Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary
Gung-Min Gie Makram Hamouda Roger Temam
Networks & Heterogeneous Media 2012, 7(4): 741-766 doi: 10.3934/nhm.2012.7.741
We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
keywords: curvilinear coordinates. Navier-Stokes equations Boundary layers singular perturbations
Boundary layers for the 2D linearized primitive equations
Makram Hamouda Chang-Yeol Jung Roger Temam
Communications on Pure & Applied Analysis 2009, 8(1): 335-359 doi: 10.3934/cpaa.2009.8.335
In this article, we establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension $2$. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of the boundary. Two cases are considered, the subcritical and supercritical modes depending on the fact that the frequency mode is less or greater than the ratio between the reference stratified flow (around which we linearized) and the buoyancy frequency. The problem of boundary layers for the LPEs is of a new type since the corresponding limit problem displays a set of (unusual) nonlocal boundary conditions.
keywords: Primitive equations boundary layers singular perturbation analysis
Asymptotic analysis for the 3D primitive equations in a channel
Makram Hamouda Chang-Yeol Jung Roger Temam
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 401-422 doi: 10.3934/dcdss.2013.6.401
In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non-local boundary conditions for the non-viscous PEs.
keywords: Primitive equations boundary layers singular perturbation analysis.
Boundary layers in smooth curvilinear domains: Parabolic problems
Gung-Min Gie Makram Hamouda Roger Témam
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1213-1240 doi: 10.3934/dcds.2010.26.1213
The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.
keywords: heat equation singular perturbations boundary layers curvilinear coordinates.

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