Spectral Portraits of the Orr--Sommerfeld Operator with Large Reynolds Numbers

A. A. Shkalikov

Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 3, Tome 3 (2003), pp. 89-112

Résumé

The model problem $-i\varepsilon y''+q(x)y=\lambda y$, $y(-1)=y(1)=0$ is associated with the Orr–Sommerfeld operator well-known in hydrodynamics. Here $\lambda$ is the spectral parameter, $\varepsilon$ is the small parameter which is proportional to the viscosity of the liquid and to the reciprocal of the Reynolds number, and $q(x)$ is the velocity of the stationary flow of the liquid in the channel $|x|\leqslant1$. We study the behavior of the spectrum of the corresponding model operator as $\varepsilon\to0$ with linear, quadratic, and monotonic analytic functions. We show that the sets of accumulation points of the spectra (the limit spectral graphs) of the model and corresponding Orr–Sommerfeld operators coincide just as the main terms of the eigenvalue counting functions along the curves of the graphs do.