The one absolutely essential piece of the stress-energy tensor for describing accretion disks is the fluid part, . The fluid density, enthalpy, and pressure, as well as other fluid characteristics, are linked by the first law of thermodynamics, , which we write in the form,
Sometimes we may wish to consider a two temperature fluid, where the temperature and molecular weight of the ions are different from those of the electrons ( and ). For such a case7). Two-temperature fluids are also important when one considers radiation , as it is the ions that are generally heated by dissipative processes in the disk, while it is generally the electrons that radiate. Ions and electrons normally exchange energy via Coulomb collisions. As this process is generally not very efficient, the electrons in the inner parts of accretion flows are usually much cooler than the ions (Coulomb collisions are not able to heat the electrons as fast as they radiate or advect into the black hole). However, there have been suggestions that more efficient processes may couple the ions and electrons , such as plasma waves  or kinetic instabilities . At this point, this remains an open issue in plasma physics, so it is difficult to know how much heating electrons experience.
In the case of a perfect fluid, the whole stress-energy tensor (44) is given by its fluid part (45), and all other parts vanish, i.e., . In this particular case, one can use , and similarly derived , to prove that7c), which is a constant of geodesic motion.
In the stress part , the shear tensor is a kinematic invariant (cf. Footnote 12). It is defined as
In the standard hydrodynamical description (e.g. ), the viscous stress tensor, , is proportional to the shear tensor,
For the case of purely circular motion, where , the kinematic invariants are60) and (62) one deduces that for purely circular motion, the rates of energy and angular momentum transport are related as
As we mentioned in Section 1, the viscosity in astrophysical accretion disks can not come from ordinary molecular viscosity, as this is orders of magnitude too weak to explain observed phenomena. Instead, the source of stresses in the disk is likely turbulence driven by the magneto-rotational instability (MRI, described in Section 8.2). Even so, one can still parametrize the stresses within the disk as an effective viscosity and use the normal machinery of standard hydrodynamics without the complication of magnetohydrodynamics (MHD). This is sometimes desirable as analytic treatments of MHD can be very difficult to work with and full numerical treatments can be costly.
For these reasons, the Shakura–Sunyaev “alpha viscosity” prescription  still finds application today. It is an ad hoc assumption based on dimensional arguments. Shakura and Sunyaev realized that if the source of viscosity in accretion disks is turbulence, then the kinematic viscosity coefficient has the form,, so that we now know the appropriate pressure to be . Typical values of estimated from magnetohydrodynamic simulations are close to 0.02 , while observations suggest a value closer to 0.1 (see  and references therein).
Magnetic fields may play many interesting roles in black hole accretion disks. Large scale magnetic fields threading a disk may exert a torque, thereby extracting angular momentum . Similarly, large scale poloidal magnetic fields threading the inner disk, ergosphere, or black hole, have been shown to be able to carry energy and angular momentum away from the system, and power jets . Weak magnetic fields can tap the differential rotation of the disk itself to amplify and trigger an instability that leads to turbulence, angular momentum transport, and energy dissipation (exactly the processes that are needed for accretion to happen) [26, 27].
In most black hole accretion disks, it is reasonable to assume ideal MHD, whereby the conductivity is infinite, and consequently the magnetic diffusivity is zero. Whenever this is true, magnetic field lines are effectively frozen into the fluid. A corollary to this is that parcels of fluid are restricted to moving along field lines, like “beads” on a wire. In ideal MHD, the Faraday tensor obeys the homogeneous Maxwell’s equation67) give the induction equation
We mentioned in Section 3.2 that a hydrodynamic treatment of accretion requires an internal viscous stress tensor of the form . However, we also pointed out that ordinary molecular viscosity is too weak to provide the necessary level of stress. Another possible source is turbulence. The mean stress from turbulence always has the property that , and so it can act as an effective viscosity. As we will explain in Section 8.2, weak magnetic fields inside a disk are able to tap the shear energy of its differential rotation to power turbulent fluctuations. This happens through a mechanism known as the magneto-rotational (or “Balbus–Hawley”) instability [26, 118, 27]. Although the non-linear behavior of the MRI and the turbulence it generates is quite complicated, its net effect on the accretion disk can, in principle, be characterized as an effective viscosity, possibly making the treatment much simpler. However, no such complete treatment has been developed at this time.
Radiation is important in accretion disks as a way to carry excess energy away from the system. In geometrically thin, optically thick (Shakura–Sunyaev) accretion disks (Section 5.3), radiation is highly efficient and nearly all of the heat generated within the disk is radiated locally. Thus, the disk remains relatively cold. In other cases, such as ADAFs (Section 7), radiation is inefficient; such disks often remain geometrically thick and optically thin.
In the optically thin limit, the radiation emissivity has the following components: bremsstrahlung , synchrotron , and their Comptonized parts and . In the optically thick limit, one often uses the diffusion approximation with the total optical depth coming from the absorption and electron scattering optical depths. In the two limits, the emissivity is then[288, 66]. Often, though, the solution of the grey problem obtained by Hubeny  can serve reliably: BHSPEC, color temperature corrections in the optically thick case (the “hardening factor”) are often applied .
In the remaining parts of this section we give explicit formulae for the bremsstrahlung and synchrotron emissivities and their Compton enhancements. These sections are taken almost directly from the work of Narayan and Yi . Additional derivations and discussions of these equations in the black hole accretion disk context may be found in [299, 295, 225, 87].
Thermal bremsstrahlung (or free-free emission) is caused by the inelastic scattering of relativistic thermal electrons off (nonrelativistic) ions and other electrons. The emissivity (emission rate per unit volume) is . The ion-electron part is given by 
Assuming the accretion environment is threaded by magnetic fields, the hot (relativistic) electrons can also radiate via synchrotron emission. For a relativistic Maxwellian distribution of electrons, the formula is 
The hot, relativistic electrons can also Compton up-scatter the photons emitted via bremsstrahlung and synchrotron radiation. The formulae for this are