  The long-wavelength effective field theory of world-volume fluctuations of
black D3-branes is shown to be a hydrodynamical system to leading order in a
gradient expansion. We study the system on a fiducial `cutoff' surface: the
fluctuating geometry imprints its dynamics on the surface via an induced stress
tensor whose conservation encapsulates the hydrodynamical description. For a
generic non-extremal D3-brane, as we move our cutoff surface from the
asymptotically flat near-boundary region to the near-horizon region, this
hydrodynamical system interpolates between a non-conformal relativistic fluid
and a non-relativistic incompressible fluid. We also consider the dependence on
the deviation from extremality of the D3-branes. In the near-extremal case we
recover the description in terms of a conformal relativistic fluid encountered
in the AdS/CFT context. We argue that this system allows us therefore to
explore the various connections that have hitherto been suggested relating the
dynamics of gravitational systems and fluid dynamics. In particular, we go on
to show that the blackfold effective field theory approach allows us to capture
this hydrodynamical behaviour and moreover subsumes the constructions
encountered in the fluid/gravity correspondence and the black hole membrane
paradigm, providing thereby a universal language to explore the effective
dynamics of black branes.
