Ahh, Accretion Discs. These are astrophysical bodies which are interesting for several reasons - one reason being that they are incredibly persnickity about which equations they obey.
What are they?
As siren correctly says, they are a disc of matter falling onto a central body such as a star or a black hole. This matter was usually taken from another nearby star, and so is made up of hot plasma, and has some angular momentum relative to the fixed central star which it has to lose. So, we see the fluid heating itself up as it flows around the star, losing energy and falling inwards to lower orbits - but it has to conserve this angular momentum, which it can't do by falling inwards.
So it passes it outwards. Accretion discs (unless they hit something at the outer edge) will have most of their matter falling inwards, with an outer edge spreading away to carry off the unwanted angular momentum.
Well, the most obvious reason is that these discs can release an amazing amount of energy. Every piece of matter has an energy associated with it - you've heard it before:
E = m c ²
This is a lot of energy for each little bit of mass. It's usually inaccessible, but in an accretion disc falling into a black hole, some back-of-the-envelope calculations show that you'd expect over half of this energy to be converted from matter into heat. This heat makes things hot (unsurprisingly) and so these discs are like furnaces - furnaces in which interesting physical processes take place. To understand the processes, we need to understand the furnaces first.
There's another reason, in that hard problems are inherently interesting. Occasionally the disc appears to just decide to stop being a disc and shoots jets of matter out of its sides - which is bizarre. They also have huge amounts of turbulence inside, which makes them even harder to model - along with the fact that they also have their own dynamos (magnetic fields which they generate themselves, and don't die away).
The shape, viewed from above, of an accretion disc, will be more or less circular. But we need to consider the thickness of the disc - or, if we're not that interested in the internal structure, we need to claim that the disc is thin, so that, if the thickness of the disc is H (or 2*H depending on your convention) a function of R and θ, we have
H << R
for all R, θ. This is not the whole story, since discs don't have to be symmetric top and bottom - indeed, they usually aren't. If the angular momentum in the disc isn't aligned with the angular momentum of the central body (in other words, if they aren't spinning in the same plane) then the inner edge of the disc will become warped. This warping will then propagate outwards as what are called warp waves (like on a Slinky that you've sharply tugged to the side at one end).
Where will the disc start and end? Well, starting is easy. It'll start where-ever the mass is coming from - usually the distant sun in a binary system. What I said above about the outer edge spreading out isn't quite true, since any mass trying to spread out past the distant sun will (I believe) just get sucked back onto the distant sun via gravity. The discussion on where the innermost edge is will be discussed briefly in each section below.
Around Black Holes
When we're close to a black hole, we have to start worrying about general relativity. As I'm no general relativist, I have to take the following on trust: black holes can be characterised by 3 properties alone: Their mass (M), their dimensionless angular momentum (a), and their charge(Q). The charge is believed to be zero, in general, so we just worry about the first two. As a good approximation we can write down a classical potential which takes account of both M and a.
Φ = - G M / (r + |a|)
Here, r is the distance to the centre of the black hole, the event horizon is around 2GM (being distorted by the spinning) and the ISCO (Innermost Stable Circular Orbit) is around 3GM (in units where I've set the speed of light c = 1).
This ISCO is where the innermost edge of the disc must be (unless there's something making it be even further out). We could calculate κ, the precession frequency and see that it becomes imaginary beyond this point - particles attempting to follow a stable circular orbit find their orbit growing exponentially erratic, and quickly fall in. Note that this inner edge is far outside the event horizon.
When the central object is a star we see a different behaviour at the inner edge. Plasma is free to just fall into a black hole, and devil take the consequences. On the surface of a star, we must have continuity in the fluid velocity (the outer layers of the star aren't allowed to just slide over the inner layers - there has to be a thin boundary layer between them), and so we see a hot region where the disc matter is decellerated onto the surface of the star.
I say decellerated because if the star was spinning faster than the inner edge of the disc, the outer layers of the star would just break off and fly away. You can see this by thinking about little fluid parcels in the disc and then on the star. If a fluid parcel was going around faster than it would simply due to orbiting the central mass on the surface of the star, it would just fly off - and end up in the disc.
We can ask how hot this boundary layer is in relation to the rest of the disc by writing down the following dimensionless quantity:
B = (1 - Ωs / Ωd )α
Here Ω is the angular frequency of the disc and the star's outer edge respectively, and α is some dimensionless exponent (which turns out, for a Keplerian disc, to be 2). Note that, as expected, if the disc's inner edge is rotating perfectly with the star's outer edge then we won't see a boundary layer; there is no need to accelerate the fluid.
If, as I stated above is usual, we have a binary system, then we see some interesting behaviour. I'll call the star which is more massive A and I'll call the other one B. In the derivations I've seen we assume A to be much more massive than B, and so we can assume that A is nearly fixed as it was above.
Just as, on Earth, the Moon rotating around the Earth causes tides, so does B orbiting around A cause tides in A. It also sets up resonance behaviour in the disc at specific radii, the values of which can be easily calculated. If B is supplying the matter to A (as in Cygnus X-1, with the blue giant supplying mass to the black hole) and the outer edge of the disc is at the radius of B, then these resonant regions start close to the center of the system and spread out in a fairly uninteresting way.
If we have the mass coming from outside of both A and B (from another source such as a third nearby star or from a nebula) then the resonances occur at radii which (as 'n', the number used to label each resonance, tends to infinity) tends to a finite radius inside the disc. At this finite radius we would have an infinite number of resonant regions within an infinitessimal distance, and plainly our model has broken down. Nevertheless, we might expect the disc to have an inner edge here due to the tidal effects of B disrupting the disc.
There is a rather extreme difficulty in the modeling of the evolution of an accretion disc. They are made up of ionised plasma - the compounds contained within have split into their constituent ions and electrons, and so the fluid can support electric currents and magnetic fields. These magnetic fields lead immediately to the magnetorotational instability in which even a tiny, tiny magnetic field is enough to make the flow become turbulent.
This instability is called a 'local' instability - it doesn't care about anything that's happening far away; only things happening nearby. We can define a 'lengthscale' - sort of the size of the region the instability cares about, and find that it is extremely small when compared with, say, the radius of the disc. Let's pluck a number out of the air and say it's a million times smaller; that's not unreasonable.
This means that if we want to model the disc as a whole on a computer, we need a very, very fine grid everywhere - the physical gaps between our data points must be small enough so that we pick up this very important magnetorotational instability. But to cover the whole disc, we'll need lots of them - a million in the radial direction! That is far, far too many; a good, publishable simulation might have 256 grid points in the radial direction, and still take an age to run. Our computers simply aren't big enough in 2009 to handle both the tiny scale of the instability and the huge scale of the disc radius.
So what can we do?
Well, we can either forget entirely about the magnetic field and simulate the entirety of somewhat unrealistic discs. This is unsatisfactory but do-able; if we're more interested in how planets form in a young solar system, and we hope that the magnetic field doesn't play too big a role, then we'd be justified in taking this route.
Most of the current focus lies on examining the scale of this instability and really trying to get a grip on it. When magnetic fields are swept around by a fluid they collide and re-connect and intermingle and absorb energy and release energy and slow the fluid down in a complicated way; there's a lot of work to be done on this.
Once we've understood the small scale (the instability), the next step will be zooming out to the large scale (the entire disc). We can't just plonk our small scale into our large scale simulations - that won't solve the problem of how large our simulation would have to be. We need to find approximations which can capture on the large scale the interesting effects of the complicated motion on the small scale without having to worry about what those motions are in each and every case.
This has been attempted, to some extent. We can say that the disc loses kinetic energy roughly proportional to the fourth power of the magnetic field, looking at the output of our simulations (known as the ad hoc 'alpha parameter'). But how to evolve this magnetic field in time without knowledge of the small scales? In what direction should it point? There remain many unanswered questions.
So, we have a system which will always be turbulent; observations show us that there is some kind of fluid dynamo occurring; the magnetic field grows and shrinks with time, and lasts far longer than it could do if there wasn't some kind of fluid phenomenon replenishing it - Ohmic dissipation would kill it off eventually otherwise. My comment about about 'in what direction should the magnetic field point' was not a throwaway comment; if we expect this turbulence to create an effective magnetic and viscous diffusivity which enables the disc to accrete, then we expect that these diffusivities might be anistropic. After all, we have two distinguished directions; the local direction of the magnetic field, and the large-scale gravity gradient which will point mostly towards the central object.
So we have a time-dependent magnetic field which is creating (slow-timescale) time dependent turbulence, which is creating some kind of time dependent effective anisotropic viscosity and diffusivity. Currently, we don't understand the magnetic field; we don't understand turbulence; we don't understand what forms these effective diffusivities should take.
There are a lot of unanswered questions.