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!

 

[blue_raver22]

I find this topic very interesting and relevant to many areas of research, such as fluid dynamics and granular material flows. The equations you have mentioned are commonly used in the study of collisions between particles in a granular gas. These equations are derived from the conservation of momentum and energy during a collision.

To calculate the change in velocity for the colliding particles, we use the equation you have provided:

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

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

Here, v* and w* represent the post-collision velocities of the particles, while v and w are the pre-collision velocities. The term (v-w)n represents the relative velocity of the particles along the normal direction of collision. The restitution coefficient, e, is a measure of how much energy is conserved during the collision.

To calculate the post-collision velocities, we simply substitute the pre-collision velocities and the restitution coefficient into the equation. This will give us the change in velocity for the colliding particles. It is important to note that this equation assumes a perfectly elastic collision, where there is no loss of energy. In real-world scenarios, there is always some energy loss during a collision due to factors such as friction and deformation of the particles.

I hope this helps in understanding the calculation of post-collision velocities in granular material flows. If you have any further questions, please feel free to ask. Good luck with your research!