  The properties of an anisotropic fluid outside a star or a black hole
embedded in an expanding universe are investigated. One finds that, in
Painleve-Gullstrand coordinates, the heat flux of the cosmological fluid
vanishes, in spite of the nonzero value of a non-diagonal component of the
stress tensor. The pressures and energy density of the fluid are regular at $r
= 2m$, divergent at $r = 0$ and change sign at certain values of their
arguments.
