  We study the mathematical properties of a nonequilibrium Langevin dynamics
which can be used to estimate the shear viscosity of a system. More precisely,
we prove a linear response result which allows to relate averages over the
nonequilibrium stationary state of the system to equilibrium canonical
expectations. We then write a local conservation law for the average
longitudinal velocity of the fluid, and show how, under some closure
approximation, the viscosity can be extracted from this profile. We finally
characterize the asymptotic behavior of the velocity profile, in the limit
where either the transverse or the longitudinal friction go to infinity. Some
numerical illustrations of the theoretical results are also presented.
