On the Model for the Orr--Sommerfeld Equation with Quadratic Profile

TL;DR

This paper thoroughly investigates the spectral behavior of the Orr--Sommerfeld equation with a quadratic profile on a finite interval, especially for large Reynolds numbers, identifying eigenvalue concentration curves and counting functions.

Contribution

It provides a complete analysis of eigenvalue distribution and asymptotic behavior for the Orr--Sommerfeld model with quadratic profile at high Reynolds numbers.

Findings

Eigenvalues concentrate along specific limit curves.

Explicit eigenvalue counting functions are derived.

Spectral behavior is characterized for large Reynolds numbers.

Abstract

The model for Orr--Sommerfeld equation with quadratic profile on the finite interval is considered. The behavior of the spectrum of this problem is completely investigated for large Reynolds numbers. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.