  A class of A.L.E. time discretisations which inherit key energetic properties
(nonlinear dissipation in the absence of forcing and long-term stability under
conditions of time dependent loading), irrespective of the time increment
employed, is established in this work. These properties are intrinsic to real
flows and the conventional Navier-Stokes equations.
  A description of an incompressible, Newtonian fluid, which reconciles the
differences between the various schools of A.L.E. thought in the literature is
derived for the purposes of this investigation. The issue of whether these
equations automatically inherit the afore mentioned energetic properties must
first be resolved. In this way natural notions of nonlinear, exponential-type
dissipation in the absence of forcing and long-term stability under conditions
of time dependent loading are also formulated.
  The findings of this analysis have profound consequences for the use of
certain classes of finite difference schemes in the context of deforming
references. It is significant that many algorithms presently in use do not
automatically inherit the fundamental qualitative features of the dynamics.
  The main conclusions are drawn on in the simulation of a driven cavity flow,
a driven cavity flow with various, included rigid bodies, a die-swell problem,
and a Stokes second order wave. The improved, second order accuracy of a new
scheme for the linearised approximation of the convective term is proved for
the purposes of these simulations. A somewhat novel method to generate finite
element meshes automatically about included rigid bodies, and which involves
finite element mappings, is also described.
