The solution of steady-state free surface problems by the finite element method

Bibliography: pages 64-68.This thesis is concerned with the development of a variational formulation for the problem of viscous incompressible free surface flows, and with the development and implementation of algorithms for the solution of this problem by finite elements. The study is restricted to two-dimensional steady problems. The approach differs from those in current use, in that it is based on a two-stage strategy suggested by theoretical (existence) studies of the problem. In the first stage the free surface is kept fixed and the resulting so-called auxiliary problem is solved. In the second stage the equation for the normal stress on the free surface is used to update the free surface. Both the auxiliary problem and the normal stress equation are formulated variationally; in the case of the latter problem the unknown variable is actually the slope of the free surface. Finite element approximations are used in both problems. Algorithms are developed for determining solutions at the two stages, and for the overall problem. The key example treated is the dieswell problem, for the plane and axisymmetric cases. Solutions obtained by the present method are presented, and compared with the solutions of others where available