  While conventional hydrodynamics incorporating dissipative effects is hard to
derive from an action principle, it is nevertheless possible to construct
classical actions when the dissipative terms are switched off. In this note we
undertake a systematic exploration of such constructions from an effective
field theory approach and argue for the existence of non-trivial second order
non-dissipative hydrodynamics involving pure energy-momentum transport. We find
these fluids to be characterized by five second-order transport coefficients
based on the effective action (a three parameter family is Weyl invariant). On
the other hand since all flows of such fluids are non-dissipative, they entail
zero entropy production; one can therefore understand them using the entropy
current formalism which has provided much insight into hydrodynamic transport.
An analysis of the most general stress tensor with zero entropy production
however turns out to give a seven parameter family of non-dissipative
hydrodynamics (a four parameter sub-family being Weyl invariant). The
non-dissipative fluids derived from the effective action approach are a special
case of the fluid dynamics constrained by conservation of the entropy current.
We speculate on the reasons for the mismatch and potential limitations of the
effective action approach.
