Symmetry transformations of an ideal steady fluid flow determined by a potential function

Abstract

First, we consider a transformation Ξ of 3D trajectories (fluid particle paths) of inviscid steady flows using the dual stream function approach for the (local) representation of velocity fields u→(x,y,z)=∇ λ×∇ μ. This enables to derive the equation governing the deformation of trajectories by the gradient field ξ→=∇ μ along the surface λ(x, y, z) = λ0. In fact, Ξ is a symmetry transformation and it looks formally like the filament motion which preserves the curvature. Then, we investigate in detail a fine structure of a Lie algebra associated with an extension of the transformation Ξ which creates a visual appearance of sliding stream surfaces λ(x, y, z) = λ0 along itself. The minimal set of generating differential invariants is found. This set consists of a single invariant which coincides with a Hamiltonian function.