The spectra of the oscillating shear flows

TL;DR

This paper investigates the spectral properties of high-frequency oscillating shear flows in inviscid incompressible fluids, providing asymptotic expansions for unstable eigenvalues when the limit problem exhibits multiple eigenvalues.

Contribution

It introduces asymptotic expansions for unstable eigenvalues in oscillating shear flows with multiple eigenvalues, advancing understanding of their spectral stability.

Findings

Derived asymptotic formulas for eigenvalues

Analyzed spectral stability of oscillating shear flows

Extended results to cases with multiple eigenvalues

Abstract

We study the spectral problems for the spatially periodic flows of inviscid incompressible fluid. The basic flows under consideration are the shear flows whose profiles oscillate on high frequencies. For such flows, we present asymptotic expansions of the unstable eigenvalues in the case when the limit spectral problem has multiple eigenvalues.