The forward motion of an unsymmetric surface-piercing cylinder: the solvability of a nonlinear problem in the supercritical case
Carlo Domenico Pagani · The Quarterly Journal of Mechanics and Applied Mathematics · 2001
We consider the wave-resistance problem for a ‘slender’ cylinder semisubmerged in a heavy fluid and moving at uniform, supercritical speed in the direction orthogonal to its generators. By a hodograph transformation, the problem (originally set up in a domain with a free boundary) reduces to the determination of a function, holomorphic in a fixed domain, satisfying some nonlinear boundary conditions depending on two (unknown) parameters. The problem in the hodograph plane is solved via the implicit function theorem; then, the two parameters are fixed by the requirement that the free boundary and the cylinder profile (which is assumed convex and reasonably smooth) form a single smooth (C1) streamline. Furthermore, the free boundary is monotone increasing downstream, monotone increasing upstream and lies under the level of calm water.