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

  • M. Frewer
  • M. Oberlack
  • V. N. GrebenevEmail author


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.


Euler equations Dual stream function Symmetries Beltrami fields Equivalence transformation Shape classification 

Mathematics Subject Classifications (2010)

76F05 76F55 53B21 53B50 58J70 


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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institute of Computational TechnologiesRussian Academy of ScienceNovosibirskRussia

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