General stability criterion of two-dimensional inviscid parallel flow

Abstract

General stability criterions of two-dimensional inviscid parallel flow are obtained analytically for the first time. First, a criterion for stability is found as $\frac{U&#39;&#39;}{U-U_s}>-μ_1$ everywhere in the flow, where $U_s$ is the velocity at inflection point, $μ_1$ is eigenvalue of Poincaré&#39;s problem. Second, we also prove a principle that the flow is stable, if and only if all the disturbances with $c_r=U_s$ are neutrally stable. Finally, following this principle, a criterion for instability is found as $\frac{U&#39;&#39;}{U-U_s}<-μ_1$ everywhere in the flow. These results extend the former theorems obtained by Rayleigh, Tollmien and Fjørtoft and will lead future works to investigate the mechanism of hydrodynamic instability.