chornedsnorkack

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?

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?