Dynamics of Inelastic Binary Collisions

In this section the dynamics of inelastic binary particle collisions are examined. The theory represents a semi-empirical extension of the binary collision theory 

Consider an inelastic collision between two smooth identical spherical particles 1 and 2, of mass m and diameter dp = d 2 [32] [60]. If J12 is the impulse of the force exerted by particle 1 on particle 2, the linear momentum balances over a collision relate the velocity vectors of the center of each sphere just before and after the collision through  

The relative velocities of the centers of the spheres immediately before and after a collision are still given by (2.116). 

For these inelastic particles it is required that the relative velocity component normal to the plane of contact, g2i k (before collision) and C21 k (after collision) satisfy the empirical relation (2.123) [31]. If the restitution coefficient therein is equal to one, the collision is elastic, which means that there is no energy loss during collision. Otherwise the collision is inelastic, which means that there is energy dissipation during collision. 

It is required that the component of the relative velocity perpendicular to the apse line should be unchanged in a collision, thus the impulse J12 must act entirely in the k direction. On this demand, J12 can be determined from (4.21), (4.22) and (2.123). The impulse of the force exerted by particle 1 on particle 2 is given by  

In this section the dynamics of inelastic binary particle collisions are examined. The theory represents a semi-empirical extension of the binary collision theory of elastic particles described in Sect. 2.4.2. The aim is to determine expressions for the total change in the first and second moments of the particle velocity to be used deriving expressions for the collisional source (4.27) and flux (4.26) terms. 

The relative velocities of the centers of the spheres immediately before and after a collision are still given by (2.111). For these inelastic particle collisions it is required that the relative particle velocity component normal to the plane of contact, g2i k (before collision) and k (after collision) satisfy the empirical relation (2.110) [68], If the restitution coefficient therein is equal to one, the collision is elastic, which means that there is no energy loss during collision. Otherwise the collision is inelastic, which means that there is energy dissipation during collision. It is required that the component of the relative velocity perpendicular to the apse line should be unchanged in a collision, thus the impulse J12 must act entirely in the k direction. On this demand, J12 can be determined from (4.47), (4.48) and (2.110). 

The impulse of the force exerted by particle 1 on particle 2 is given by  

The particle velocities just after collision can be expressed in terms of those just before collision in accordance with (4.47) and (4.48) [102]  

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