  This paper is concerned with the problem of shape optimization of
two-dimensional flows governed by the time-dependent Navier-Stokes equations.
We derive the structures of shape gradients with respect to the shape of the
variable domain for time-dependent cost functionals by using the state
derivative with respect to the shape of the fluid domain and its associated
adjoint state. Finally we apply a gradient type algorithm to our problem and
numerical examples show that our theory is useful for practical purpose and the
proposed algorithm is feasible in low Reynolds number flow.
