Paczyński and his collaborators developed, in the late 1970s and early 1980s, a very general method of constructing perfect fluid equilibria of matter orbiting around a Kerr black hole [139, 236, 235, 234]. They assumed for the stress energy tensor and four velocity,81) is the gradient of a scalar function, and thus the left-hand side must also be the gradient of a scalar, which is possible if and only if von Zeipel theorems, found by a number of authors [51, 29, 1, 156].
In real flows, the function is determined by dissipative processes that have timescales much longer than the dynamical timescale, and are not yet fully understood. Paczyński realized that instead of deriving from unsure assumptions about viscosity that involve a free function fixed ad hoc (e.g., by assuming const), one may instead assume the result, i.e., assume . Assuming is not self-consistent, but neither is assuming const.
In Boyer–Lindquist coordinates, the equation for the equipressure surfaces, const, may be written as , with the function given by2 (Eqs. 21 and 22), one can integrate Eq. (83) to get the equipressure surfaces. A description of how to do this for both Schwarzschild and Kerr black holes is given in . Figure 4 illustrates the simplest (and important) case of .
Another useful way to think about thick disks is from the relativistic analog of the Newtonian effective potential ,
Before leaving the topic of Polish doughnuts, we should point out that, starting with the work of Hawley, Smarr, and Wilson , this simple, analytic solution has been the most commonly used starting condition for numerical studies of black hole accretion.
Komissarov  was able to extend the Polish doughnut solution by adding a purely azimuthal magnetic field to create a magnetized torus. This is possible because a magnetic field of this form only enters the equilibrium solution as an additional pressure-like term. For example, the extended form of Eq. (81) is84) becomes  gives a procedure for solving the case of a barotropic magnetized torus with constant angular momentum ( const.).