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 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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