Solvability of a plane elliptic problem for the flow in a channel with a surface-piercing obstacle

Let us consider the three-dimensional problem of the 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 the two dimensional 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≠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.