  In this study, a shape optimization problem for the two-dimensional
stationary Navier--Stokes equations with an artificial boundary condition is
considered. The fluid is assumed to be flowing through a rectangular channel,
and the artificial boundary condition is formulated so as to take into account
the possibility of ill-posedness caused by the usual do-nothing boundary
condition. The goal of the optimization problem is to maximize the vorticity of
the said fluid by determining the shape of an obstacle inside the channel.
Meanwhile, the shape variation is limited by a perimeter functional and a
volume constraint. The perimeter functional was considered to act as a Tikhonov
regularizer and the volume constraint is added to exempt us from topological
changes in the domain. The shape derivative of the objective functional was
formulated using the rearrangement method, and this derivative was later on
used for gradient descent methods. Additionally, an augmented Lagrangian method
and a class of solenoidal deformation fields were considered to take into
account the goal of volume preservation. Lastly, numerical examples based on
the gradient descent and the volume preservation methods are presented.
