Formal derivation of boundary layers in fluid mechanics
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. Greenspan} [The theory of rotating fluids (Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge, University Press) (1968; Zbl 0182.28103)]. The following cases are studied: Ekman layers, vertical layers, Hartmann layers, horizontal and vertical layers, MHD layers (here the basic equations are developed for small magnetic Reynolds numbers), Munk and Stommel layers. Time layers are briefly mentioned.
scientific article; zbMATH DE number 2176389