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On the motion of vortices in two dimensions. I: Existence of the Kirchhoff-Routh function. II: Some further investigations on the Kirchhoff-Routh function. (English) Zbl 0063.03560

Author’s introduction: G. Kirchhoff [Vorlesungen über mathematische Physik. Erster Band: Mechanik. 4. Aufl. Leipzig: B. G. Teubner (1897; JFM 28.0603.01)] was the first to establish the existence of “a stream function giving the motion of vortices” in an unbounded region. Later, in 1881, E. J. Routh [Proc. Lond. Math. Soc. 12, 73–89 (1881; JFM 13.0720.01)] enunciated a theorem regarding the transformation of a function of that nature for the case of a single vortex moving in a bounded region. No proof was given and the existence of such a stream function was not established. In 1921, M. Lagally [Math. Z. 10, 231–239 (1921; JFM 48.0949.04)] established the existence of this “Routh’s stream function” and the more general “Kirchhoff’s path function” for a simply connected bounded region. An independent proof for the case of a single vortex in such a region was also given by A. Masotti in 1931 [Atti Pontif. Accad. Sci. Nuovi Lincei 84, 209–216 (1931; Zbl 0003.08203; JFM 57.0577.02), et al.], by using the Green function of the first kind. However, the most general case of the motion of a number of vortices in a multiply connected region is not covered by their work.
The present article establishes the existence of the “Kirchhoff-Routh function” for the general case. The proof makes use of a generalized Green function and may therefore be regarded as a generalization of Masotti’s work. Explicit formula of this function is also given. A more detailed treatment of this work will appear elsewhere (see Part II).
In Part II we investigate the behavior of the Kirchhoff-Routh function (whose existence we have established in Part I under a conformal transformation of fluid motion.

MSC:

76B47 Vortex flows for incompressible inviscid fluids
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