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On the equation satisfied by a steady Prandtl-Munk vortex sheet. (English) Zbl 1080.76018

Summary: We show that the vorticity distribution obtained by minimizing the induced drag on a wing, the so-called Prandtl-Munk vortex sheet, is not a travelling-wave weak solution of the Euler equations, contrary to what has been claimed by a number of authors. Indeed, it is a weak solution of a inhomogeneous Euler equation, where the forcing term represents a “tension” force applied to the tips. This is consistent with a heuristic argument due to P. G. Saffman [Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge: Cambridge University Press (1992; Zbl 0777.76004)]. Thus, the notion of weak solution captures the correct physical behavior of this case.

MSC:

76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 0777.76004
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