At first, we shall study the thermal partition function of quantum field theories on arbitrary stationary background spacetime, and with arbitrary stationary background gauge fields, in the long wavelength expansion. We shall demonstrate that the equations of relativistic hydrodynamics are significantly constrained by the requirement of consistency with any partition function. In specific examples, at low orders in the derivative expansion, we shall demonstrate that these constraints coincide precisely with the equalities between hydrodynamic transport coefficients that follow from the local form of the second law of thermodynamics. We shall then go on to construct non-trivial classical action for second order non-dissipative hydrodynamics involving pure energy-momentum transport. Comparison of our results with non-dissipative hydrodynamics that follows from entropy current analysis, reveals the existence of additional constrains, when the effective theory of hydrodynamics follows from an action principle.