In this paper, we derive the optimal boundary conditions of the Navier-Stokes fluid in a bounded domain with a thin layer. We get the effects of the layer by investigating the limiting δ, the thickness of the layer shrinking to zero. In the two-dimensional case, we derive effective boundary conditions by the limit of u and p on the boundary of the uncoated body.
We consider the mathematical problem arising from fluids by applying a thin layer of special area to a body to be protected. The body is assumed to be a two-dimensional smooth and bounded domain, denoted by \(\Omega_{1}\). The layer, denoted by \(\Omega_{2}\), has a thickness of \(\delta\ll1\). Then the whole domain \(\Omega=\bar{\Omega}_{1}\cup\Omega_{2}\).
We study the Navier-Stokes fluid of the coated body
where \(u^{\delta}\) and \(p^{\delta}\) are the velocity and the pressure of the fluid, respectively. \(f(x,t)\) is the force term. \(\mathcal{A}\) is a \(2\times2\) matrix, which is positive and symmetry. For simplicity, we assume the interior body \(\Omega_{1}\) is homogeneous. More precisely, we assume that
where σ and μ are positive constants, \({\boldsymbol{\nu}}=\nabla d(x)\) is the unit outward normal to ∂Ω and \(\tau\perp{\boldsymbol{\nu}}\) is a unit tangent vector. Note that (3) implies
This model describes an incompressible Navier-Stokes fluid in a composed domain, which contains two different parts, a microscopic one (\(\Omega_{2}\)) and a macroscopic one (\(\Omega_{1}\)). In the macroscopic part, the fluid is considered to be isotropic. The microscopic part is anisotropic and surrounds the macroscopic part and has the thickness δ, which is small when compared to the size of \(\Omega_{1}\). The real world applications may include lake or sea shore protection and desert stabilization: to protect a shoreline from erosion, along the shoreline in the water aquatic plants and trees may be planted, and large rocks and concrete blocks may be placed (see Figure 1); similarly, along the edge of a desert trees may be planted to prevent the growth of the desert. If these barriers are placed periodically and parallel to the shoreline/desert edge, then after homogenization the viscous tensor in the thin layer \(\Omega_{2}\) takes the form of \(\mathcal{A}(x)\nabla u^{\delta}\) with \(\mathcal{A}(x)\) satisfying (4).
Figure 1
A model of a lake or a sea near the shore.
In the three-dimensional case of an anisotropic fluid, the viscous tensor, as a symmetric \(3\times3\) matrix, has three orthogonal eigenvectors and three eigenvalues. Each of these eigenvalues measures the impact rate of the anisotropic fluid in the corresponding eigen-direction. To protect \(\partial\Omega _{1}\) from eroding, it is desirable that the eigenvector corresponding to the smallest eigenvalue is orthogonal to the boundary of the body in order to directly confront the ambient effect. The other eigenvalues (in the tangential directions) may not need to be small. Motivated by these considerations, in this paper, we introduce the notion of ‘optimally aligned layer’ (a similar conception as in [1]), by which we mean that at every point \(x\in\Omega_{2}\), the vector normal to \(\partial\Omega_{1}\) at the projection of x is always an eigenvector of the viscous tensor \(\mathcal{A}(x)\).
The static case of such a problem was studied by Rosencrans et al. [1–4] and Li [5] for the heat equation on a coated body under different boundary conditions. In fluid mechanics, Miksis and Davis [6] studied a Navier-Stokes fluid flows over another one over rough and coated surfaces. Jäger and Mikelic [7] considered the boundary conditions on the contact interface between a porous medium and a free fluid. Kohn and Vogelius [8, 9] studied the effective models in a thin plate with rapidly varying thickness, he found that those models are different according to the different scaling. Braides et al. [10] got the effective results on inhomogeneous thin layers in 2D and 3D. Ansini and Braides [11] applied their effective results on oscillating boundary to thin films. Other important work on the domain coarsening in thin layers was done by Niethammer and Otto [12]. Moreover, the effect of the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid was recently studied in [13]. For other work, we refer the interested reader to [14–16] and the references therein.
In this paper, we considered an incompressible fluid in a bounded domain with a thin layer in 2D. The aim of this paper is to find the homogenized model of (1) and corresponding optimal boundary conditions as the thickness δ tends to zero. The main difficulty we met here is to investigate the optimal boundary conditions for the homogenized model. To this end, we define a boundary layer equations between the macroscopic domain and the coating. We obtained the optimal boundary conditions by using a special test function constructed by the boundary layer solution.
The paper is organized as follows. In Section 2, we review some classical results and state our main results. In Section 3, we give the local parameter transformation. In Section 4, we prove the main results in this paper.
In this paper, we assume that the boundaries of \(\partial\Omega_{1}\) and Ω are both smooth enough when \(\delta>0\) is sufficiently small. Then we assume that the distance function \(d(x)\), the unit normal vector \(\nu(x)\), and the unit tangent vector \(\tau(x)\) are also smooth enough on \(\bar{\Omega}_{2}\). We denote the time-space domain by \(\Omega_{T}=\Omega\times(0,T)\).
Let \(\mathcal{D}(\Omega)\) be the space of \(\mathcal{C}^{\infty}\) functions with compact support contained in Ω. We define the space
$$\begin{aligned}& \mathcal{V}=\bigl\{ u\in\mathcal{D}(\Omega), \operatorname{div} u=0\bigr\} , \\& \mathcal {H}=\bigl\{ \text{the closure of } \mathcal{V} \text{ in } L^{2}(\Omega)\bigr\} , \\& \mathcal{W}=\bigl\{ \text{the closure of } \mathcal{V} \text{ in } H^{1}_{0}(\Omega)\bigr\} . \end{aligned}$$
Throughout the paper, we make assumptions on Ω, \(\mathcal{A}(x)\) and \(f(x)\) as follows:
1.
Ω is bounded, open and convex with smooth boundary (at least \(C^{2}\));
2.
\(\mathcal{A}(x)\in C^{2}(\Omega)\);
3.
\(f(x)\in L^{2}(\Omega_{T})\).
Definition 2.1
A function pair \(\{u^{\delta},p^{\delta}\}\in L^{2}(0,T;\mathcal{W})\times L^{2}(0,T;H^{1}(\Omega))\) is said to be a weak solution to (1) if
for all \(\eta\in L^{2}(0,T;\mathcal{V})\) with \(\eta(x,t)=0\) at \(t=0,T\).
Lemma 2.1
Let\(\Omega\subset R^{2}\)be an open bounded domain. \(u^{\delta}_{0}\in L^{2}(\Omega)\)and\(f\in L^{2}(\Omega_{T})\). Then up to a constant, (1) determines a unique weak solution pair\(\{u^{\delta},p^{\delta}\}\)in the sense of Definition 2.1and the following estimates hold:
It is obvious that the tensor \(\mathcal{A}(x)\) is positive, symmetric, and coercive. Note that ∂Ω is sufficiently smooth, the energy estimates can be proved formally multiplying (1) by \(u^{\delta}\) for (i), \(u^{\delta}_{t}\) for (ii), and \(tu^{\delta}_{t}\) for (iii). To prove (iv), we use the div and curl decomposition. Let P and Q be two orthogonal operators. For all f, we have
$$ f=P(f)+Q(f), $$
where P maps the divergence-free functions to themselves. Then we can write (1) in the following way:
The classical result (see [17]) shows that \(u^{\delta}_{t}\) is bounded in \(L^{2}(0,T;\mathcal{W}')\). By (1), we see that ∇p is bounded in \(L^{2}(0,T;H^{-1}(\Omega))\). Finally, by using the Necćas inequality (see[18], Chapter 1, Proposition 1.2), we see that p is bounded in \(L^{2}(\Omega\times(0,T))\) and (iv) holds.
From those a priori estimates, by using the Faedo-Galerkin method, we can establish the existence result. □
Now, we state our main results in this paper.
Theorem 2.1
Under the hypotheses of Lemma 2.1, suppose\(\mathcal{A}(x)\)is given by (2) or (3) and\(u^{\delta}_{0}\)converges strongly to\(u_{0}(x)\)in the corresponding space. We also assume that\(\sigma(\delta)\geq\mu(\delta)\geq\delta\). Let\(\{u^{\delta},p^{\delta}\}\)be the solution pair of (1). Then up to a constant, there exists a unique function pair\(\{u,p\}\)such that
$$ u^{\delta}\rightarrow u\quad \textit{weakly in }L^{2} \bigl(0,T;H^{1}(\Omega_{1})\bigr),\qquad p^{\delta}\rightarrow p\quad \textit{weakly in } L^{2}\bigl(\Omega_{1} \times(0,T)\bigr) $$
as\(\delta\rightarrow0\), where\(\{u,p\}\)is the solution pair of the following homogenized problem:
Under the hypotheses of Lemma 2.1, suppose\(\mathcal{A}(x)\)is given by (2) or (3) and\(u^{\delta}_{0}\)converges strongly to\(u_{0}(x)\)in the corresponding space. We assume that\(\delta\leq\sigma(\delta)\leq\mu(\delta)\leq M\sigma(\delta )\)for some constants\(M>0\). Let\(\{u^{\delta},p^{\delta}\}\)be the solution pair of (1). The function pair\(\{u,p\}\)is defined in Theorem 2.1. Then (4) still holds as\(\delta\rightarrow0\)together with the following homogenized boundary conditions:
1.
If\(\lim_{\delta\rightarrow0}\frac{\mu^{\frac {3}{2}}}{\delta}=0\), the optimal boundary condition would be
The results in Theorem 2.1 show that boundary conditions on \(\partial \Omega_{1}\) mainly depend on the relationship between δ and σ. More precisely, there exists a critical index between δ and σ (\(\delta\backsim\sigma^{\frac{3}{2}}\)). The same result was found by Mikelic [19, 20], where the authors considered effective models in a domain with a thin layer between two porous media.
Definition 2.2
A function pair \(\{u,p\}\) is said to be a weak solution to (4) if \((u,p)\in L^{2}(0,T;\mathcal{W})\times L^{2}(\Omega_{1}\times(0,T))\) and satisfies
for any \(\eta\in C^{\infty}(\bar{\Omega}_{1}\times(0,T])\) with \(\operatorname {div}\eta=0\). Also, u satisfies the following weak boundary conditions:
1.
\(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=0 \) on \(\partial\Omega_{1}\) if
$$ \mathcal{L}(u,\eta)=0. $$
(15)
2.
\(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=\alpha u\cdot\nu e_{2} \) or \(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=\alpha\sqrt{\beta }\tau\cdot u e_{1}+\alpha\nu\cdot u e_{2}\) on \(\partial\Omega_{1}\) (resp. \(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=\beta\tau \cdot u e_{1}+\beta^{\frac{3}{2}}\nu\cdot u e_{2}\)) if
$$ \begin{aligned} &\mathcal{L}(u,\eta)=- \int_{0}^{l}\alpha u\cdot\nu e_{2}\eta \, dr \quad \text{or}\quad \mathcal{L}(u,\eta) =- \int_{0}^{l}(\alpha\sqrt{\beta}\tau\cdot u e_{1}+\alpha u\cdot\nu e_{2})\eta \, dr \\ &\biggl(\text{resp. } \mathcal{L}(u,\eta) =- \int_{0}^{l}\bigl(\beta\tau\cdot u e_{1}+ \beta^{\frac{3}{2}}\nu\cdot u e_{2}\bigr)\eta \, dr\biggr). \end{aligned} $$
(16)
3.
\(u=0\) or \(u\cdot\nu=0\) (resp. \(u\cdot\tau=0\)) on \(\partial \Omega_{1}\) if
$$ \begin{aligned} & \int_{0}^{l}(\sqrt{\beta}\tau\cdot u e_{1}+\nu\cdot u e_{2})\cdot\eta \, dr=0\quad \text{or} \\ & \int_{0}^{l}\nu\cdot u e_{2}\cdot\eta \, dr=0\qquad \biggl(\text{resp. } \int_{0}^{l}\tau\cdot u e_{2}\cdot\eta \, dr=0\biggr). \end{aligned} $$
(17)
Theorem 2.3
Suppose\(\partial\Omega_{1}\)is smooth enough (at least\(\partial\Omega _{1}\in C^{2}\)for example), \(u_{0}\in L^{2}(\Omega_{1})\)and\(f\in L^{2}(\Omega_{T})\). Then (4) with any of the boundary condition (5)∽(9) (resp. (10)∽(14)) has one weak solution in the sense of Definition 2.2.
Proof
To establish the existence result, it is sufficient give the a priori estimates on different boundary conditions.
1. \(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=0 \) on \(\partial\Omega_{1}\).
Multiplying (4) by u and integrating over \(\Omega_{1}\times (0,t)\), we have
We obtain the a priori estimate \(u\in L^{\infty}(0,T;L^{2}(\Omega _{1}))\cap L^{2}(0,T;H^{1}(\Omega_{1}))\) by the Young inequality and the Fredrich inequality.
2. \(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=\alpha u\cdot\nu e_{2} \) on \(\partial\Omega_{1}\).
Multiplying (4) by u and integrating over \(\Omega_{1}\times (0,t)\), we have
To obtain the a priori estimate, we only consider the second term on the right-hang side. For each \(\varepsilon_{0}>0\), by the boundary trace imbedding theorem (see [21], Chapter 5, (5.36)) and the interpolation inequality (see [21], Chapter 5, (5.2)), we have
Choosing ε small enough, we obtain the same estimate as above by Gronwall’s inequality. By the boundary condition \(\frac{\partial u}{\partial\nu}-(u\otimes u+pI)\nu=\alpha\sqrt{\beta }\tau\cdot u e_{1}+\alpha\nu\cdot u e_{2}\), we can get the same estimate following the steps as above.
3. \(u=0\) or \(u\cdot\nu=0\) on \(\partial\Omega_{1}\).
The same estimate can be obtained by multiplying u in (4) and integrating over \(\Omega_{1}\times(0,t)\).
With this estimate, we can construct an approximate solution by the Faedo-Galerkin method and follow the steps in [17] (Chapter 1, (6.4)) to obtain the existence result. □
To investigate the boundary conditions on \(\partial\Omega_{1}\), it is convenient to write the system in another coordinate system. We introduce the parameter transformation as [1].
3.1 The parameter transformation
Before proving the main theorems, we introduce the parameter transformation near \(\partial\Omega_{1}\). Let \(l=|\partial\Omega_{1}|\) be the arc-length of \(\partial\Omega_{1}\). By using the arc-length \(r\in[0,l)\), we define the mapping
where \({\boldsymbol{\nu}}(r)\) is the unit outer normal vector of \(\partial\Omega _{1}\) at \({\mathbf{p}}(r)\). It is well known that X is a diffeomorphism. We also have
We denote by \((r,s)\) the inverse of the mapping \(x=X(r,s)\). We denote by \(\tau(r)\) the unit tangent vector \(\frac{{d\mathbf{p}}}{dr}\) of \(\partial\Omega_{1}\) at \({\mathbf{p}}(r)\), and write the curvature of \(\partial\Omega_{1}\) at \({\mathbf{p}}(r)\) as \(\kappa(r)\). Then we have the Frenet equations,
By Lemma 2.1(i) and the parameter transformation, the lemma can be proved. □
3.2 Boundary layer problem
A classical way of finding boundary conditions is by using the matched asymptotic method (see [22]). The main idea is to supplement the problem by an inner system in which the independent variables are stretched out in order to capture the behavior in the neighborhood of the boundary. As for our problem, this approach uses the limit rule, by which asymptotic behavior of the outward in the neighborhood of the boundary has to be equal to asymptotic behavior of the inner.
Let \(g(r,t)\in C_{0}^{\infty}(\bar{\Omega}_{1}\times[0,+\infty))\) be a solenoidal field with compact support. We extend it to the domain \(\Omega\times[0,+\infty)\) via the functions φ and π defined by the following system:
In this section, we put our emphasis on the proofs of Theorem 2.1 and Theorem 2.2. They contain two parts:
1.
Recover the system in \(\Omega_{1}\).
2.
Investigate the optimal boundary conditions on \(\partial\Omega_{1}\).
Proof of the first part in Theorem 2.1 and Theorem 2.2
By Lemma 2.1 and the standard compactness theorem, we have \(u^{\delta}\rightharpoonup u\) weakly in \(L^{2}(0,T;H^{1}(\Omega_{1}))\) and \(p^{\delta}\rightharpoonup p\) weakly in \(L^{2}(\Omega_{1}\times(0,T))\) as \(\delta\rightarrow0\). Next, we choose the test function
where \(\varphi=(\varphi_{1},\varphi_{2})^{\tau}\) is the solution of the boundary layer problem. Note that \(\operatorname{div}_{x_{1},x_{2}}\eta(r, s,t)=0 \) in \(\Omega_{1}\cup\Omega_{2}\) and \(\eta(r,s,t)|_{\partial\Omega}=0\). Multiplying (1) by \(\eta(r,s,t)\) and integrating over \(\Omega \times(0,T)\), we have
To avoid the influence on \(\partial\Omega_{1}\), we set \(g=0\) on \(\partial \Omega_{1}\), then (15) admits a unique solution \(\varphi=0\). Then we have
Note that (25) and (26) are not determined because of lacking of boundary conditions. In the following, we investigate the boundary conditions on \(\partial\Omega_{1}\) satisfied by u.
Now we consider the limit to (29), (30), (36), and (37) as \(\delta\rightarrow0\). We study three cases:
1. If \(\lim_{\delta\rightarrow0}\frac{\sigma^{\frac {3}{2}}}{\delta}=0\), in this case, \(\lim_{\delta\rightarrow 0}\sigma=\lim_{\delta\rightarrow0}\mu=0\), we have
2. If \(\lim_{\delta\rightarrow0}\frac{\sigma^{\frac {3}{2}}}{\delta}=\alpha\in(0,+\infty)\), in this case, \(\lim_{\delta\rightarrow0}\sigma=\lim_{\delta\rightarrow 0}\mu=0\), we consider two different subcases.
2(i). If \(\lim_{\delta\rightarrow0}\frac{\mu}{\sigma}=0\), we have
2(ii). If \(\lim_{\delta\rightarrow0}\frac{\mu}{\sigma}=\beta\in (0,1]\), we also have \(B_{2}^{\delta}\rightarrow0\) and \(B_{3}^{\delta}\rightarrow0\),
Under the conditions in Theorem 2.2, (29), (33), and (34) are still satisfied. To obtain the boundary condition as \(\delta \rightarrow0\), we consider three cases:
1. If \(\lim_{\delta\rightarrow0}\frac{\mu^{\frac{3}{2}}}{\delta }=0\), then \(\lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\mu }}{\delta} =\lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\sigma}}{\delta }=0\), \(\lim_{\delta\rightarrow0}\sigma=\lim_{\delta \rightarrow0}\mu\)=0, we have
2. If \(\lim_{\delta\rightarrow0}\frac{\mu^{\frac{3}{2}}}{\delta }=\alpha\in(0,+\infty)\), then \(\lim_{\delta\rightarrow0}\sigma =\lim_{\delta\rightarrow0}\mu=0\), and we consider two subcases.
2(i). If \(\lim_{\delta\rightarrow0}\frac{\sigma}{\mu}=0\), then we have \(\lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\sigma }}{\delta}\leq \lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\mu}}{\delta} =\lim_{\delta\rightarrow0}(\frac{\mu\sqrt{\mu}}{\delta}\times \frac{\sigma}{\mu})=0\), passing to the limit, we have
2(ii). If \(\lim_{\delta\rightarrow0}\frac{\sigma}{\mu}=\beta\in (0,1]\), then we have \(\lim_{\delta\rightarrow0}\frac{\sigma\sqrt {\sigma}}{\delta}= \lim_{\delta\rightarrow0}(\frac{\mu\sqrt{\mu}}{\delta}\times \frac{\sigma\sqrt{\sigma}}{\mu\sqrt{\mu}})=\alpha\beta^{\frac{3}{2}}\), \(\lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\mu}}{\delta} =\lim_{\delta\rightarrow0}(\frac{\mu\sqrt{\mu}}{\delta}\times \frac{\sigma}{\mu})=\alpha\beta\), passing to the limit, we have
Then we obtain the optimal boundary condition on u,
$$ \frac{\partial u}{\partial\nu}-(u\otimes u+pI)\cdot\nu=\alpha\beta \tau\cdot u e_{1}+\alpha\beta^{\frac{3}{2}}\nu\cdot u e_{2}\quad \text{on } \partial\Omega_{1}. $$
3. If \(\lim_{\delta\rightarrow0}\frac{\mu^{\frac{3}{2}}}{\delta }=+\infty\), then \(\lim_{\delta\rightarrow0}\frac{\sigma\sqrt{\mu }}{\delta}=+\infty\). We consider two subcases.
3(i). If \(\lim_{\delta\rightarrow0}\frac{\sigma}{\mu}=0\), then we have
$$ \frac{\delta}{\sigma\sqrt{\mu}} B_{1}^{\delta}=\frac{\delta}{\sigma\sqrt {\mu}}B_{11}^{\delta}+ \frac{\delta}{\sigma\sqrt{\mu}}B_{12}^{\delta}\rightarrow- \int_{0}^{l}(\tau\cdot ue_{1})\cdot g\, dr. $$
The limit of \(B_{3}^{\delta}\) can be derived as \(B_{2}^{\delta}\) and we have \(\frac{\delta}{\sigma\sqrt{\sigma}}|B_{3}^{\delta}|\rightarrow0\). Then we obtain the optimal boundary condition on u,
As stated in the introduction, to protect \(\partial\Omega_{1}\) from eroding, it is desirable that the eigenvector corresponding to the smallest eigenvalue is orthogonal to the boundary of the body in order to directly confront the ambient effect. In our problem, it requires \(\delta\leq\sigma\leq\mu\). The optimal boundary conditions to our problem are given in Theorem 2.2. We must acknowledge that we need the condition \(\delta\leq\sigma\leq\mu\leq M\sigma\) for some constants \(M>0\) so that we can ensure \(0< h\leq1\) as δ is small enough. If this condition does not hold, the estimate (22) will not be satisfied. The case that \(h>1\) we will consider in the future.
This work is supported by the research of high schools of Guangzhou (No. 1201420925) and GDNSF (No. 2014A030313526).
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