The standard model of accretion disc falls in a broader class of disc-like solutions of accretion flows, the so-called slim discs. Models from this class are derived under assumption of axial symmetry, stationarity and geometrical thinness. The last assumption is not quantified straightforwardly; it is implemented by means of verticaly averaged quantities and equations and its validity needs to be considered in each individual case.
The topic of slim discs is not covered by the textbook by V. Karas; if available, the third edition of the textbook by Frank, King & Raine is the suitable resource (the first edition very likely lacks this topic; I have no idea about the second one). Some equations may be taken from another set of my hand-notes. The numbered set of equations on page labelled AGN18 is similar to that what could be seen in derivation of the equations governing the standard Shakura-Sunyaev model. The key differences are in eq. (4) where radial velocity gradient and radial pressure gradient are considered and in eq. (5) which allows for advection of heat through the disc medium towards the centre. I.e., in contrary to the Shakura-Sunyaev model, heat energy dissipated by the viscous forces is not necessarily radiated away (at the place where it is produced), but may be transported with the disc medium inwards. Note that the terms denoted and are those that have been considered in the standard model. The advected heat, is commonly parametrised with parameter that is supposed to be approximately constant (though it may be a function of radius in reality).
On page AGN19, a particular model of "Advection Dominated Accretion Flow" (ADAF) is presented. This rather simple model assumes that the amount of energy radiated away through the disc surface is negligible, i.e., the energy ballance equation holds: . Furthermore, the solution of the set of equations is searched for in the power-law form. In this case, a simple argumentation that all terms in each equation need to have the same power of leads quickly to the solution. Note that, e.g., even though the soultion has an angular velocity , the disc is not Keplerian. Also, in contrary to the standard disc, ADAF solution has a non-negligible radial velocity (which is consistent with the assumption that the heat is trapped in the matter and dragged towards the centre).