Three Important Theorems for Fluid Dynamics

Abstract

The new proposed "energy gradient theory," which physically explains the phenomena of flow instability and turbulent transition in shear flows and has been shown to be valid for parallel flows, is extended to curved flows in this study. Then, three important theorems for fluid dynamics are deduced. These theorems are (1) Potential flow (inviscid and irrotational) is stable. (2) Inviscid rotational (nonzero vorticity) flow is unstable. (3) Velocity profile with an inflectional point is unstable when there is no work input or output to the system, for both inviscid and viscous flows. These theorems are, for the first time, deduced, and are of great significance for the understanding of generation of turbulence and the explanation of complex flows. From these results, it is concluded that the classical Rayleigh theorem (1880) on inflectional velocity instability of inviscid flows is incorrect which has last for more than a century. It is demonstrated that existence of inflection point on velocity profile is a sufficient condition, but not a necessary condition for flow instability, for both inviscid and viscous flows. The paradox of dual role of viscosity has been resolved: viscosity has only stable role. Why the paradox appeared is due to that the Rayleigh Theorem of inviscid flows by linear analysis is incorrect.