  Motivated by the fluid/gravity correspondence, we consider the Penrose
inequality in the framework of fluid dynamics. In general relativity, the
Penrose inequality relates the mass and the entropy associated with a
gravitational background. If the inequality is violated by some Cauchy data, it
suggests a creation of a naked singularity, thus providing means to consider
the cosmic censorship hypothesis. The analogous inequality in the context of
fluid dynamics can provide a valuable tool in the study of finite-time blowups
in hydrodynamics. We derive the inequality for relativistic and nonrelativistic
fluid flows in general dimension. We show that the inequality is always
satisfied at the ideal fluid order. At the leading viscous order, the
inequality may be violated by relativistic fluid flows, while it is always
satisfied by nonrelativistic incompressible flows. The inequality may be
violated at the next to leading viscous order by both relativistic and
nonrelativistic flows.
