  We study an optimization problem that aims to determine the shape of an
obstacle that is submerged in a fluid governed by the Stokes equations. The
mentioned flow takes place in a channel, which motivated the imposition of a
Poiseuille-like input function on one end and a do-nothing boundary condition
on the other. The maximization of the vorticity is addressed by the $L^2$-norm
of the curl and the {\it det-grad} measure of the fluid. We impose a Tikhonov
regularization in the form of a perimeter functional and a volume constraint to
address the possibility of topological change. Having been able to establish
the existence of an optimal shape, the first order necessary condition was
formulated by utilizing the so-called rearrangement method. Finally, numerical
examples are presented by utilizing a finite element method on the governing
states, and a gradient descent method for the deformation of the domain. On the
said gradient descent method, we use two approaches to address the volume
constraint: one is by utilizing the augmented Lagrangian method; and the other
one is by utilizing a class of divergence-free deformation fields.
