Pressure-driven flow in thin domains

Abstract

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure.

Keywords

Stokes equation pressure boundary condition two-scale convergence thin domain Bogovskii operator Korn inequality

1. Introduction

We consider the flow of an incompressible Newtonian fluid in a thin domain $Ω^{ε}$ in $R^{m + l}$ whose characteristic length ε in the l directions is much smaller than those in the other m directions. Such thin structures allow the following representation of the boundary: $\partial Ω^{ε} = Γ_{N}^{ε} \cup Γ_{D}^{ε}$ , where the measure of $Γ_{N}^{ε}$ (Neumann boundary) is of the order $ε^{l}$ while the measure of $Γ_{D}^{ε}$ (Dirichlet boundary) is of order unity. The Dirichlet part is modelled as a solid wall by imposing a no-slip condition for the velocity field. The Neumann part is modelled as a penetrable inlet/outlet boundary, by prescribing the normal stress. Thus, the flow is driven by an external pressure gradient alone, which is applied as a surface force along $Γ_{N}^{ε}$ . That is, no body forces or moving boundaries are contributing to the motion of the fluid, although such effects can be superimposed. Since the flow is assumed to be governed by the Stokes equation, inertial effects are also neglected.

Our aim is to obtain a precise asymptotic description of the flow by letting the thickness parameter ε tend to zero. A dimensional reduction of the problem is accomplished by a technique called two-scale convergence for thin domains, which was developed by Marušić and Marušić-Paloka [11]. This framework offers an elegant approach compared to previous methods which have relied mostly on rescaling techniques.

As a result we obtain a generalized form of the Reynolds–Poiseuille law. The original laws were formulated by Reynolds [16] in the context of lubrication theory (moving boundaries) and Poiseuille [15] who studied flow of water through thin glass tubes and the effects of pressure drop. The first rigorous mathematical derivation of Reynolds work was done by Bayada and Chambat [1]. Starting from the Stokes system and assuming that the velocity field satisfies a Dirichlet condition on the whole boundary, the authors prove that the limit pressure satisfies the classical Reynolds equation with a Neumann condition. In real-life applications, however, it is more common that the Reynolds equation is solved by prescribing the boundary values of the pressure [13]. Under present assumptions we obtain the Reynolds equation with a Dirichlet boundary condition for the pressure, which is in better agreement with engineering practice. Furthermore, this illustrates the general manner in which boundary conditions are preserved for asymptotic limits:

A Dirichlet condition for the velocity field in the original problem, posed in a thin domain, becomes a Neumann condition for the pressure in the limit problem, posed in a domain of unit thickness.

A Neumann condition for the stress tensor in the original problem becomes a Dirichlet condition for the pressure in the limit problem.

A similar result holds for Stokes flow in a porous medium [5].

Let us also mention that there seems to be no precise definition of the term “pressure-driven flow” in the scientific literature. From a physical point of view one can argue that the stress vector (traction) should be continuous across the interface between two similar fluids, see e.g. Lamb [9, Art. 327]. It therefore makes sense to prescribe all components of the stress vector on some part of the boundary provided that the surrounding fluid is of the same constitutive type. The situation is more complicated for the interface between two dissimilar fluids, say a droplet of water surrounded by air, which is characterized by a jump in the density and continuity of the tangential velocity. For a general discussion regarding boundary conditions at fluid-fluid interfaces we refer to Ch. 5 of Panton [12]. Although the pressure of an incompressible fluid has a precise mathematical definition, it does not have a clearly defined physical meaning – one can measure only the normal component of the total stress (including the viscous part). We therefore choose to prescribe the boundary values for the pressure implicitly as a Neumann condition for the stress tensor. As shown in the works of Glowinski [7,8] and Pironneau [14], the resulting boundary value problem is well-posed. In particular, for the mixed boundary value problem both pressure and velocity are uniquely determined. Nevertheless, alternative formulations exist. For example, it is possible to impose an explicit boundary condition for the pressure on one part of the boundary as shown in [3,10], provided that the tangential component of the velocity is prescribed on the same part. Both formulations could be legitimate – the preferred choice depends on the boundary data available. Suppose that one could measure only the average stresses at the inlet and the outlet of a finite pipe connecting two large reservoirs of fluid, where the streamlines bend slightly at both ends of the pipe. Then the normal stress condition seems like the most realistic alternative.

2. Problem formulation

2.1. Thin geometry

Following [11], let ω be a bounded domain in $R^{m}$ with a Lipschitz boundary $\partial ω$ and let ${S (x^{1})}_{x^{1} \in \overline{ω}}$ be a family of domains in $R^{l}$ such that:

For each $x^{1} \in \overline{ω}$ , the set $S (x^{1})$ is contained in the unit cube ${(- 1 / 2, 1 / 2)}^{l}$ in $R^{l}$ .

There exists $α > 0$ such that the ball $\begin{matrix} B (0, α) = {y \in R^{l} : | y | < α} \end{matrix}$ of radius α is contained in $S (x^{1})$ for each $x^{1} \in \overline{ω}$ .

The set $\begin{matrix} Ω = {(x^{1}, y) \in R^{m + l} : x^{1} \in ω, y \in S (x^{1})} \end{matrix}$ forms a Lipschitz domain in $R^{m + l}$ .

The boundary of Ω is divided into two disjoint parts

\begin{matrix} \partial Ω = Γ_{D} \cup Γ_{N} \\ Γ_{D} = {(x^{1}, y) \in R^{m + l} : x^{1} \in ω, y \in \partial S (x^{1})} \\ Γ_{N} = {(x^{1}, y) \in R^{m + l} : x^{1} \in \partial ω, y \in S (x^{1})} . \end{matrix}

The thin geometry

Ω^{ε}

is defined by anisotropic scaling of the domain Ω using the parameter

0 < ε ⩽ 1

\begin{matrix} (1) & \begin{matrix} Ω^{ε} = {(x^{1}, x^{2}) \in R^{m + l} : x^{1} \in ω, x^{2} \in ε S (x^{1})} \\ Γ_{D}^{ε} = {(x^{1}, x^{2}) \in R^{m + l} : x^{1} \in ω, x^{2} \in ε \partial S (x^{1})} \\ Γ_{N}^{ε} = {(x^{1}, x^{2}) \in R^{m + l} : x^{1} \in \partial ω, x^{2} \in ε S (x^{1})} . \end{matrix} \end{matrix}

In particular, for

m + l = 3

there are two possible configurations: the thin film and the thin pipe (see Fig. 1 and 2 respectively).

Fig. 1.

Structure of Ω (thin film).

Fig. 2.

Structure of Ω (thin pipe).

2.2. Function spaces

Let $n = m + l$ . We shall work mainly with the following spaces $\begin{array}{l} W (Ω^{ε}) = {φ \in H^{1} (Ω^{ε}; R^{n}) : φ = 0 on Γ_{D}^{ε}} \\ V (Ω^{ε}) = {φ \in W (Ω^{ε}) : div φ = 0 in Ω^{ε}} . \end{array}$ Clearly $V (Ω^{ε}) \subset W (Ω^{ε})$ , both being closed subspaces of $H^{1} (Ω^{ε}; R^{n})$ . Since $Γ_{D}^{ε}$ has positive surface measure (by assumption) we can equip $W (Ω^{ε})$ with the norm $\begin{matrix} ‖ v ‖_{W (Ω^{ε})} = ‖ \nabla v ‖_{L^{2} (Ω^{ε})} = {(\int_{Ω^{ε}} | \nabla v |^{2} d x)}^{1 / 2}, \end{matrix}$ which is equivalent to the usual $H^{1}$ -norm by Friedrichs’ inequality.

2.3. Problem

In $Ω^{ε}$ we pose the following boundary value problem for the velocity field $u^{ε}$ and the pressure field $p^{ε}$ : $\begin{matrix} (2) & \{\begin{matrix} div (- p^{ε} I + 2 μ e (\nabla u^{ε})) = 0 & in Ω^{ε} (2a) \\ div u^{ε} = 0 & in Ω^{ε} (2b) \\ (- p^{ε} I + 2 μ e (\nabla u^{ε})) \hat{n} = - p_{b} \hat{n} & on Γ_{N}^{ε} (2c) \\ u^{ε} = 0 & on Γ_{D}^{ε} (2d) \end{matrix} \end{matrix}$ Here $\hat{n}$ is the outward unit normal of $\partial Ω^{ε}$ and $\begin{matrix} e (\nabla u^{ε}) = \frac{1}{2} (\nabla u^{ε} + {(\nabla u^{ε})}^{T}) \end{matrix}$ denotes the symmetric part of the velocity gradient. Thus, (2) becomes the Stokes system, driven by an external pressure $p_{b}$ , which is implicitly imposed by condition (2c). The scalar function $p_{b} \in H^{1} (Ω^{ε})$ is assumed to depend only on the variable $x^{1} \in R^{m}$ . Such a restriction seems reasonable as it excludes circular or re-entrant flow in the vicinity of $Γ_{N}^{ε}$ . Note also that the system (2) depends only the boundary values of $p_{b}$ restricted to $Γ_{N}^{ε}$ .

2.4. Existence and uniqueness

Let the normalized pressure $q^{ε}$ be defined by $\begin{matrix} q^{ε} = p^{ε} - p_{b} . \end{matrix}$ Then a weak formulation of (2) is given by $\begin{matrix} (3) & \int_{Ω^{ε}} - q^{ε} div φ + 2 μ e (\nabla u^{ε}) : \nabla φ + \nabla p_{b} \cdot φ d x = 0, \end{matrix}$ for all $φ \in W (Ω^{ε})$ , where $\nabla p_{b} = (\partial p_{b} / \partial x_{1}^{1}, \dots, \partial p_{b} / \partial x_{m}^{1}, 0, \dots, 0)$ . By standard arguments, such as the Korn inequality and the Lax–Milgram theorem, it can be shown that the weak formulation (3) has a unique solution $(u^{ε}, q^{ε})$ in $V (Ω^{ε}) \times L^{2} (Ω^{ε})$ . For the details we refer to [8, Ch. 4] or [4]. However, to prove the homogenization result (see Theorem 3.4 below) we need careful a priori estimates for the velocity and the pressure. This question is addressed in Section 4.

3. Main result

The notion of two-scale convergence for thin domains was introduced by Marušić and Marušić-Paloka in [11]. We recall here the definition.

Definition 3.1.
Following the notation in Section 2.1 we say that a sequence ${u^{ε}}_{ε > 0}$ in $L^{2} (Ω^{ε})$ two-scale converges to u in $L^{2} (Ω)$ provided that $\begin{matrix} (4) & lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} u^{ε} (x) φ (x^{1}, x^{2} / ε) d x = \frac{1}{| Ω |} \int_{Ω} u (x^{1}, y) φ (x^{1}, y) d x^{1} d y \end{matrix}$ for all φ in $L^{2} (Ω)$ and we write $u^{ε} \overset{2}{⇀} u$ . Moreover, we say that $u^{ε}$ two-scale converges strongly to u if $\begin{matrix} (5) & lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| u^{ε} (x) - u (x^{1}, x^{2} / ε) |}^{2} d x = 0 \end{matrix}$ and we write $u^{ε} \overset{2}{\to} u$ (strongly).

Next, we introduce the concept of permeability function for thin domains.
Definition 3.2.
The solution ϕ of the boundary value problem $\begin{matrix} (6) & \{\begin{matrix} - Δ_{y} ϕ = 1 & in Ω \\ ϕ = 0 & on Γ_{D}, \end{matrix} \end{matrix}$ is called the permeability function of Ω. That is, for each $x^{1} \in ω$ , $ϕ (x^{1}, \cdot)$ is the solution of $\begin{matrix} \{\begin{matrix} - Δ_{y} ϕ (x^{1}, \cdot) = 1 & in S (x^{1}) \\ ϕ (x^{1}, \cdot) = 0 & on \partial S (x^{1}), \end{matrix} \end{matrix}$ where $\begin{matrix} Δ_{y} = \frac{\partial^{2}}{\partial y_{1}^{2}} + \dots + \frac{\partial^{2}}{\partial y_{l}^{2}} . \end{matrix}$ The permeability of the domain $S (x^{1})$ is defined as $\begin{matrix} (7) & a (x^{1}) = \int_{S (x^{1})} ϕ (x^{1}, y) d y (x^{1} \in ω) . \end{matrix}$
Remark 3.3.
The regularity of the function ϕ, defined by equation (6), depends on the smoothness of the boundary $\partial Ω$ . For the main result to hold it seems that a minimum requirement would be $\begin{matrix} ess \sup_{x^{1} \in ω} \int_{S (x^{1})} | \nabla_{y} ϕ |^{2} d y < \infty \end{matrix}$ with $a \in W^{1, \infty} (ω)$ . Furthermore, hypothesis (ii) in Section 2.1 is necessary to have $1 / a \in W^{1, \infty} (ω)$ . To avoid technicalities, we assume the stronger condition that ϕ belongs to $W^{1, \infty} (Ω)$ . Indeed, this is the case for Examples 3.1 and 3.2 below (both very important from application point of view). In the general case, however, it is not clear to us if the requirement that $\partial Ω$ is of class $C^{0, 1}$ is sufficient to ensure $W^{1, \infty}$ -regularity for ϕ.
Theorem 3.4 (Main result).

For each $0 < ε ⩽ 1$ the boundary value problem ( 2 ) has a unique solution $(u^{ε}, p^{ε}) \in V (Ω^{ε}) \times L^{2} (Ω^{ε})$ such that $\begin{matrix} (8) & \frac{1}{| Ω^{ε} |^{1 / 2}} (μ {‖ ε^{- 2} u^{ε} ‖}_{L^{2} (Ω^{ε})} + μ {‖ ε^{- 1} \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ p^{ε} ‖}_{L^{2} (Ω^{ε})}) ⩽ C ‖ \nabla p_{b} ‖_{L^{2} (Ω)}, \end{matrix}$ where the constant C depends only on Ω and $Γ_{D}$ . Let ϕ denote the permeability function defined by ( 6 ). Then the following strong two-scale convergence holds, $\begin{matrix} ε^{- 2} u^{ε} \overset{2}{\to} u, p^{ε} \overset{2}{\to} p, \end{matrix}$ where u in $L^{2} (Ω; R^{n})$ is defined by $\begin{matrix} u (x^{1}, y) = - \frac{1}{μ} ϕ (x^{1}, y) \nabla p (x^{1}), \end{matrix}$ and p in $H^{1} (ω)$ is the solution of the boundary value problem $\begin{matrix} (9) & \{\begin{matrix} div (a \nabla p) = 0 & in ω (9a), \\ p = p_{b} & on \partial ω (9b), \end{matrix} \end{matrix}$ where a is defined by ( 7 ).

Note that the Dirichlet boundary condition (9b) for the limit pressure arises from the Neumann condition (2c) for the stress tensor in the original problem. By contrast, if one imposes a non-homogeneous Dirchlet condition for the velocity field on $Γ_{N}^{ε}$ , one ends up with a Neumann condition in (9) (see equation (5.7) in [1]). For practical purposes, the main difference between the two is that the limit pressure is uniquely determined by (9b), whereas it is only defined up to an additive constant for the Neumann problem. The limit velocity u is uniquely defined in both formulations.

3.1. Example 1 (Reynolds’ law)

Let $m = 2$ and $l = 1$ . For simplicity $x^{1} \in ω$ is denoted as x. Suppose ω is a Lipschitz domain in $R^{2}$ and that $S (x)$ is the line segment $\begin{matrix} S (x) = {y \in R : h^{-} (x) < y < h^{+} (x)}, \end{matrix}$ where $h^{\pm} \in C^{0, 1} (\overline{ω})$ (see Fig. 1 and Fig. 3(a)). Thus, the thickness of the film is given by the function $h = h^{+} - h^{-}$ , which is assumed to be bounded from below by the constant $2 α$ . Then the permeability function satisfies $\begin{matrix} \{\begin{matrix} - \frac{\partial^{2} ϕ}{\partial y^{2}} = 1, & if h^{-} (x) < y < h^{+} (y) \\ ϕ = 0, & if y = h^{\pm} (x), \end{matrix} \end{matrix}$ whose solution is given by $\begin{matrix} ϕ (x, y) = \frac{1}{2} (h^{+} (x) - y) (y - h^{-} (x)), a (x) = \frac{h {(x)}^{3}}{12} . \end{matrix}$ Hence, the pressure equation (9) becomes the classical Reynolds equation with a Dirichlet boundary condition: $\begin{matrix} (10) & \{\begin{matrix} div (h^{3} \nabla p) = 0 & in ω \\ p = p_{b} & on \partial ω . \end{matrix} \end{matrix}$ This equation together with $\begin{matrix} u (x, y) = - \frac{1}{2 μ} (h^{+} (x) - y) (y - h^{-} (x)) \nabla p (x) \end{matrix}$ gives the asymptotic solution for pressure-driven incompressible flow in a thin film of variable thickness $h (x)$ .

Fig. 3.

Fluid domains.

3.2. Example 2 (Poiseuille’s law)

Let $m = 1$ and $l = 2$ . As above, $x^{1} \in ω$ is denoted as x. Suppose $ω \subset R$ is the line segment $0 < x < L$ and that $S (x)$ is the disc $\begin{matrix} S (x) = {y \in R^{2} : | y | < h (x)}, \end{matrix}$ where $h \in C^{0, 1} (\overline{ω})$ (see Fig. 2 and Fig. 3(b)). The radius $h (x)$ is assumed to be bounded from below by the constant α. Switching to polar coordinates, $y = (r cos θ, r sin θ)$ , we see that the permeability function ϕ satisfies $\begin{matrix} \{\begin{matrix} - \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial ϕ}{\partial r}) - \frac{1}{r^{2}} \frac{\partial^{2} ϕ}{\partial θ^{2}} = 1, & if 0 ⩽ r < h (x) \\ ϕ = 0, & if r = h (x) . \end{matrix} \end{matrix}$ Solving this equation gives $\begin{matrix} ϕ (x, y) = \frac{h {(x)}^{2} - | y |^{2}}{4}, a (x) = \frac{π h {(x)}^{4}}{8} . \end{matrix}$ Hence, (9) becomes the classical Poiseuille equation with a Dirichlet boundary condition: $\begin{matrix} (11) & \{\begin{matrix} \frac{d}{d x} (h^{4} \frac{d p}{d x}) = 0, & in 0 < x < L \\ p = p_{b}, & on x = 0, x = L . \end{matrix} \end{matrix}$ Thus, the last equation together with $\begin{matrix} u (x, y) = - \frac{1}{4 μ} (h {(x)}^{2} - | y |^{2}) \nabla p (x) \end{matrix}$ gives the asymptotic solution for pressure-driven incompressible flow in a thin circular pipe of length L and of variable radius $h (x)$ .

4. Estimates

4.1. Korn inequality

To obtain precise a priori estimates of the velocity field $u^{ε}$ in (3) we need to control the constant in the Korn inequality for the domain $Ω^{ε}$ as ε tends to zero. The following estimates show how the constant depends on the parameter ε.

Theorem 4.1.
Let $K_{1}^{ε}$ and $K_{2}^{ε}$ denote the best constants in the inequalities $\begin{array}{l} (12) & ‖ v ‖_{L^{2} (Ω^{ε})} ⩽ K_{1}^{ε} {‖ e (\nabla v) ‖}_{L^{2} (Ω^{ε})} \\ (13) & ‖ \nabla v ‖_{L^{2} (Ω^{ε})} ⩽ K_{2}^{ε} {‖ e (\nabla v) ‖}_{L^{2} (Ω^{ε})} \end{array}$ for all $v \in W (Ω^{ε})$ . Then there exist constants $K_{1}$ and $K_{2}$ depending only on Ω and $Γ_{D}$ such that $\begin{matrix} sup_{0 < ε ⩽ 1} ε^{- 1} K_{1}^{ε} ⩽ K_{1}, sup_{0 < ε ⩽ 1} K_{2}^{ε} ⩽ K_{2} . \end{matrix}$
Proof.
For a proof see Appendix A. □

4.2. Velocity estimates

Theorem 4.2.
For the velocity $u^{ε}$ , the following estimates hold $\begin{matrix} (14) & \begin{matrix} {‖ ε^{- 1} \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ \frac{K_{1} K_{2}}{2 μ} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} \\ {‖ ε^{- 2} u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ \frac{K_{1}^{2}}{2 μ} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})}, \end{matrix} \end{matrix}$ where $K_{1}$ and $K_{2}$ are the constants in Theorem 4.1 .
Proof.
By choosing $v = u^{ε}$ in (3), using $div u^{ε} = 0$ , we establish $\begin{matrix} (15) & \int_{Ω^{ε}} 2 μ {| e (\nabla u^{ε}) |}^{2} d x = - \int_{Ω^{ε}} \nabla p_{b} \cdot u^{ε} d x . \end{matrix}$ By the Korn inequality (12), the right-hand side can be estimated as $\begin{matrix} - \int_{Ω^{ε}} \nabla p_{b} \cdot u^{ε} d x & ⩽ ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})} \\ ⩽ ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} {‖ e (\nabla u^{ε}) ‖}_{L^{2} (Ω^{ε})} . \end{matrix}$ Thus, we obtain $\begin{matrix} 2 μ {‖ e (\nabla u^{ε}) ‖}_{L^{2} (Ω^{ε})} ⩽ ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} . \end{matrix}$ Applying the Korn inequality anew (to the left side) gives $\begin{matrix} 2 μ {‖ u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ ε^{2} K_{1}^{2} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})}, \\ 2 μ {‖ \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ ε K_{1} K_{2} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} . \end{matrix}$ □
4.3. The Bogovskiĭ operator

Let $W / V (Ω^{ε})$ denote the quotient space of $W (Ω^{ε})$ by $V (Ω^{ε})$ . Thus, $W / V (Ω^{ε})$ consists of equivalence classes ${v}$ of elements $v \in W (Ω^{ε})$ modulo the space of divergence free elements. By a result of Bogovskiĭ [2] (see also [4,6,17]) the operator $\begin{matrix} A : W / V (Ω^{ε}) \to L^{2} (Ω^{ε}) \end{matrix}$ defined by $A {v} = div v$ is an isomorphism, provided that $Γ_{N}^{ε}$ has positive surface measure and non-empty interior. The so called Bogovskiĭ operator $\begin{matrix} B : L^{2} (Ω^{ε}) \to W / V (Ω^{ε}), \end{matrix}$ is defined as the inverse of the operator A. It follows that the dual operator $\begin{matrix} B^{'} : V^{⊥} (Ω^{ε}) \to L^{2} (Ω^{ε}) \end{matrix}$ is an isomorphism called the pressure operator (or de Rham–Tartar operator). Subsequently we write $B = B_{Ω^{ε}}$ to emphasize the dependence of B on the domain $Ω^{ε}$ . To obtain a $L^{2}$ -bound for the pressure, we estimate the operator norm $\begin{matrix} ‖ B_{Ω^{ε}}^{'} ‖ = ‖ B_{Ω^{ε}} ‖ = sup_{\begin{array}{c} p \in L^{2} (Ω^{ε}) \\ ‖ p ‖ = 1 \end{array}} inf_{\begin{array}{c} v \in W (Ω^{ε}) \\ div v = p \end{array}} {(\int_{Ω^{ε}} | \nabla v |^{2} d x)}^{1 / 2} . \end{matrix}$

Theorem 4.3.
Let $\begin{matrix} B_{Ω^{ε}} : L^{2} (Ω^{ε}) \to W / V (Ω^{ε}), \end{matrix}$ denote the Bogovskiĭ operator. Then there exist positive constants $C_{1}$ and $C_{2}$ depending only on Ω and $Γ_{D}$ such that $\begin{matrix} C_{1} ⩽ inf_{0 < ε ⩽ 1} ε ‖ B_{Ω^{ε}} ‖ ⩽ sup_{0 < ε ⩽ 1} ε ‖ B_{Ω^{ε}} ‖ ⩽ C_{2} . \end{matrix}$
Proof.
For a proof see Appendix B. □
Remark 4.4.
We would like to point out that the lower bound in Theorem 4.3 for the norm of the Bogovskiĭ operator is not needed to prove the main result, nor is it used anywhere in the analysis. It is included only to show that the norm does indeed “blow up” as $ε \to 0$ and that an upper bound of the form $‖ B_{Ω^{ε}} ‖ ⩽ C_{2} ε^{- 1}$ is the best possible.

4.4. Pressure estimate

Theorem 4.5.
The normalized pressure $q^{ε}$ satisfies the estimate $\begin{matrix} (16) & {‖ q^{ε} ‖}_{L^{2} (Ω^{ε})} ⩽ 2 C_{2} K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})}, \end{matrix}$ where $K_{1}$ is the constant in Theorem 4.1 and $C_{2}$ is the constant in Theorem 4.3 .
Proof.
Let $F^{ε} \in W^{'} (Ω^{ε})$ be defined by $\begin{matrix} ⟨ F^{ε}, φ ⟩ = \int_{Ω^{ε}} 2 μ e (\nabla u^{ε}) : \nabla φ + \nabla p_{b} \cdot φ d x . \end{matrix}$ From (3), we see that $F^{ε}$ belongs to $V^{⊥} (Ω^{ε})$ . Since $\begin{matrix} | ⟨ F^{ε}, φ ⟩ | & ⩽ 2 μ {‖ e (\nabla u^{ε}) ‖}_{L^{2} (Ω^{ε})} ‖ \nabla φ ‖_{L^{2} (Ω^{ε})} + ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} ‖ φ ‖_{L^{2} (Ω^{ε})} \\ ⩽ ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} ‖ \nabla φ ‖_{L^{2} (Ω^{ε})} + ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} ‖ \nabla φ ‖_{L^{2} (Ω^{ε})} \\ = 2 ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} ‖ \nabla φ ‖_{L^{2} (Ω^{ε})}, \end{matrix}$ we deduce $\begin{matrix} {‖ F^{ε} ‖}_{W^{'} (Ω^{ε})} ⩽ 2 ε K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} . \end{matrix}$ Let $q^{ε}$ in $L^{2} (Ω^{ε})$ be defined by the pressure operator, i.e. $\begin{matrix} q^{ε} = B_{Ω^{ε}}^{'} F^{ε} ⟺ \int_{Ω^{ε}} q^{ε} div φ d x = ⟨ F^{ε}, φ ⟩ \forall φ \in W (Ω^{ε}) . \end{matrix}$ From Theorem 4.3 we infer $\begin{matrix} {‖ q^{ε} ‖}_{L^{2} (Ω^{ε})} & ⩽ ‖ B_{Ω^{ε}}^{'} ‖ {‖ F^{ε} ‖}_{W^{'} (Ω^{ε})} \\ ⩽ 2 ε K_{1} ‖ B_{Ω^{ε}}^{'} ‖ ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} \\ ⩽ 2 C_{2} K_{1} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} . \end{matrix}$ □
5. Two-scale convergence

5.1. Convergence result

Since $p_{b} \in H^{1} (ω)$ , we have $\begin{matrix} \frac{1}{| Ω^{ε} |^{1 / 2}} ‖ \nabla p_{b} ‖_{L^{2} (Ω^{ε})} = \frac{1}{| Ω |^{1 / 2}} ‖ \nabla p_{b} ‖_{L^{2} (Ω)} . \end{matrix}$ By Theorems 4.2 and 4.5, the solution $(u^{ε}, q^{ε})$ of (3) satisfies $\begin{matrix} (17) & \frac{1}{| Ω^{ε} |^{1 / 2}} ({‖ ε^{- 2} u^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ ε^{- 1} \nabla u^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ q^{ε} ‖}_{L^{2} (Ω^{ε})}) ⩽ C \end{matrix}$ for some constant C that does not depend on ε. Indeed, C can be chosen as $\begin{matrix} C = (\frac{K_{1}^{2} + K_{1} K_{2}}{2 μ} + 2 C_{2} K_{1}) \frac{1}{| Ω |^{1 / 2}} ‖ \nabla p_{b} ‖_{L^{2} (Ω)} . \end{matrix}$

Lemma 5.1.
For any sequence ${(u^{ε}, q^{ε})}_{0 < ε ⩽ 1}$ in $W (Ω^{ε}) \times L^{2} (Ω^{ε})$ satisfying the bound ( 17 ), there exists $u \in L^{2} (Ω; R^{n})$ with $\nabla_{y} u \in L^{2} (Ω; R^{n \times l})$ , where $\begin{matrix} \nabla_{y} = (\frac{\partial}{\partial y_{1}}, \dots, \frac{\partial}{\partial y_{l}}), \end{matrix}$ and $q \in L^{2} (Ω)$ such that, up to a subsequence,
$\begin{matrix} ε^{- 2} u^{ε} \overset{2}{⇀} u = (u_{1}, u_{2}), \end{matrix}$ where $u_{1}$ denotes the m first components of u and $u_{2}$ denotes the l last components of u,

$\begin{matrix} ε^{- 1} \nabla u^{ε} \overset{2}{⇀} D (u) = (\begin{matrix} 0 & \nabla_{y} u_{1} \\ 0 & \nabla_{y} u_{2} \end{matrix}) and u = 0 on Γ_{D}, \end{matrix}$

$\begin{matrix} ε^{- 1} e (\nabla u^{ε}) \overset{2}{⇀} \frac{1}{2} (D (u) + D {(u)}^{T}) = \frac{1}{2} (\begin{matrix} 0 & \nabla_{y} u_{1} \\ {(\nabla_{y} u_{1})}^{T} & 2 e (\nabla_{y} u_{2}) \end{matrix}), \end{matrix}$

$\begin{matrix} q^{ε} \overset{2}{⇀} q . \end{matrix}$

Proof.
For the sake of completeness we provide the proof which follows the same ideas as in [11]. In view of (17) and Theorem 1 in [11] there exists $u \in L^{2} (Ω; R^{n})$ such that, up to a subsequence, $\begin{matrix} lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} ε^{- 2} u^{ε} \cdot φ (x^{1}, x^{2} / ε) d x = \frac{1}{| Ω |} \int_{Ω} u (x^{1}, y) \cdot φ (x^{1}, y) d x^{1} d y \end{matrix}$ for any $φ \in L^{2} (Ω; R^{n})$ . Furthermore, there exists $δ \in L^{2} (Ω; R^{n \times n})$ such that, up to a subsequence, $\begin{matrix} lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} ε^{- 1} \nabla u^{ε} : Φ (x^{1}, x^{2} / ε) d x = \frac{1}{| Ω |} \int_{Ω} δ (x^{1}, y) : Φ (x^{1}, y) d x^{1} d y \end{matrix}$ for any $Φ \in L^{2} (Ω; R^{n \times n})$ .

Since $u^{ε}$ vanishes on $Γ_{D}$ we have $\begin{matrix} (18) & \begin{matrix} 0 & = \int_{\partial Ω^{ε}} u^{ε} \cdot Φ (x^{1}, x^{2} / ε) \hat{n} d S \\ = \int_{Ω^{ε}} \nabla u^{ε} : Φ (x^{1}, x^{2} / ε) + u^{ε} \cdot ({div}_{x^{1}} + ε^{- 1} {div}_{y}) Φ (x^{1}, x^{2} / ε) d x \end{matrix} \end{matrix}$ for all $Φ \in H^{1} (Ω; R^{n \times n})$ such that $Φ \hat{n} = 0$ on $Γ_{N}$ . Multiplying (18) with $ε^{- 1} / | Ω^{ε} |$ and passing to the limit $ε \to 0$ gives $\begin{matrix} (19) & 0 = \int_{Ω} δ (x^{1}, y) : Φ (x^{1}, y) + u (x^{1}, y) \cdot {div}_{y} Φ (x^{1}, y) d x^{1} d y . \end{matrix}$ Choosing $Φ \in C_{c}^{\infty} (Ω; R^{n \times n})$ we see that u is weakly differentiable w.r.t. to the variable y, with $(0, \nabla_{y} u) = δ \in L^{2} (Ω; R^{n \times n})$ . Hence the trace of u on $Γ_{D}$ is well defined. Applying the divergence theorem to (19) we obtain $\begin{matrix} 0 = \int_{Γ_{D}} u (x^{1}, y) \cdot Φ (x^{1}, y) \hat{n} d S (x^{1}, y) \end{matrix}$ for all $Φ \in H^{1} (Ω; R^{n \times n})$ such that $Φ \hat{n} = 0$ on $Γ_{N}$ . We conclude that u satisfies the Dirichlet condition $u = 0$ on $Γ_{D}$ . Convergence of the symmetric part of the gradient follows by linearity. □

5.2. Conservation of volume

Lemma 5.2.
Let ${u^{ε}}_{0 < ε ⩽ 1}$ be as in Lemma 5.1 with the additional condition that $div u^{ε} = 0$ in $Ω^{ε}$ . Then any subsequential two-scale limit $u = (u_{1}, u_{2})$ satisfies $\begin{matrix} (20) & {div}_{y} u_{2} = 0 in Ω \end{matrix}$ and $\begin{matrix} (21) & {div}_{x^{1}} (\int_{S (x^{1})} u_{1} (x^{1}, y) d y) = 0 in ω . \end{matrix}$
Proof.
The proof is very similar to the proof of [11, Proposition 4]. We include it only for the sake of completeness. The divergence free condition for $u^{ε}$ can be stated as $\begin{matrix} \int_{\partial Ω^{ε}} φ u^{ε} \cdot \hat{n} d S = \int_{Ω^{ε}} \nabla φ \cdot u^{ε} d x \end{matrix}$ for all scalar test-functions $φ \in H^{1} (Ω^{ε})$ . It follows that $\begin{matrix} (22) & 0 = \int_{Ω^{ε}} \nabla_{x^{1}} φ (x^{1}, x^{2} / ε) \cdot u_{1}^{ε} + ε^{- 1} \nabla_{y} φ (x^{1}, x^{2} / ε) \cdot u_{2}^{ε} d x \end{matrix}$ for all $φ \in H^{1} (Ω)$ such that $φ = 0$ on $Γ_{N}$ . Multiplying (22) with $ε^{- 1} / | Ω^{ε} |$ and passing to the limit $ε \to 0$ we obtain $\begin{matrix} 0 = \int_{Ω} \nabla_{y} φ (x^{1}, y) \cdot u_{2} (x^{1}, y) d x^{1} d y, \end{matrix}$ hence (20).

Choosing $φ (x^{1}, y) = φ (x^{1})$ such that $φ = 0$ on $\partial ω$ in (22) and multiplying with $ε^{- 2} / | Ω^{ε} |$ , the limit equation becomes $\begin{matrix} 0 = \int_{Ω} \nabla_{x^{1}} φ (x^{1}) \cdot u_{1} (x^{1}, y) d x^{1} d y = \int_{ω} \nabla_{x^{1}} φ (x^{1}) \cdot (\int_{S (x^{1})} u_{1} (x^{1}, y) d y) d x^{1}, \end{matrix}$ hence (21). □
5.3. Momentum equation

Lemma 5.3 (The thin film equations).

Let $u = (u_{1}, u_{2})$ and q be as in Lemma 5.1 . Then the first component $u_{1}$ and q satisfy $\begin{matrix} (23) & \{\begin{matrix} - \nabla_{x^{1}} q + μ Δ_{y} u_{1} = \nabla_{x^{1}} p_{b} & in Ω \\ - \nabla_{y} q = 0 & in Ω \\ u_{1} = 0 & on Γ_{D} \\ (\begin{matrix} - q & μ \nabla_{y} u_{1} \\ 0 & - q \end{matrix}) \hat{n} = 0 & on Γ_{N}, \end{matrix} \end{matrix}$ whereas the second component $u_{2} = 0$ .

Proof.
By taking φ in (3) of the form $\begin{matrix} φ (x) = (φ_{1} (x^{1}, x^{2} / ε), ε φ_{2} (x^{1}, x^{2} / ε)) \end{matrix}$ we obtain $\begin{array}{l} \int_{Ω^{ε}} - q^{ε} ({div}_{x^{1}} φ_{1} + {div}_{y} φ_{2}) \\ (24) & + 2 μ e (\nabla u^{ε}) : (\begin{matrix} \nabla_{x^{1}} φ_{1} & ε^{- 1} \nabla_{y} φ_{1} \\ ε \nabla_{x^{1}} φ_{2} & \nabla_{y} φ_{2} \end{matrix}) + \nabla_{x^{1}} p_{b} \cdot φ_{1} d x = 0 \end{array}$ for all $(φ_{1}, φ_{2}) \in W (Ω)$ . Letting $ε \to 0$ gives $\begin{array}{l} \int_{Ω} - q ({div}_{x^{1}} φ_{1} + {div}_{y} φ_{2}) \\ + μ (D (u) + D {(u)}^{T}) : (\begin{matrix} 0 & \nabla_{y} φ_{1} \\ 0 & 0 \end{matrix}) + \nabla_{x^{1}} p_{b} \cdot φ_{1} d x^{1} d y = 0, \end{array}$ where $D (u)$ is defined in Lemma 5.1. Consequently $\begin{matrix} (25) & \int_{Ω} - q ({div}_{x^{1}} φ_{1} + {div}_{y} φ_{2}) + μ \nabla_{y} u_{1} : \nabla_{y} φ_{1} + \nabla_{x^{1}} p_{b} \cdot φ_{1} d x^{1} d y = 0 \end{matrix}$ for all $(φ_{1}, φ_{2}) \in W (Ω)$ . Clearly, the last equality is the weak form of the homogenized problem (23).

By taking φ in (3) of the form $\begin{matrix} φ (x) = (0, φ_{2} (x^{1}, x^{2} / ε)), \end{matrix}$ where $φ_{2}$ is a smooth function such that ${div}_{y} φ_{2} = 0$ , we obtain $\begin{matrix} \int_{Ω^{ε}} 2 μ e (\nabla u^{ε}) : (\begin{matrix} 0 & 0 \\ \nabla_{x^{1}} φ_{2} & ε^{- 1} \nabla_{y} φ_{2} \end{matrix}) d x = 0 \end{matrix}$ Letting $ε \to 0$ gives $\begin{matrix} \int_{Ω} 2 μ e (\nabla_{y} u_{2}) : \nabla_{y} φ_{2} d x^{1} d y = 0 . \end{matrix}$ By density, this relation is valid also for $φ_{2} = u_{2}$ . Thus, from Korn’s inequality we conclude that $u_{2} = 0$ , due to the Dirichlet condition on $Γ_{D}$ . □
Lemma 5.4.
The class of test functions $φ = (φ_{1}, φ_{2}) \in W (Ω)$ for which ( 25 ) holds can be enlarged to $W_{y} (Ω)$ which consists of all $φ = (φ_{1}, φ_{2}) \in L^{2} (Ω; R^{n})$ such that
$\nabla_{y} φ \in L^{2} (Ω; R^{n \times l})$ ,

$φ = 0$ on $Γ_{D}$ ,

${div}_{x^{1}} (\int_{S (x^{1})} φ_{1} (x^{1}, y) d y) \in L^{2} (ω)$ .
Moreover, $u = (u_{1}, 0)$ belongs to $W_{y} (Ω)$ .
Proof.
Since $\nabla_{y} q = 0$ and $φ_{1}$ can be extended by zero outside $Γ_{D}$ , we can write the first term in (25) as $\begin{matrix} \int_{Ω} q {div}_{x^{1}} φ_{1} d x^{1} d y & = \int_{ω} q (x^{1}) \int_{S (x^{1})} {div}_{x^{1}} φ_{1} d y d x^{1} \\ = \int_{ω} q \int_{R^{l}} {div}_{x^{1}} φ_{1} d y d x^{1} \\ = \int_{ω} q {div}_{x^{1}} (\int_{R^{l}} φ_{1} d y) d x^{1} \\ = \int_{ω} q {div}_{x^{1}} (\int_{S (x^{1})} φ_{1} d y) d x^{1} . \end{matrix}$ Thus (25) can be written as $\begin{array}{l} \int_{ω} - q {div}_{x^{1}} (\int_{S (x^{1})} φ_{1} d y) \\ (26) & + (\int_{S (x^{1})} μ \nabla_{y} u_{1} : \nabla_{y} φ_{1} + \nabla_{x^{1}} p_{b} \cdot φ_{1} d y) d x^{1} = 0 . \end{array}$ This relation holds for all $φ = (φ_{1}, φ_{2}) \in W_{y} (Ω)$ , since $W (Ω)$ is dense in $W_{y} (Ω)$ . □
5.4. Pressure equation

Let $ϕ \in W^{1, \infty} (Ω)$ be the permeability function defined by (6). Upon substituting $p = p_{b} + q$ it is readily seen that $\begin{matrix} u^{1} (x^{1}, y) = - \frac{1}{μ} ϕ (x^{1}, y) \nabla_{x^{1}} p (x^{1}) \end{matrix}$ is a solution of (23) in the sense of distributions. There remains to prove $H^{1}$ -regularity for p and to find the boundary value problem that this function satisfies. To this end we take in (26), $\begin{matrix} φ_{1} (x^{1}, y) = ϕ (x^{1}, y) Φ (x^{1}), φ_{2} = 0, \end{matrix}$ where Φ is an arbitrary vector field in $H^{1} (ω; R^{m})$ . Thus we obtain $\begin{array}{l} \int_{ω} - q {div}_{x^{1}} (\int_{S (x^{1})} ϕ Φ d y) \\ + (\int_{S (x^{1})} μ \nabla_{y} u_{1} : Φ {(\nabla_{y} ϕ)}^{T} + \nabla_{x^{1}} p_{b} \cdot (ϕ Φ) d y) d x^{1} = 0 . \end{array}$ Using (6) we obtain $\begin{matrix} (27) & 0 = \int_{ω} - q {div}_{x^{1}} (a Φ) + (a \nabla_{x^{1}} p_{b} + μ {\bar{u}}_{1}) \cdot Φ d x^{1}, \end{matrix}$ where a is defined by (7) and $\begin{matrix} {\bar{u}}_{1} (x^{1}) = \int_{S (x^{1})} u_{1} (x^{1}, y) d y . \end{matrix}$ Relation (27) proves that q belongs to $H_{0}^{1} (ω)$ with $\begin{matrix} - \nabla_{x^{1}} q = \frac{μ}{a} {\bar{u}}_{1} + \nabla_{x^{1}} p_{b}, \end{matrix}$ provided that $1 / a \in W^{1, \infty} (ω)$ (see Remark 3.3), which is equivalent to $\begin{matrix} - a \nabla_{x^{1}} p = μ {\bar{u}}_{1} in ω . \end{matrix}$ Using the divergence free condition for ${\bar{u}}_{1}$ (Lemma 5.2) we see that $p = q + p_{b}$ belongs to $H^{1} (ω)$ and is a solution of (9). Hence p and u are uniquely determined and we conclude that the convergence holds for the whole sequence.

6. Strong convergence

There remains to prove the strong two-scale convergence of the velocity and the pressure. By Lemma 1, part (i) of [11] it suffices to prove that $\begin{matrix} lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| ε^{- 2} u^{ε} |}^{2} d x = \frac{1}{| Ω |} \int_{Ω} | u |^{2} d x \\ lim_{ε \to 0} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| p^{ε} |}^{2} d x = \frac{1}{| Ω |} \int_{Ω} | p |^{2} d x . \end{matrix}$ Choosing $φ = u^{ε}$ in (3), we obtain $\begin{array}{rcl} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| ε^{- 1} e (\nabla u^{ε}) |}^{2} d x & = & - \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} \frac{1}{2 μ} \nabla p_{b} \cdot (ε^{- 2} u^{ε}) d x \\ \to & - \frac{1}{| Ω |} \int_{Ω} \frac{1}{2 μ} \nabla p_{b} \cdot u_{1} d x^{1} d y \end{array}$ as $ε \to 0$ . Using (26) with test function $(φ_{1}, φ_{2}) = (u_{1}, 0)$ (Lemma 5.4) we see that $\begin{matrix} - \int_{Ω} \nabla p_{b} \cdot u_{1} d x^{1} d y = \int_{Ω} μ | \nabla_{y} u_{1} |^{2} d x^{1} d y . \end{matrix}$ This proves the strong two-scale convergence $\begin{matrix} (28) & ε^{- 1} e (\nabla u^{ε}) \overset{2}{\to} \frac{1}{2} (\begin{matrix} 0 & \nabla_{y} u_{1} \\ {(\nabla_{y} u_{1})}^{T} & 0 \end{matrix}) . \end{matrix}$

6.1. Strong two-scale convergence of the velocity

To prove strong two-scale convergence of $ε^{- 2} u^{ε}$ , consider the auxiliary problem $\begin{matrix} (29) & \{\begin{matrix} div (- ρ^{ε} I + 2 μ e (\nabla z^{ε})) = ε^{- 2} u^{ε} & in Ω^{ε} \\ div z^{ε} = 0 & in Ω^{ε} \\ (- ρ^{ε} I + 2 μ e (\nabla z^{ε})) \hat{n} = 0 & on Γ_{N}^{ε} \\ z^{ε} = 0 & on Γ_{D}^{ε} . \end{matrix} \end{matrix}$ For each $0 < ε ⩽ 1$ this problem has a unique solution $(z^{ε}, ρ^{ε}) \in V (Ω^{ε}) \times L^{2} (Ω^{ε})$ such that $\begin{matrix} \frac{1}{| Ω^{ε} |^{1 / 2}} ({‖ ε^{- 2} z^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ ε^{- 1} \nabla z^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ ρ^{ε} ‖}_{L^{2} (Ω^{ε})}) ⩽ C, \end{matrix}$ where the constant C does not depend on ε. The derivation of this inequality is analogous to the derivation of (17), so is the limit process when $ε \to 0$ . We therefore omit all arguments that are similar to those used in Section 5 and state merely the conlusions. Extracting subsequences such that $\begin{matrix} ε^{- 2} z^{ε} \overset{2}{⇀} z = (z_{1}, z_{2}), ε^{- 1} \nabla z^{ε} \overset{2}{⇀} (0, \nabla_{y} z), ρ^{ε} \overset{2}{⇀} ρ, \end{matrix}$ we find that the limit $(z, ρ)$ satisfies $\begin{matrix} \int_{Ω} - ρ ({div}_{x^{1}} φ_{1} + {div}_{y} φ_{2}) + μ \nabla_{y} z_{1} : \nabla_{y} φ_{1} + u_{1} \cdot φ_{1} d x^{1} d y = 0 \end{matrix}$ for all $(φ_{1}, φ_{2}) \in W (Ω)$ and by density (see Lemma 5.4) we have $\begin{array}{l} \int_{ω} - ρ {div}_{x^{1}} (\int_{S (x^{1})} φ_{1} d y) \\ (30) & + (\int_{S (x^{1})} μ \nabla_{y} z_{1} : \nabla_{y} φ_{1} + u_{1} \cdot φ_{1} d y) d x^{1} = 0 \end{array}$ for all $(φ_{1}, φ_{2}) \in W_{y} (Ω)$ .

Using the weak formulation of (29), with test function $u^{ε}$ , and the strong convergence (28), we can pass to the limit in $\begin{array}{rcl} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| ε^{- 2} u^{ε} |}^{2} d x & = & - \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} 2 μ ε^{- 2} e (\nabla z^{ε}) : e (\nabla u^{ε}) d x \\ \to & - \frac{1}{| Ω |} \int_{Ω} μ \nabla_{y} z_{1} : \nabla_{y} u_{1} d x^{1} d y \end{array}$ as $ε \to 0$ . Choosing $φ_{1} = u_{1}$ in (30), using Lemma 5.2, we deduce $\begin{matrix} - \int_{Ω} μ \nabla_{y} z_{1} : \nabla_{y} u_{1} d x^{1} d y = \int_{Ω} | u_{1} |^{2} d x^{1} d y . \end{matrix}$ Hence $ε^{- 2} u^{ε} \overset{2}{\to} u$ strongly.

6.2. Strong two-scale convergence of the pressure

To prove the strong convergence of $p^{ε}$ , we consider another auxiliary problem: $\begin{matrix} (31) & \{\begin{matrix} div (- ρ^{ε} I + 2 μ e (\nabla z^{ε})) = 0 & in Ω^{ε} \\ div z^{ε} = p^{ε} & in Ω^{ε} \\ (- ρ^{ε} I + 2 μ e (\nabla z^{ε})) \hat{n} = 0 & on Γ_{N}^{ε} \\ z^{ε} = 0 & on Γ_{D}^{ε} . \end{matrix} \end{matrix}$ For each $0 < ε ⩽ 1$ this problem has a unique solution $(z^{ε}, ρ^{ε}) \in W (Ω^{ε}) \times L^{2} (Ω^{ε})$ such that $\begin{matrix} \frac{1}{| Ω^{ε} |^{1 / 2}} ({‖ z^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ ε \nabla z^{ε} ‖}_{L^{2} (Ω^{ε})} + {‖ ε^{2} ρ^{ε} ‖}_{L^{2} (Ω^{ε})}) ⩽ C, \end{matrix}$ where the constant C does not depend on ε. By compactness, as in Lemma 5.1, we can extract a subsequence such that $\begin{matrix} z^{ε} \overset{2}{⇀} z = (z_{1}, z_{2}), ε \nabla z^{ε} \overset{2}{⇀} (0, \nabla_{y} z), ε^{2} ρ^{ε} \overset{2}{⇀} ρ, \end{matrix}$ with $z = 0$ on $Γ_{D}$ and $\nabla_{y} z \in L^{2} (Ω; R^{n \times l})$ . Here there is no need to find the limit problem satisfied by $(z, ρ)$ , but we need to show that $z \in W_{y} (Ω)$ . From the second and fourth equation of (31) it follows that $\begin{matrix} (32) & 0 = \int_{Ω^{ε}} (\nabla_{x^{1}} φ + ε^{- 1} \nabla_{y} φ) (x^{1}, x^{2} / ε) \cdot z^{ε} + φ (x^{1}, x^{2} / ε) p^{ε} d x \end{matrix}$ for all scalar test functions $φ \in H^{1} (Ω)$ such that $φ = 0$ on $Γ_{N}$ . Multiplying (32) with $ε / | Ω^{ε} |$ and passing to the limit $ε \to 0$ we obtain $\begin{matrix} 0 = \int_{Ω} \nabla_{y} φ (x^{1}, y) \cdot z_{2} (x^{1}, y) d x^{1} d y, \end{matrix}$ so ${div}_{y} z_{2} = 0$ in Ω. Choosing $φ (x^{1}, y) = φ (x^{1})$ such that $φ = 0$ on $\partial ω$ in (32) and multiplying with $1 / | Ω^{ε} |$ , the limit equation becomes $\begin{array}{l} 0 & = \int_{Ω} \nabla_{x^{1}} φ (x^{1}) \cdot z_{1} (x^{1}, y) + φ (x^{1}) p (x^{1}) d x^{1} d y \\ = \int_{ω} \nabla_{x^{1}} φ (x^{1}) \cdot (\int_{S (x^{1})} z_{1} (x^{1}, y) d y) + φ (x^{1}) p (x^{1}) | S (x^{1}) | d x^{1}, \end{array}$ so $\begin{matrix} {div}_{x^{1}} (\int_{S (x^{1})} z_{1} (x^{1}, y) d y) = p (x^{1}) | S (x^{1}) | in ω . \end{matrix}$ This proves that z belongs to the extended class of test functions $W_{y} (Ω)$ defined in Lemma 5.4.

Taking $φ = z^{ε}$ in the weak formulation (3) with $p_{b} = p$ , gives $\begin{array}{rcl} \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} {| p^{ε} |}^{2} d x & = & \frac{1}{| Ω^{ε} |} \int_{Ω^{ε}} p p^{ε} + 2 μ e (\nabla u^{ε}) : \nabla z^{ε} + \nabla_{x^{1}} p \cdot z^{ε} d x \\ \to & \frac{1}{| Ω |} \int_{Ω} | p |^{2} + μ \nabla_{y} u_{1} : \nabla_{y} z_{1} + \nabla_{x^{1}} p \cdot z_{1} d x^{1} d y, \end{array}$ thanks to the strong convergence of $e (\nabla u^{ε})$ as $ε \to 0$ . Choosing $φ_{1} = z_{1}$ in (26) with $p_{b} = p$ (note that in this case $q = 0$ ), we see that $\begin{matrix} \int_{Ω} μ \nabla_{y} u_{1} : \nabla_{y} z_{1} + \nabla_{x^{1}} p \cdot z_{1} d x^{1} d y = 0 \end{matrix}$ Hence $p^{ε} \overset{2}{\to} p$ strongly.

Footnotes

Acknowledgement

The authors wish to thank an anonymous referee for many thoughtful comments and suggestions which significantly improved the final version of this paper.

Proof of Theorem 4.1

First we show how the constant in the Korn inequality behaves under isotropic scaling of the domain. To this end, let $□^{ε} = {(- ε / 2, ε / 2)}^{n}$ denote the cube in $R^{n}$ of length ε confined between the hyperplanes $\begin{matrix} Γ_{\pm}^{ε} = {(x_{1}, \dots, x_{n}) \in R^{n} : x_{n} = \pm ε / 2} . \end{matrix}$ For $ε = 1$ we omit the superscripts.

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 the assumptions stated in Section 2.1, we can assume, without loss of generality, that $\begin{matrix} S (x^{1}) = {(- 1 / 2, 1 / 2)}^{l} \forall x^{1} \in \overline{ω} \end{matrix}$ so that $Ω = ω \times {(- 1 / 2, 1 / 2)}^{l}$ . Indeed, due to the Dirichlet condition on $\partial S (x^{1})$ , we can extend any $v \in W (Ω)$ by zero to $ω \times {(- 1 / 2, 1 / 2)}^{l}$ which is confined between the hyperplanes $Γ_{\pm}$ . Moreover, we choose $L > 0$ large enough so that the set $\begin{matrix} \tilde{Ω} = {(- L / 2, L / 2)}^{m} \times {(- 1 / 2, 1 / 2)}^{l} \end{matrix}$ satisfies $\begin{matrix} Ω \subset \tilde{Ω} . \end{matrix}$

Extension operator.

Covering argument. For $z \in R^{m}$ let $Q (z, ε)$ denote the cube in $R^{m}$ of length ε centered at z, i.e. $\begin{matrix} Q (z, ε) = {y \in R^{m} : | y_{i} - z_{i} | < ε / 2, i = 1, \dots, m} . \end{matrix}$ For each $0 < ε ⩽ 1$ , there exists a finite collection of points ${z^{i}}$ in $ε Z^{m}$ such that $\begin{matrix} ω \subset \overline{⋃_{i} Q (z^{i}, ε)} \subset {(- L / 2, L / 2)}^{m} . \end{matrix}$ Set $\begin{matrix} □_{i}^{ε} = Q (z^{i}, ε) \times {(- ε / 2, ε / 2)}^{l} . \end{matrix}$ Then $\begin{matrix} Ω^{ε} \subset \overline{⋃_{i} □_{i}^{ε}} \subset \overline{\tilde{Ω}} \end{matrix}$ for all $0 < ε ⩽ 1$ . Since each cube $□_{i}^{ε}$ is a translation of the cube $□^{ε} = {(- ε / 2, ε / 2)}^{n}$ we can use the inequality (35) locally. Proof of Theorem 4.1.

Suppose v in $W (Ω^{ε})$ . In order to use the extension operator in Lemma A.2, let $\bar{v}$ in $W (Ω)$ be defined by $\begin{matrix} \bar{v} = \{\begin{matrix} v & in Ω^{ε} \\ 0 & in Ω - Ω^{ε} . \end{matrix} \end{matrix}$ Thus we obtain $\begin{matrix} \int_{Ω^{ε}} | v |^{2} d x & = \int_{Ω^{ε}} | E \bar{v} |^{2} d x \\ ⩽ \sum_{i} \int_{□_{i}^{ε}} | E \bar{v} |^{2} d x \\ ⩽ \sum_{i} ε^{2} K_{1}^{2} \int_{□_{i}^{ε}} {| e (\nabla (E \bar{v})) |}^{2} d x \\ ⩽ ε^{2} K_{1}^{2} \int_{\tilde{Ω}} {| e (\nabla (E \bar{v})) |}^{2} d x, \end{matrix}$ or $\begin{matrix} ‖ v ‖_{L^{2} (Ω^{ε})} ⩽ ε K_{1} {‖ e (\nabla (E \bar{v})) ‖}_{L^{2} (\tilde{Ω})} . \end{matrix}$ Hence $\begin{matrix} ‖ v ‖_{L^{2} (Ω^{ε})} ⩽ ε K_{1} C_{ω} {‖ e (\nabla \bar{v}) ‖}_{L^{2} (Ω)} = ε K_{1} C_{ω} {‖ e (\nabla v) ‖}_{L^{2} (Ω^{ε})} . \end{matrix}$ This gives an upper bound for $K_{1}^{ε}$ in (12). The estimation of $K_{2}^{ε}$ in (13) is completely analogous. □

Proof of Theorem 4.3

The upper bound is proved by a simple scaling argument. For the lower bound we estimate $B ψ$ for a special function ψ.

For any $0 < ε ⩽ 1$ , let $S^{ε}$ denote the scaling operator defined by $\begin{array}{l} S^{ε} f (x) = f (x^{1}, x^{2} / ε) (f is a scalar function) \\ S^{ε} v (x) = (v_{1} (x^{1}, x^{2} / ε), ε v_{2} (x^{1}, x^{2} / ε)) (v is a vector field), \end{array}$ where $x = (x^{1}, x^{2}) \in R^{m} \times R^{l}$ and similarly for $v = (v_{1}, v_{2})$ . Clearly, $S^{ε}$ is invertible with ${(S^{ε})}^{- 1} = S^{1 / ε}$ . A simple change of variables shows that $S^{ε}$ maps $L^{2} (Ω)$ onto $L^{2} (Ω^{ε})$ with $\begin{matrix} \frac{1}{| Ω^{ε} |^{1 / 2}} {‖ S^{ε} f ‖}_{L^{2} (Ω^{ε})} = \frac{1}{| Ω |^{1 / 2}} ‖ f ‖_{L^{2} (Ω)} . \end{matrix}$ Moreover, $S^{ε}$ maps $W (Ω)$ onto $W (Ω^{ε})$ with $\begin{matrix} div (S^{ε} v) = S^{ε} (div v) in Ω^{ε} . \end{matrix}$ In particular, $S^{ε}$ acting on ${v} \in W / V (Ω)$ through $\begin{matrix} S^{ε} {v} = {S^{ε} v} \end{matrix}$ is well defined in $W / V (Ω^{ε})$ .

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