Stationary shapes for 2-d water-waves and hydraulic jumps

Abstract

A hydraulic jump is a physical phenomenon commonly observed in nature such as in open channel flows or spillways and is dependent upon the relation between the initial upstream fluid speed and a critical speed characterized by a dimensionless number F known as the Froude number. In this paper we prove the existence of hydraulic jumps for stationary water-waves as a consequence of the existence of bifurcation branches of non-flat liquid interfaces originated from each of a sequence of upstream velocities F-1 > F-2 > center dot center dot center dot > F-r > center dot center dot center dot (F-r -> 0 as r -> infinity). We further establish explicitly, for F > 0, F not equal F-r, r is an element of N, the existence and uniqueness of the solution of a perfect, incompressible, irrotational free surface flow over a flat bottom, under the influence of gravity; as well as the corresponding hydraulic jump. Published by AIP Publishing.