Solvability of a Plane Elliptic Problem for the Flow in a Channel with a Surface-Piercing Obstacle

Abstract

Let us consider the three-dimensional problem of steady flow of a heavy ideal fluid past a surface-piercing obstacle in a rectangular channel of constant depth. The flow is parallel at infinity upstream with constant velocity $c$. We discuss an approximate linear problem obtained in the limit of a “flat obstacle”. This is a boundary value problem for the Laplace equation in a three-dimensional unbounded domain, with a second order condition on part of the boundary – the Neumann–Kelvin condition. By a Fourier expansion of the potential function we reduce the three-dimensional problem to a sequence of plane problems for the Fourier coefficients. For every value of the velocity $c$ these problems can be described in terms of a two parameter elliptic problem in a strip. We discuss a well-posed formulation of such problem by a special variational approach, relying on some a priori properties of finite energy solutions. As a result, we prove unique solvability for $c\ne c_{m,k}$, where $c_{m,k}$ is a known sequence of values depending on the dimensions of the channel and on the limit length of the obstacle. Accordingly, we can prove the existence of a solution of the three-dimensional problem; the related flow has in general a non-trivial wave pattern at infinity downstream. We also investigate the regularity of the solution in a neighborhood of the obstacle. The meaning of the singular values $c_{m,k}$ is discussed from the point of view of the nonlinear theory.