Abstract

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the well-known issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. a common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. a meta theorem is then presented which covers all the cases considered here. © 2010 Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg.

Department(s)

Mathematics and Statistics

Comments

National Science Foundation, Grant None

Keywords and Phrases

Boundary Layer; Corrector; Injection and Suction; Inviscid Limit; Navier-Stokes System; Nonlinear Pipe Flow; Nonlinear Plane Parallel Channel Flow; Prandtl Theory; Spectral Constraint

International Standard Serial Number (ISSN)

1860-6261; 0252-9599

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2023 Springer, All rights reserved.

Publication Date

17 Aug 2010

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