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While modelling accretion disks around Schwarzschild black holes, various pseudo-Newtonian potentials are prescribed and then compared to general relativistic results. Though the pseudo-Newtonian potentials are useful, they do have some weak points as discussed here and here.

The general relativistic effective potential for Schwarzschild geometry (in units $G=M=c=1$) is given by $$V_{eff}=\sqrt{\left(1-\frac{2}{r}\right)\left(1+\frac{l^2}{r^2}\right)}$$ where $r$ is the radial coordinate and $l$ is the conserved angular momentum.

This form of the potential is derived by studying the particle trajectories around the black hole. However, in accretion disks we are mainly concerned about fluid flows. As we move closer to the black hole, it is usually expected that the fluid behavior will deviate (though slightly) from that of particle trajectories.

The following are my questions:

In this sense, would it be a better approach to study the accretion flow using an (exact) potential which is derived from the relativistic hydrodynamic equations in Schwarzschild geometry rather than using the potential obtained for particle trajectories?

Do such an (exact) potential exist in the literature which is obtained from hydrodynamics rather than particle dynamics in Schwarzschild geometry?

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