Isocirculational flows and their Lagrangian and energy principles

Abstract

Theorems are proved relating the equilibrium and stability of steady flows to the stationariness of the energy for isocirculational perturbations at fixed angular momentum. Similar principles are demonstrated for con­tainers forced to rotate at fixed angular velocity. The invariant concepts of the load, λ , of a vortex line and the metage, μ , of a point on that line are introduced and used to define the surfaces of constant load. It is shown that the content of Kelvin’s circulation theorem is contained in the conservation of the function C(λ) which gives the circulations around the large vortex tubes that make the surfaces of constant load. Two flows are isocirculational if and only if their C(λ) -functions are the same. A new Lagrangian for inviscid barotropic fluid mechanics that uses ρ, λ, ͘ρ, λֺ as variables is shown to yield the fluid equations in Clebsch’s form. An energy functional that yields all steady flows as its stationary points is deduced. Its minimization yields stable inviscid barotropic flows.