Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

Abstract

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 $\varepsilon^{j}$, $j = 0, 1$, where $\varepsilon$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $\varepsilon^{j}$, $j = 0, 1$, for $\varepsilon$ 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.