  In this work a finite element simulation of the motion of a rigid body in a
fluid, with free surface, is described. A completely general referential
description (of which both Lagrangian and Eulerian descriptions are special
cases) of an incompressible, Newtonian fluid is used. Such a description
enables a distorting finite element mesh to be used for the deforming fluid
domain.
  A new scheme for the linearised approximation of the convective term is
proposed and the improved, second order accuracy of this scheme is proved. A
second theorem, which provides a guideline for the artificial adjustment of the
Reynolds number when applying a continuation technique, is also proved. The
most effective means of eliminating pressure as a variable and enforcing
incompressibility are reviewed. A somewhat novel method to generate finite
element meshes automatically about included rigid bodies, and which involves
finite element mappings, is described.
  The approach taken when approximating the free surface, is that it may be
treated as a material entity, that is, the material derivative of the free
surface is assumed zero. Euler's equations and conservation of linear momentum
are used to determine the motion of the rigid body. A predictor-corrector
method is used to solve the combined sub-problems. The resulting model is
tested in the context of a driven cavity flow, a driven cavity flow with
various, included rigid bodies, a die-swell problem, and a Stokes second order
wave.
