Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions

Iftimie, Dragoş; Planas, Gabriela

doi:10.1088/0951-7715/19/4/007

Cited by 120 publications

(110 citation statements)

References 19 publications

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“…This is known to be true in the absence of material boundaries, see [17,1] for the two dimensional case and [9,20] for the three dimensional case. Also, if the boundary conditions are of Navier type, see [2,16,8,24], noncharacteristic, see [21] or for certain symmetric 2D flows, see [18,14,15], convergence to an Euler solution remains valid. The most relevant case from the physical point of view corresponds to no slip boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

Iftimie

Filho

Lopes

2008

Commun. Math. Phys.

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Abstract. In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the NavierStokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis:first, that the initial exterior domain velocity converges strongly in L 2 to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [4,5,6]

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Section: Introductionmentioning

confidence: 99%

Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

Iftimie

Filho

Lopes

2008

Commun. Math. Phys.

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“…Recently, the vanishing viscosity limits problems was established on different physical models with the slip boundary conditions, such as [5,6,17,26,27,29].…”

Section: )mentioning

confidence: 99%

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

Chen

Xiao

Zhang

2017

Math Methods in App Sciences

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In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition.

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“…These boundary conditions, originally defined by Navier, have recently received considerable attention from fluid mechanics as a physically motivated replacement for Dirichlet boundary conditions, as they allow a thorough characterization of the boundary layer. See for instance [Clopeau et al 1998;Lopes Filho et al 2005;Kelliher 2006; Iftimie and Planas 2006;Iftimie and Sueur 2006]. We also discuss Neumann boundary conditions for the velocity and for the vorticity, and Robin boundary conditions for the vorticity.…”

Section: Introductionmentioning

confidence: 98%

Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane

Kelliher¹

2010

Pacific J. Math.

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We show that the k-th eigenvalue of the Dirichlet Laplacian is strictly less than the k-th eigenvalue of the classical Stokes operator (equivalently, of the clamped buckling plate problem) for a bounded domain in the plane having a locally Lipschitz boundary. For a C 2 boundary, we show that eigenvalues of the Stokes operator with Navier slip (friction) boundary conditions interpolate continuously between eigenvalues of the Dirichlet Laplacian and of the classical Stokes operator.

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Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions

Cited by 120 publications

References 19 publications

Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane

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