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 $\frac{U''}{U-U_s}>-\mu_1$ everywhere in the flow, where $U_s$ is the velocity at inflection point, $\mu_1$ is eigenvalue of Poincar\'{e}'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 $\frac{U''}{U-U_s}<-\mu_1$ everywhere in the flow. These results extend the former theorems obtained by Rayleigh, Tollmien and Fj{\o}rtoft and will lead future works to investigate the mechanism of hydrodynamic instability.