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 if c is not equal to cm,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.
Cite this article
S. Gatti, D. Pierotti, Solvability of a Plane Elliptic Problem for the Flow in a Channel with a Surface-Piercing Obstacle. Z. Anal. Anwend. 22 (2003), no. 2, pp. 357–381DOI 10.4171/ZAA/1150