝑘)0𝑏𝑙𝑥𝜀=,𝑡𝑘𝑗=1𝜀𝑗+2𝐮𝑗(0)𝑥𝜀,𝑝,𝑡(𝑘)0𝑏𝑙𝑥𝜀=,𝑡𝑘𝑗=1𝜀𝑗+1𝑝𝑗(0)𝑥𝜀,𝑑,𝑡(𝑘)0±𝑏𝑙𝑥𝜀=,𝑡𝑘𝑗=0𝜀𝑗+𝛾𝑑(0)(±)𝑗𝑥𝜀.,𝑡(4.7) From the definition of the truncation function 𝜂 it follows that the corrector in 𝑥=𝟎 appears in the expression of the asymptotic solution (4.4) only in 𝜀{𝑥11/4,𝑥21/4}, which represents a neighborhood of 𝐷𝑟𝜀 with constant viscosity and where the forces are equal to zero.

Remark 4.1. The boundary layer method is close to the well-known method of matching of asymptotic expansions (see [14]). However there is a difference between these two methods. In the boundary layer method the regular expansion is reexpanded with respect to the fast variable (as in the matching method), but then the new expansion is multiplied by a cutting function. This product being inserted into the equation gives a discrepancy in the right-hand side which is then compensated by an appropriate boundary layer corrector. So, we get finally a unique expression for the asymptotic approximation in each point of the domain. In the matching approach the reexpanded regular ansatz (outer expansion) coexists with the so-called inner expansion defined near the boundary, so that there are several overlapping expressions for an asymptotic approximation in different parts of the domain. We emphasize that this difference is not too important and corresponds more to the form of presentation of the result.

4.2. The Determination of the Asymptotic Solution

This subsection is devoted to the resolution of the problems satisfied by the three different components of the asymptotic solution presented in the previous subsection. Since some computations are different with respect to the values of 𝛾, we will analyse the problems and the order of solving them for 𝛾>3 and for 𝛾=3. Moreover, in each case we will specify the leading term of the asymptotic expansion.

We begin the approach with the problems for the boundary layer correctors corresponding to 𝑥1=1 and 𝑥2=1 since the study of these problems is the same both for 𝛾>3 and for 𝛾=3. To fix the ideas, we obtain in the sequel the problems for the corrector corresponding to the end 𝑥1=1. As we noticed before, the term containing this corrector is not equal to zero only in a neighborhood of the boundary 𝑥1=1. So, the problems and the other relations corresponding to this corrector are obtained substituting the asymptotic solution in (2.6)1,2,3,6,7,11. We obtain two separate problems: one for the velocity-pressure correctors and the other one for the displacements correctors.

Since in this neighborhood the viscosity is constant, the problem for (𝐮𝑗(1),𝑝𝑗(1)) has constant coefficients. Denoting by Π1 the semi-infinite rectangle (,0)×(1,1) and imposing for the velocity and for the pressure the condition of decay at , we obtain for (𝐮𝑗(1),𝑝𝑗(1)) the problem:𝜈0Δ𝜉𝜉𝐮𝑗(1)+𝜉𝑝𝑗(1)=𝜌𝑓𝜕𝐮(1)𝑗2𝜕𝑡inΠ1×(0,𝑇),div𝜉𝐮𝑗(1)=0inΠ1×𝐮(0,𝑇),𝑗(1)𝜉1=,±1,𝑡𝜕𝑑(1)(±)𝑗+2𝛾𝜉𝜕𝑡1𝐞,𝑡2𝐮in(,0)×(0,𝑇),𝑗(1)0,𝜉2𝜉,𝑡=𝜓2𝛿,𝑡𝑗0𝐞1𝑢11,𝑗1,𝜉2𝐞,𝑡1𝑢12,𝑗11,𝜉2𝐞,𝑡2𝐮in(1,1)×(0,𝑇),𝑗(1)𝟎,𝑝𝑗(1)0,uniformlywhen𝜉1.(4.8) The compatibility condition for (4.8) reads:11𝑢11,𝑗1,𝜉2,𝑡d𝜉2=dd𝑡0𝑑(1)(+)𝑗+2𝛾𝑑(1)()𝑗+2𝛾𝜉1,𝑡d𝜉1+𝛿𝑗011𝜓𝜉2,𝑡d𝜉2.(4.9) For any 𝛾3 the right-hand side of (4.8)3 is known; so the boundary layer correctors for the velocity and for the pressure corresponding to 𝑥1=1 are uniquely determined from (4.8) (see [2, Section  4]). The condition (4.9) represents a relation for the regular part of the asymptotic solution.

The boundary layer correctors for the displacements exponentially stabilizing to zero at are obtained as the unique solution of the following problems:𝜕4𝑑(1)(±)𝑗𝜕𝜉41𝜕=𝜌2𝑑(1)(±)𝑗4𝛾𝜕𝑡2𝜕𝜇5𝑑(1)(±)𝑗𝛾𝜕𝜉41𝜕𝑡±𝑝|(1)𝑗5𝜉2=±1𝜕in(,0)×(0,𝑇),𝑎𝑑(1)(±)𝑗𝜕𝜉𝑎10,uniformly,whenonly𝜉1,𝑎{0,1,2,3}.(4.10)

Since at the step 𝑗 the problem (4.10) gives both 𝑑(1)(±)𝑗 and 𝑑(1)(±)𝑗+1, introducing the asymptotic solution into (2.6)7 we obtain two boundary conditions for the regular part of the asymptotic solution for the displacements:𝑑1(±)𝑗(1,𝑡)=𝑑(1)(±)𝑗(0,𝑡),𝜕𝑑1(±)𝑗𝜕𝑥1(1,𝑡)=𝜕𝑑(1)(±)𝑗+1𝜕𝜉1(0,𝑡).(4.11)

In a similar way, we obtain the boundary layer correctors corresponding to the end 𝑥2=1. The boundary layers for the velocity-pressure are defined on Π2×(0,𝑇), with Π2=(1,1)×(,0), and the boundary layers for the displacements are defined also on (,0)×(0,𝑇).

We study next the problems for the regular parts of the asymptotic solution. The results are obtained for the regular part corresponding to 𝐷1𝜀; the regular part corresponding to 𝐷2𝜀 may be obtained from the previous with some obvious changes.

Introducing (4.5) into (2.6)1,2,3,11 and collecting together the terms of the same order with respect to 𝜀 we are leaded to consider the following problem for (𝑢11,𝑗,𝑢12,𝑗,𝑝1𝑗,𝑞1𝑗,𝑑1(±)𝑗):𝑥𝜈1𝜕2𝑢11,𝑗𝜕𝜉22+𝜕𝑞1𝑗𝜕𝑥1=𝑓𝛿𝑗0𝜌𝑓𝜕𝑢11,𝑗2𝜕𝜕𝑡+2𝜕𝑥1𝜈𝑥1𝜕𝑢11,𝑗2𝜕𝑥1𝑥+𝜈1𝜕2𝑢12,𝑗2𝜕𝜉2𝜕𝑥1𝜕𝑝1𝑗1𝜕𝑥1,𝜕𝑝1𝑗𝜕𝜉2=𝜌𝑓𝜕𝑢12,𝑗3+𝜕𝜕𝑡𝜕𝑥1𝜈𝑥1𝜕𝑢12,𝑗3𝜕𝑥1+𝜕𝜕𝑥1𝜈𝑥1𝜕𝑢11,𝑗1𝜕𝜉2𝑥+2𝜈1𝜕2𝑢12,𝑗1𝜕𝜉22,𝜕𝑢11,𝑗𝜕𝑥1+𝜕𝑢12,𝑗𝜕𝜉2𝜕=0in(0,1)×(1,1)×(0,𝑇),4𝑑1(+)𝑗𝜕𝑥41𝑞1𝑗=𝑔+𝛿𝑗0+𝑝1|𝑗1𝜉2=1𝜕𝜌2𝑑1(+)𝑗𝛾𝜕𝑡2𝜕𝜇5𝑑1(+)𝑗𝛾𝜕𝑥41𝜕𝜕𝑡in(0,1)×{1}×(0,𝑇),4𝑑1()𝑗𝜕𝑥41+𝑞1𝑗=𝑔𝛿𝑗0𝑝1|𝑗1𝜉2=1𝜕𝜌2𝑑1()𝑗𝛾𝜕𝑡2𝜕𝜇5𝑑1()𝑗𝛾𝜕𝑥41𝐮𝜕𝑡in(0,1)×{1}×(0,𝑇),1𝑗𝑥1=,±1,𝑡𝜕𝑑1(±)𝑗𝛾+3𝑥𝜕𝑡1𝐞,𝑡2in(0,1)×{±1}×(0,𝑇).(4.12) The two cases, 𝛾>3 and 𝛾=3, appear because of the last relation of the previous system. We can see, indeed, that for 𝛾>3 the unknown of this relation is 𝐮1𝑗, while for 𝛾=3, (4.12)6 contains two unknowns. From this point, the computations are different with respect to the values of 𝛾.

We introduce the functions:𝑁1𝜉2=12𝜉221,𝑁2𝜉2=𝜉21𝑁1(𝜏)d𝜏,(4.13) with the properties: 𝑁1=1,𝑁1(±1)=0 and 𝑁2(1)=2/3.

We also use the notations:𝑈1𝑗1𝑥1,𝜉2,𝑡=𝜌𝑓𝜕𝑢11,𝑗2𝜕𝜕𝑡2𝜕𝑥1𝜈𝑥1𝜕𝑢11,𝑗2𝜕𝑥1𝑥𝜈1𝜕2𝑢12,𝑗2𝜕𝜉2𝜕𝑥1+𝜕𝑝1𝑗1𝜕𝑥1,𝒟1(±)𝑗1𝑥1,𝑡=±𝑝1|𝑗1𝜉2=1𝜕𝜌2𝑑1(±)𝑗𝛾𝜕𝑡2𝜕𝜇5𝑑1(±)𝑗𝛾𝜕𝑥41,𝐷𝜕𝑡1𝐹𝜉21𝐹𝑥1𝐷,𝜏d𝜏,2𝐹𝜉21𝜃1𝐹𝑥11,𝜏d𝜏d𝜃2𝜉2+111𝜃1𝐹𝑥1𝐼,𝜏d𝜏d𝜃,𝑥𝑘1𝐹𝑥10𝑎𝑘10𝑎10𝐹𝑠,𝜉2𝐽,𝑡d𝑠,𝑥1𝐹𝐼𝑥41(𝐹)+𝑥31210𝐼𝑥31(𝐹)10𝐼𝑥21(𝐹)𝑥2310𝐼𝑥31(𝐹)10𝐼𝑥21.(𝐹)(4.14)

4.2.1. The Order of Solving the Problems for 𝛾>3

The regular part corresponding to 𝐷1𝜀 is computed by integrating (4.12), as stated below.

Proposition 4.2. The unknowns 𝐮1𝑗,𝑞1𝑗,𝑝1𝑗,𝑑1(±)𝑗 are determined from (4.12), up to nine functions of 𝑡.

Proof. Integrating twice (4.12)1 from −1 to 𝜉2 and using the boundary conditions (4.12)6 we get: 𝑢11,𝑗𝑥1,𝜉2=1,𝑡𝜈𝑥1𝐷2𝑈1𝑗1+1𝜈𝑥1𝜕𝑞1𝑗𝜕𝑥1𝑓𝛿𝑗0𝑁1𝜉2,(4.15) which contains as unknowns 𝑢11,𝑗 and 𝑞1𝑗. The other functions contained by this relation are either known from previous computations or equal to zero.
We integrate next the incompressibility condition (4.12)3 with respect to 𝜉2 with the boundary condition (4.12)6 for 𝜉2=1 and we obtain𝑢12,𝑗𝑥1,𝜉2=,𝑡𝜕𝑑1()𝑗𝛾+3𝜕𝜕𝑡𝜕𝑥11𝜈𝑥1𝜕𝑞1𝑗𝜕𝑥1𝑓𝛿𝑗0𝑁2𝜉2𝐷1𝐷2𝜕𝜕𝑥11𝜈𝑥1𝑈1𝑗1.(4.16) The previous two relations give the components of the velocity 𝐮1𝑗 with respect to 𝑞1𝑗.
The pressure approximations are determined from (4.12)2, supposing that the integration functions, depending on 𝑥1, 𝑡 are equal to zero, since we consider that any function depending only on 𝑥1, 𝑡 could be contained in 𝑞1𝑗+1. 𝑝1𝑗=𝐷1𝜌𝑓𝜕𝑢12,𝑗3+𝜕𝜕𝑡𝜕𝑥1𝜈𝑥1𝜕𝑢12,𝑗3𝜕𝑥1+𝜕𝜕𝑥1𝜈𝑥1𝜕𝑢11,𝑗1𝜕𝜉2𝑥+2𝜈1𝜕2𝑢12,𝑗1𝜕𝜉22.(4.17) Taking 𝜉2=1 in (4.16) and using the boundary condition (4.12)6 for 𝜉2=1 we obtain the following second order differential equation for the function 𝑞1𝑗: 𝜕𝜕𝑥11𝜈𝑥1𝜕𝑞1𝑗𝜕𝑥1𝑓𝛿𝑗0=32𝜕𝑑𝜕𝑡1(+)𝑗𝛾+3𝑑1()𝑗𝛾+3+3211𝐷2𝜕𝜕𝑥11𝜈𝑥1𝑈1𝑗1d𝜉2.(4.18) Integrating (4.18) from 𝑥1 to 1, we express 𝜕𝑞1𝑗/𝜕𝑥1 by means of (𝜕𝑞1𝑗/𝜕𝑥1)(1,𝑡), which represents the only unknown of this expression. This function of 𝑡 is obtained as follows: we take 𝑥1=1 in (4.15) and we introduce the result into (4.9). Hence, we determined the expression of 𝜕𝑞1𝑗/𝜕𝑥1 in (0,1)×(0,𝑇), which is: 1𝜈𝑥1𝜕𝑞1𝑗𝜕𝑥1𝑓𝛿𝑗03=2𝛿𝑗011𝜓𝜉2,𝑡d𝜉232dd𝑡0𝑑(1)(+)𝑗+2𝛾𝑑(1)()𝑗+2𝛾𝜉1,𝑡d𝜉132𝜕𝜕𝑡1𝑥1𝑑1(+)𝑗𝛾+3𝑑1()𝑗𝛾+33(𝑠,𝑡)d𝑠+211𝐷21𝜈𝑥1𝑈1𝑗1d𝜉2.(4.19) Introducing (4.19) into (4.15) and (4.18) into (4.16) we determine 𝐮1𝑗 in (0,1)×(1,1)×(0,𝑇).
We integrate next 𝜈(𝑥1) · (4.19) from 0 to 𝑥1 and we get 𝑞1𝑗 determined up to the function of 𝑡, 𝑞1𝑗(0,𝑡): 𝑞1𝑗𝑥1,𝑡=𝛿𝑗0𝑥103𝑓(𝑠,𝑡)d𝑠2𝑥10𝜈(𝜃)1𝜃𝜕𝑑𝜕𝑡1(+)𝑗𝛾+3𝑑1()𝑗𝛾+33(𝑠,𝑡)d𝑠d𝜃2dd𝑡0𝑑(1)(+)𝑗+2𝛾𝑑(1)()𝑗+2𝛾𝜉1,𝑡d𝜉1+𝛿𝑗011𝜓𝜉2,𝑡d𝜉2𝑥10𝜈+3(𝑠)d𝑠2𝑥1011𝐷2𝑈1𝑗1𝑠,𝜉2,𝑡d𝜉2d𝑠+𝑞1𝑗(0,𝑡).(4.20) The functions 𝑑1(±)𝑗 satisfy the fourth-order differential equations: 𝜕4𝑑1(±)𝑗𝜕𝑥41=±𝑞1𝑗+𝑔±𝛿𝑗0+𝒟1(±)𝑗1,(4.21) with 𝑞1𝑗 given by (4.20). Writting 𝑞1𝑗 as 𝑄1𝑗(𝑥1,𝑡)+𝑞1𝑗(0,𝑡) and integrating four times (4.21) with respect to 𝑥1 we obtain the following expressions for 𝑑1(±)𝑗: 𝑑1(+)𝑗𝑥1,𝑡=𝑑1(+)𝑗(0,𝑡)13𝑥12+2𝑥13+𝜕𝑑1(+)𝑗𝜕𝑥1𝑥(0,𝑡)12𝑥12+𝑥13+𝑑1(+)𝑗(1,𝑡)3𝑥122𝑥13+𝜕𝑑1(+)𝑗𝜕𝑥1(1,𝑡)𝑥12+𝑥13+𝑞1𝑗𝑥(0,𝑡)12𝑥2413+𝑥121424+𝐽𝑥1𝑄1𝑗+𝐽𝑥1𝑔+𝛿𝑗0+𝐽𝑥1𝒟1(+)𝑗1,𝑑1()𝑗𝑥1,𝑡=𝑑1()𝑗(0,𝑡)13𝑥12+2𝑥13+𝜕𝑑1()𝑗𝜕𝑥1𝑥(0,𝑡)12𝑥12+𝑥13+𝑑1()𝑗(1,𝑡)3𝑥122𝑥13+𝜕𝑑1()𝑗𝜕𝑥1(1,𝑡)𝑥12+𝑥13𝑞1𝑗𝑥(0,𝑡)12𝑥2413+𝑥121424𝐽𝑥1𝑄1𝑗+𝐽𝑥1𝑔𝛿𝑗0+𝐽𝑥1𝒟1()𝑗1.(4.22) Hence, the regular part of the asymptotic solution corresponding to 𝐷1𝜀 is determined up to the functions 𝑞1𝑗(0,𝑡), 𝑑1(±)𝑗(0,𝑡), (𝜕𝑑1(±)𝑗/𝜕𝑥1)(0,𝑡), 𝑑1(±)𝑗(1,𝑡), (𝜕𝑑1(±)𝑗/𝜕𝑥1)(1,𝑡), which achieves the proof.

In a similar way we express the regular part of the asymptotic solution corresponding to 𝐷2𝜀 depending on 9 unknown functions of 𝑡.

We continue the construction with the problems for the corrector in 𝑥=𝟎. As we previously said, the term of the asymptotic solution containing this corrector is not equal to zero only in a neighborhood of 𝐷𝑟𝜀. In this neighborhood, the expression of the asymptotic solution reduces to (𝐮1(𝑘)(𝑥1,𝜉2,𝑡)𝜒(𝜉1)+𝐮2(𝑘)(𝜉1,𝑥2,𝑡)𝜒(𝜉2)+𝐮(𝑘)0𝑏𝑙(𝜉,𝑡),𝑝1(𝑘)(𝑥1,𝜉2,𝑡)𝜒(𝜉1)+𝑝2(𝑘)(𝜉1,𝑥2,𝑡)𝜒(𝜉2𝑝)+(𝑘)0𝑏𝑙(𝜉,𝑡),𝑑(𝑘)±1(𝑥1,𝑡)𝜒(𝜉1)+𝑑(𝑘)±2(𝑥2,𝑡)𝜒(𝜉2)+𝑑(𝑘)0±𝑏𝑙(𝜉,𝑡)). For obtaining the problems satisfied by the correctors in 𝑥=𝟎, we introduce the previous expression of the asymptotic solution in (2.6)1,2,3,4,5,6,11, with 𝜈=𝜈0 and 𝐟=𝐠=𝟎 in (2.6)1,3; for derivating the terms which contain the two types of variables 𝑥𝑖 and 𝜉𝑖 we proceed as follows: we replace 𝜕/𝜕𝑥𝑖 by (1/𝜀)(𝜕/𝜕𝜉𝑖), we replace 𝑥𝑖 by 𝜀𝜉𝑖 and we expand the functions as a Taylor expansion with respect to 𝑥𝑖=𝜀𝜉𝑖. We introduce the notations =𝐷𝑟{(𝜉1,𝜉2)𝜉12,𝜉2(1,1)}{(𝜉1,𝜉2)𝜉1(1,1),𝜉22},𝐴±𝐵±=(1/𝜀)𝐴𝜀±𝐵𝜀± and we obtain for (𝐮𝑗(0),𝑝𝑗(0)) the following nondivergence free problem:𝜈0Δ𝜉𝜉𝐮𝑗(0)+𝜉𝑝𝑗(0)=𝜌𝑓𝜕𝐮(0)𝑗2𝜕𝑡+𝐅𝑗+𝜉1𝜒1𝜉1𝑞1𝑗+1(0,𝑡)𝐞1+𝜉