  It is demonstrated that with using Painlev{\'e}-Gullstrand coordinates in
their quasi-Cartesian variant, the Hamiltonian functional for relativistic
perfect fluid hydrodynamics near a non-rotating black hole differs from the
corresponding flat-spacetime Hamiltonian just by a simple term. Moreover, the
internal region of the black hole is then described uniformly together with the
external region, because in Painlev{\'e}-Gullstrand coordinates there is no
singularity at the event horizon. An exact solution is presented which
describes stationary accretion of an ultra-hard matter ($\varepsilon\propto
n^2$) onto a moving black hole until reaching the central singularity. Equation
of motion for a thin vortex filament on such accretion background is derived in
the local induction approximation. The Hamiltonian for a fluid having
ultra-relativistic equation of state $\varepsilon\propto n^{4/3}$ is calculated
in explicit form, and the problem of centrally-symmetric stationary flow of
such matter is solved analytically.
