  We study the rheological signatures of departure from equilibrium in
two-dimensional viscous fluids with and without internal spin. Under the
assumption of isotropy, we provide the most general linear constitutive
relations for stress and couple stress in terms of the velocity and spin
fields. Invoking Onsager's regression hypothesis for fluctuations about steady
states, we derive the Green-Kubo formulae relating the transport coefficients
to time correlation functions of the fluctuating stress. In doing so, we verify
the claim that one of the non-equilibrium transport coefficients, the
odd-viscosity, requires time reversal symmetry breaking in the case of systems
without internal spin. However, the Green-Kubo relations for systems with
internal spin also show that there is a possibility for non-vanishing odd
viscosity even when time reversal symmetry is preserved. Furthermore, we find
that breakdown of equipartition in non-equilibrium steady states results in the
decoupling of the two rotational viscosities relating the vorticity and the
internal spin.
