  This paper focuses on the so-called Weighted Inertia-Dissipation-Energy
(WIDE) variational approach for the approximation of unsteady Leray-Hopf
solutions of the incompressible Navier-Stokes system. Initiated in [56], this
variational method is here extended to the case of non-Newtonian fluids with
power-law index $r\geq11/5$ in three space dimensions and large nonhomogeneous
data. Moreover, boundary conditions are not imposed on some parts of
boundaries, representing, e.g., outflows. Correspondingly, natural boundary
conditions arise from the minimization. In particular, at walls we recover
boundary conditions of Navier-slip type. At outflows and inflows, we obtain the
condition
$-\frac12|\boldsymbol{v}|^2\boldsymbol{n}+\mathbb{T}\boldsymbol{n}=0$. This
provides the first theoretical explanation for the onset of such boundary
conditions.
