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

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.

Keywords:

• Three-dimensional flow
• free surface flow
• Neumann–Kelvin condition
• linearized problem
• supercritical

AMS Subject Classifications:

• 35R35
• 76B07
• 76B45

Acknowledgments

The authors would like to express their gratitude to Chérif Amrouche, Professor of Mathematics at: the Laboratory of Mathematics and its Applications (LMAP), Université de Pau et des Pays de l'Adour, France, for his advice and availability during the stay of Nadjiba Foukroun at the LMAP laboratory.

Disclosure statement

No potential conflict of interest was reported by the author(s).