The Dual Stream Function Representation of an Ideal Steady Fluid Flow and its Local Geometric Structure

Abstract

Using the methodology of Lie groups and Lie algebras we determine new symmetry and equivalence classes of the stationary three-dimensional Euler equations by introducing potential functions that are based on the so-called dual stream function representation of the steady state velocity field u(x, y, z) = ∇λ(x, y, z) × ∇μ(x, y, z), which itself can only be defined locally. In particular an infinite dimensional Lie algebra for Beltrami fields is gained. We show that this Lie algebra generates canonical transformations of a Hamiltonian flow for the dual pair of variables $\lambda$ and $\mu$. It enables us to make the classification of a two-dimensional Riemannian manifold $M^{2}$ wherein $(\lambda ,\mu )$ presents the local coordinates of $M^{2}$. Furthermore the local geometry of this manifold is explored in detail. As a result an infinite set of locally conserved currents and charges in the context of a conformal field theory is finally observed.