  We consider the problem of finding optimal shapes of fluid domains. The fluid
obeys the Navier--Stokes equations. Inside a holdall container we use a phase
field approach using diffuse interfaces to describe the domain of free flow. We
formulate a corresponding optimization problem where flow outside the fluid
domain is penalized. The resulting formulation of the shape optimization
problem is shown to be well-posed, hence there exists a minimizer, and first
order optimality conditions are derived.
  For the numerical realization we introduce a mass conserving gradient flow
and obtain a Cahn--Hilliard type system, which is integrated numerically using
the finite element method. An adaptive concept using reliable, residual based
error estimation is exploited for the resolution of the spatial mesh.
  The overall concept is numerically investigated and comparison values are
provided.
