Singularities of Rayleigh equation

TL;DR

This paper analyzes the solutions of the Rayleigh equation near critical points, elucidating their boundary and asymptotic behaviors, which are crucial for understanding shear flow stability and vorticity dynamics.

Contribution

It provides a local description of Rayleigh equation solutions near degenerate critical points and connects boundary values with behavior at infinity.

Findings

Characterization of solutions near degenerate critical points

Linking boundary values to asymptotic behavior

Insights into shear flow stability analysis

Abstract

The Rayleigh equation, which is the linearized Euler equations near a shear flow in vorticity formulation, is a key ingredient in the study of the long time behavior of solutions of linearized Euler equations, in the study of the linear stability of shear flows for Navier-Stokes equations and in particular in the construction of the so called Tollmien-Schlichting waves. It is also a key ingredient in the study of vorticity depletion. In this article we locally describe the solutions of Rayleigh equation near critical points of any order of degeneracy, and link their values on the boundary with their behaviors at infinity.