Formal derivation of boundary layers in fluid mechanics
type: Item
instance of: scholarly article
title: Formal derivation of boundary layers in fluid mechanics
zbMATH Open document ID: 1080.35071
published in: Journal of Mathematical Fluid Mechanics
publication date: 2005-06-16T00:00:00Z
review text: A 2002 article by the author and \textit{A. Grenier} [RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 96, No. 3, 401--410 (2002;Zbl 1185.76517)] provides a good introduction to the present paper. Of particular importance are fluid flows incompatible with given boundary conditions, which results in the formation of boundary layers. Basic are Navier-Stokes equations with a Coriolis term, if such flows represent problems of geophysics. Such equations are of the form: \(\partial_tu+ u\nabla u+ (e\times u)/\varepsilon+ p/\varepsilon- E\Delta u/\varepsilon\), with zero divergence \(\nabla\cdot u= 0\). The parameters \(E\) and \(\varepsilon\) are called, respectively, the Ekman and the Rossby numbers. The author comments on various computational approaches to such flows, including the popular technique of matching expansions, conveying the well known truth that these are serious and ugly computations. The author proposes to simplify such computations by using of algebraic computations. The equations of the system are rewritten in the form \(A^\varepsilon U^\varepsilon+ Q^\varepsilon(U^\varepsilon)= F^\varepsilon\). Here \(A^\varepsilon\) is a (linear) matrix differential operator, \(Q^\varepsilon\) is a nonlinear operator, taking care of the nonlinear terms, \(F^\varepsilon\) is the forcing term. Past practice indicated that the nonlinear terms appear in higher-order perturbations, and they do not create the existence of the boundary layer. Thus the first approximation considers the linear terms only \(A^\varepsilon U^\varepsilon= F^\varepsilon\). To find approximations to the boundary layers the author studies \(A^\varepsilon U^\varepsilon= 0\). The author argues that to find the sizes of boundary layers it suffices to find asymptotic solutions of \(a^\varepsilon= 0\), where \(a^\varepsilon\) is the determinant of \(A^\varepsilon\). He points out that vanishing of this determinant is an algebraic form of the classical pressure equation of \textit{H. P. Greensp…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D10
zbMATH DE Number: 2176389
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: Ekman layers
zbMATH Keywords: vertical layers
zbMATH Keywords: MHD layers
zbMATH Keywords: Munk layers
zbMATH Keywords: Stommel layers
zbMATH Keywords: time layers
zbMATH Keywords: Hartmann layers
author: David Gérard-Varet
MaRDI profile type: Publication
full work available at URL: s00021-004-0115-9
OpenAlex ID: W2123006368
DOI: 10.1007/S00021-004-0115-9
description: scientific article; zbMATH DE number 2176389