Steady inviscid rotational flows with free surfaces

Abstract

An inviscid fluid in steady two-dimensional motion in a region bounded by a closed streamline must have constant vorticity. We solve here for some such flows where the boundary is in part free, the fluid velocity magnitude being constant on the free boundary. A trivial example of such a flow is a circular cylinder of fluid rotating about its axis as if rigid, for which the whole circular boundary is free, irrespective of its radius. We now ask what happens to that flow when it comes into contact with solid boundaries. There is no steady flow when the contact is with a finite segment of a single plane wall, but a unique solution exists when the rotating fluid mass is in contact with some concave boundaries. Computed results are obtained for vortices lying inside a parabolic dish, or in a corner between two planes.