Novel discoveries on the mathematical foundation of linear hydrodynamic stability theory

TL;DR

This paper uncovers fundamental mathematical limitations in classical hydrodynamic stability theory, revealing that traditional equations and models fail to accurately predict certain stability behaviors in inviscid and high Reynolds number flows.

Contribution

It introduces new insights into the mathematical foundations, challenging existing assumptions and highlighting the inadequacy of classical equations for certain stability analyses.

Findings

Linearized Euler equations do not approximate inviscid stability.

Eigenvalue instability does not capture dominant growth at high Reynolds numbers.

Rayleigh and Orr-Sommerfeld equations cannot fully describe the nature of the full differentials.

Abstract

We present some new discoveries on the mathematical foundation of linear hydrodynamic stability theory. The new discoveries are: 1. Linearized Euler equations fail to provide a linear approximation on inviscid hydrodynamic stability. 2. Eigenvalue instability predicted by high Reynolds number linearized Navier-Stokes equations cannot capture the dominant instability of super fast growth. 3. As equations for directional differentials, Rayleigh equation and Orr-Sommerfeld equation cannot capture the nature of the full differentials.