Spectral portraits of the Orr--Sommerfeld operator with large Reynolds numbers

Abstract

A model problem of the form -iεy''+q(x)y=λy, y(-1)=y(1)=0, is associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here (λ) is the spectral parameter, (ε) is the small parameter which is proportional to the viscocity 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|\leqslant 1). We study the behaviour of the spectrum of the corresponding model operator as (ε\to 0) with linear, quadratic and monotonous analytic functions. We show that the sets of the accumulation points of the spectra (the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld operators coincide as well as the main terms of the counting eigenvalue functions along the curves of the graphs.