On variational features of vortex flows |
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Authors: | V L Berdichevsky |
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Institution: | (1) Department of Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA |
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Abstract: | Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity
over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones,
the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of
motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible
fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle
for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997;
Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set
to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property.
This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and
the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational
principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion
is discussed.
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Keywords: | Vortex line Variational principle Steady vortex flow |
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