Abstract
It is proved that a steady-state flow has an extremal kinetic energy in comparison with ″equivorticity″ flows. This result is applied to investigate the stability of steady-state flows: if the extremum is a minimum or a maximum, then the flow is stable, i.e. a small change in the initial velocity field causes only a small change in the velocity field for all time. To determine the nature of the extremum (maximum, minimum, etc.) a second variation is explicitly calculated. For the case of plane flows, sufficient conditions of the stability with respect to small finite perturbations are found. These conditions are close to the necessary ones.
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© 1965 Nauka, Moscow
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Arnold, V.I. (1965). A Variational principle for three-dimensional steady flows of an ideal fluid. In: Givental, A., Khesin, B., Varchenko, A., Vassiliev, V., Viro, O. (eds) Vladimir I. Arnold - Collected Works. Vladimir I. Arnold - Collected Works, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31031-7_1
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DOI: https://doi.org/10.1007/978-3-642-31031-7_1
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31030-0
Online ISBN: 978-3-642-31031-7
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