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.