Solvability of a three-dimensional free surface flow problem over an obstacle

Abstract

ABSTRACT In this paper, we consider a problem of a three-dimensional free surface flow over an obstacle lying on the bottom of an infinite channel. The flow is supercitical, irrotational, stationary and the fluid is ideal and incompressible. We linearize the problem, then by a Fourier expansion of the velocity potential, we reduce the three-dimensional problem to a sequence of two-dimensional problems for the Fourier coefficients. We take into account of the gravity and we neglect the effects of the superficial tension. These problems are formulated by the Laplace operator and the Neumann–Kelvin condition on the free surface of the fluid domain. Some a priori properties of the solution are given, they allow us to construct a space, where we can use Lax–Milgram's theorem, to prove the existence and the uniqueness of the solution of the two-dimensional problem. Then, this sequence of solutions of 2D-problems, gives us the unique solution of the considered three-dimensional problem.