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Helicity conservation laws

  • Zbigniew PeradzyŃSki
Conference paper
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 71)

Abstract

Using the language of differential forms on a space-time, one can write the equation of an ideal fluid in a form similar to the Maxwell equations. Vorticity current plays then the role of the source term and the Euler equations can be interpreted as the generalisation, to the whole space-time, of the well-known fact that the number of vortex lines passing through any two-dimensional surface spanned on a closed contour can be expressed by a circulation associated with this contour. A similar procedure can be used for the ideal MHD. It appears that by using this formulation various helicity conservation theorems may be derived in the natural and straightforward manner.

Keywords

Euler Equation Differential Form Vortex Line Exterior Derivative Vector Notation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arnold, V.I., Khesin, B.A. 1998 Topological Methods in Hydrodynamics. Springer-Verlag, New York-Berlin-Heidelberg.zbMATHGoogle Scholar
  2. Flanders, H. 1963 Differential Forms with applications to physical sciences. Academic Press, New York-London.zbMATHGoogle Scholar
  3. Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech.35, part 1, 117–129.zbMATHADSCrossRefGoogle Scholar
  4. Peradzyński, Z. 1988 Properly posed boundary condition and the existence theorem insuperfluid He4. In Trends in Applications of Mathematics to Mechanics, Proc. 7th Symposium (ed. J.F. Besseling & W. Eckhaus), pp. 224–237. Springer Verlag.Google Scholar
  5. Peradzyński, Z. 1990 Helicity theorem and vortex lines in superfluid He4. Int. J. Theoret. Physics29, 1277–1284.zbMATHCrossRefGoogle Scholar
  6. Peradzyński, Z. 1991 Differential forms and fluid dynamics. Arch. Mech.43, 653–661.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Zbigniew PeradzyŃSki
    • 1
    • 2
  1. 1.Institute of Applied Mathematics and MechanicsWarsaw UniversityWarsaw
  2. 2.Institute of Fundamental Technological ResearchWarsawPoland

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