  This paper is concerned with the optimal shape design of the newtonian
viscous incompressible fluids driven by the stationary nonhomogeneous
Navier-Stokes equations. We use three approaches to derive the structures of
shape gradients for some given cost functionals. The first one is to use the
Piola transformation and derive the state derivative and its associated adjoint
state; the second one is to use the differentiability of a minimax formulation
involving a Lagrangian functional with a function space parametrization
technique; the last one is to employ the differentiability of a minimax
formulation with a function space embedding technique. 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.
