23 research outputs found
Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array
In this paper we investigate the asymptotic validity of boundary layer
theory. For a flow induced by a periodic row of point-vortices, we compare
Prandtl's solution to Navier-Stokes solutions at different numbers. We
show how Prandtl's solution develops a finite time separation singularity. On
the other hand Navier-Stokes solution is characterized by the presence of two
kinds of viscous-inviscid interactions between the boundary layer and the outer
flow. These interactions can be detected by the analysis of the enstrophy and
of the pressure gradient on the wall. Moreover we apply the complex singularity
tracking method to Prandtl and Navier-Stokes solutions and analyze the previous
interactions from a different perspective
Vanishing Viscous Limits for 3D Navier-Stokes Equations with A Navier-Slip Boundary Condition
In this paper, we investigate the vanishing viscosity limit for solutions to
the Navier-Stokes equations with a Navier slip boundary condition on general
compact and smooth domains in . We first obtain the higher order
regularity estimates for the solutions to Prandtl's equation boundary layers.
Furthermore, we prove that the strong solution to Navier-Stokes equations
converges to the Eulerian one in and
L^\infty((0,T)\times\o), where is independent of the viscosity, provided
that initial velocity is regular enough. Furthermore, rates of convergence are
obtained also.Comment: 45page
Uniform regularity for the Navier-Stokes equation with Navier boundary condition
We prove that there exists an interval of time which is uniform in the
vanishing viscosity limit and for which the Navier-Stokes equation with Navier
boundary condition has a strong solution. This solution is uniformly bounded in
a conormal Sobolev space and has only one normal derivative bounded in
. This allows to get the vanishing viscosity limit to the
incompressible Euler system from a strong compactness argument
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co