  We study the existence and properties of the non-equilibrium steady state
which arises by putting two copies of systems at different temperatures into a
thermal contact. We solve the problem for the relativistic systems that are
described by the energy-momentum of a perfect hydro with general equation of
state (EOS). In particular, we examine several simple examples: a hydro with a
linear EOS, a holographic CFT perturbed by a relevant operator and a barotropic
fluid, i.e., P = P(E). Our studies suggest that the formation of steady state
is a universal result of the hydro regime regardless of the kind of fluid.
