View Full Version : Viscous disc paradox

2015-Nov-01, 09:50 AM
Which way does friction in a viscous disc operate?

Imagine a ring consisting of ringlets. First consider a case of a pair of nearby ringlets - in the same plane, both circular orbits.
If all particles of both ringlets are in circular orbits in the same plane, then they can never collide and therefore never exert any force. There is then zero temperature and zero viscosity.
Now suppose the viscosity is nonzero. What then?
By Third Law of Kepler, the inner ringlet should move faster.
Therefore, the inner ringlet should propel the outer ringlet ahead: the outer ringlet should expand and the inner one shrink.
But the problem is that the particles of rings, whether dust grains or gas molecules, are severally subject to Newton´s laws... and therefore also laws of Kepler. Including the Second.
While particles of inner ringlets are indeed faster than outer ringlet, they are so while they are in the inner ringlet, and do not meet outer ringlet.
The particles which can and do collide are those on elliptical orbit.
Considering two neighbouring circular ringlets and an elliptical ringlet tangent to both at its apsides.
The outer ringlet is slower than the inner, as per 3rd law - but the elliptical orbit at its apoapse is even slower, as per 2nd.
Therefore, the outer ringlet should be slowed down and shrink. And inner ringlet, by the same reasoning, should encounter the faster part of elliptical ringlet at periapse, speed up and expand.

So what´s the solution of the paradox? How should a viscous disc behave?

2015-Nov-01, 05:45 PM
Good question.

I didn't know the answer, so I typed the words "viscosity accretion disk" into the Google search engine. The results


provide a number of very informative articles. Note that the second item on the list is called "Accretion Disks for Beginners". Why don't you read it, and then come back here and tell us what you learned?

Ken G
2015-Nov-01, 06:52 PM
So what´s the solution of the paradox? How should a viscous disc behave?It's probably better just to look at a bunch of particles all at one place, rather than thinking about the orbits of individual particles. The distribution of velocities will be pretty isotropic, so there's a range of ellipticities there, but without collisions, all those particles will return to the same point eventually. However, if you put in collisions, or other forms of viscosity, you get a kind of mixmaster effect, where some of the particles pick up speed and go further out, and others fall further in. So it ends up being a kind of combination of both things you are talking about-- it works like diffusion. Whether the net result is inward or outward depends on the density profile, just as it would for diffusion.