  The equation of state (EOS) of a Fermi superfluid is investigated in the
BCS-BEC crossover at zero temperature. We discuss the EOS based on Monte-Carlo
(MC) data and asymptotic expansions and the EOS derived from the extended BCS
(EBCS) mean-field theory. Then we introduce a time-dependent density
functional, based on the bulk EOS and Landau's superfluid hydrodynamics with a
von Weizs\"acker-type correction, to study the free expansion of the Fermi
superfluid. We calculate the aspect ratio and the released energy of the
expanding Fermi cloud showing that MC EOS and EBCS EOS are both compatible with
the available experimental data of $^6$Li atoms. We find that the released
energy satisfies an approximate analytical formula that is quite accurate in
the BEC regime. For an anisotropic droplet, our numerical simulations show an
initially faster reversal of anisotropy in the BCS regime, later suppressed by
the BEC fluid.
