  This paper focuses on the analysis of an optimal control problem governed by
a nonsmooth quasilinear partial differential equation that models a stationary
incompressible shear-thickening fluid. We start by studying the directional
differentiability of the non-smooth term within the state equation as a prior
step to demonstrate the directional differentiability of the solution operator.
Thereafter, we establish a primal first order necessary optimality condition
(Bouligand (B) stationarity), which is derived from the directional
differentiability of the solution operator. By using a local regularization of
the nonsmooth term and carrying out an asymptotic analysis thereafter, we
rigourously derive a weak stationarity system for local minima. By combining
the B- and weak stationarity conditions, and using the regularity of the
Lagrange multiplier, we are able to obtain a strong stationarity system that
includes an inequality for the scalar product between the symmetrized gradient
of the state and the Lagrange multiplier.
