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.