  Second-order phase transitions are characterized by a divergence of the
spatial correlation length of the order parameter fluctuations. For confined
systems, this is known to lead to remarkable equilibrium physical phenomena,
including finite-size effects and critical Casimir forces. We explore here some
non-equilibrium aspects of these effects in the stationary state resulting from
the action of external forces: by analyzing a model of a correlated fluid under
shear, spatially confined by two parallel plates, we study the resulting
viscosity within the setting of (Gaussian) Landau-Ginzburg theory.
Specifically, we introduce a model in which the hydrodynamic velocity field
(obeying the Stokes equation) is coupled to an order parameter with dissipative
dynamics. The well-known Green-Kubo relation for bulk systems is generalized
for confined systems. This is shown to result in a non-local Stokes equation
for the fluid flow, due to the correlated fluctuations. The resulting effective
shear viscosity shows universal as well as non-universal contributions, which
we study in detail. In particular, the deviation from the bulk behavior is
universal, depending on the ratio of the correlation length and the film
thickness $L$. In addition, at the critical point the viscosity is proportional
to $\ell/L$, where $\ell$ is a dynamic length scale. These findings are
expected to be experimentally observable, especially for systems where the bulk
viscosity is affected by critical fluctuations.
