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Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 741-766. doi: 10.3934/nhm.2012.7.741
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Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

  • 1.
    The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405
  • Received: 01 January 2012 Revised: 01 May 2012

  • 35B25, 35C20, 35K05, 76D07, 76D10.

  • 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.

    Citation: Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary[J]. Networks and Heterogeneous Media, 2012, 7(4): 741-766. doi: 10.3934/nhm.2012.7.741

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  • 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.


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