Discrete Spectrum of Bifurcation of Exact Solutions for Stationary Longitudinal Waves in the Flow of Perfect Fluid Around a Circular Body of a Large Radius

Abstract

We consider a free boundary problem for incompressible perfect fluid circulating around a circle Γ of a large radius, i.e. within a central gravity field. 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. The centrifugal force caused by the circulation of the flow makes the fluid go outward. We show that there exist stationary waves, which are periodic oscillations of the fluid and are exact solutions corresponding to bifurcating branches emanating from trivial solution. In the frame of the problem in question, the wave number can take only the integer multiplies of 2π . This makes the spectrum discrete, which makes a crucial difference with the problem for flows in an infinite domain over the straight line or plane (i.e. the ocean problem). To prove the existence of exact solutions, the method of conformal mapping is used. By this device the free boundary problem is transformed into a boundary value problem in a fixed domain.