Colliding Particles in Granular Material Flows

[GGBCN]

Hello Everyone

I am looking at equations of colliding particles in a granular gas and wondering how to calculate them.

We assume that the grains are identical perfect spheres (in R^3) of diameter D>0, (x,v) and (x-Dn,w) are their states before a collision, where n ε S^2 is the unit vector along the centre of both spheres, and x the position vector of the centre of the first sphere, e is the restitution coefficient which relates the normal components of the particle velocities before and after collision, the post collisional velocities (v*,w*) then are such that

(v*-w*)n = -e((v-w)n)

I was wondering how from this equation do we calculate the change of velocity for the colliding particles:

v* = v- 1/2(1+e)((v-w)n)n,

w* = w+ 1/2(1+e)((v-w)n)n

Many thanks to anyone that can help!

 

[bigfooted]

First rewrite the existing velocity vectors (and angular momentum) from their native coordinate system to velocity vectors in the coordinate system connected to the collision interface. In this coordinate system, only the normal component of velocity is changed.
Then apply conservation of linear and angular momentum to get the new velocities and angular momentum and transform them back to the original coordinate system.

Look for instance in the book of Crowe, sommerfeld and Tsiu - multiphase flow with droplets and particles.
They have a nice derivation, including the effect of the angular velocity of the particles and the restitution coefficient.

There is also a result without much explanation on the wiki page for inelastic collision which you could use. This is just the rewritten conservation of momentum.

 

[bigfooted]

PS: I noticed that the wiki page on momentum has some explanation:
http://en.wikipedia.org/wiki/Momentum

Note that all these 1d equations hold true in 2d and 3d, when you transform the velocity vectors to the coordinate system connected to the collision interface

 

[GGBCN]

Sorry for the late reply, many thanks for the help!