  The accretion of test fluids flowing onto a black hole is investigated.
Particularly, by adopting a dynamical Hamiltonian approach, we are capable to
find the critical points for various cases of black hole in conformal gravity.
In these cases, we have analyzed the general solutions of accretion employing
the isothermal equations of state. The steady state and spherically symmetric
accretion of different test fluids onto the conformal gravity black hole has
been considered. Further, we have classified these flows in the context of
equations of state and the cases of conformal gravity black hole. The new
behavior of polytropic fluid accretion is also discussed in all three cases of
black hole. Black hole mass accretion rate is the most important part of this
research in which we have investigated that the Schwarzschild black hole
produce a typical signature than the conformal gravity black hole and
Schwarzschild de-Sitter black hole. The critical fluid flow and the mass
accretion rate have been presented graphically by the impact parameters
$\beta$, $\gamma$, $k$ and these parameters have great significance.
Additionally, the maximum mass rate of accretion fall near the universal and
Killing horizons and minimum rate of accretion occurs in between these regions.
Finally, the results are compared with the different cases of black hole
available in the literature.
