On the Existence of Progressive Waves in the Flow of Perfect Fluid around a Circle

Abstract

We consider a free boundary problem for incompressible perfect fluid with surface tension. The problem to be considered is as follows: A perfect fluid is circulating around a circle Γ (see Fig. 1). The outward curve γ is a free boundary to be sought. We assume that the flow, which is confined between Γ and γ is irrotational. On the free boundary, surface tension works and makes the free boundary circular. On the other hand, the centrifugal force caused by the circulation of the flow makes the fluid go outward. Hence the balance of these two kinds of forces determines the geometrical properties of the free boundary. We show that there exist progressive waves, which are periodic motions of the fluid and are the exact solutions corresponding to the solitary waves. Fig. 1