The subcritical motion of a surface-piercing cylinder: existence and regularity of waveless solutions of the linearized problem

Abstract

We consider a boundary value problem in a strip, which is obtained by linearization of the two-dimensional problem of a stream about a slender cylinder, semisubmerged in a heavy fluid of finite depth; it is assumed that the cylinder has uniform, subcritical speed in the direction orthogonal to its generators. We discuss in particular the waveless statement of the problem, characterized by the asymptotic condition of vanishing flow oscillations at infinity. By suitable variational formulations of the problem, we find two classes of solutions, differing by their regularity properties. The most regular solutions exist for particular shapes of the cylinder's section and provide a velocity field which is everywhere continuous and bounded in the strip. The solutions of the other class exist for every (reasonably smooth) symmetric cylinder's profile and have finite energy, but are singular at two points on the strip boundary, representing the points where the free surface meets the cylinder's hull. The relevance of these results for the solvability of the nonlinear problem is discussed.