  The necessary and sufficient conditions are obtained for a unit time-like
vector field $u$ to be the unit velocity of a divergence-free perfect fluid
energy tensor. This plainly kinematic description of a conservative perfect
fluid requires considering eighteen classes defined by differential
concomitants of $u$. For each of these classes, we get the additional
constraints that label the flow of a conservative energy tensor, and we obtain
the pairs of functions $\{\rho,p\}$, energy density and pressure, which
complete a solution to the conservation equations.
