On the Spectrum Localization of the Orr--Sommerfeld Problem for Large Reynolds Numbers

Abstract

The paper deals with the Orr--Sommerfeld problem and the corresponding model problem

$$ - i\varepsilon ^2 y'' - q(x)y = - \lambda y,{\text{ }}y( - 1) = y(1) = 0.$$

The functions q(x)= x and q(x)= x ² in this model correspond to the Couette and the Poiseuille profiles, respectively. Small values of the parameter ɛ correspond to large Reynolds numbers. As ɛ tends to zero, the spectrum of the model problem is localized near certain critical curves in the complex plane, whose explicit form can be determined. Moreover, there are asymptotic formulas for the eigenvalue distribution along these curves as ɛ→0. The main result of the paper is the following: as the Reynolds number tends to infinity, the spectrum of the Orr--Sommerfeld problem for the Couette and the Poiseuille flows is localized to the critical curves, which are the same as in the model problem. Moreover, the main terms of the asymptotic formulas for the eigenvalue distribution are preserved.