Dense-Phase Transport Theorem

For a dense system of hard, smooth, and elastic spherical particles, a transport theorem based on the analogy of the kinetic theory of dense gases [Reif, 1965] may be derived. Define an ensemble average of any property xjr of a particle as 

In a time duration dt, each particle changes its velocity by dv = (F/m) dt, where F is the total external force acting on an individual particle of mass m. The change in xjr can be given by 

the rate of variation in the ensemble average of xjr in dr is equal to 

A change of xjr in dr is also possible through a net flux of particles entering the volume element. This increase rate is simply given by 

The change of xjr due to collisions may be related to the changes of velocities of colliding particles in the form 

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Finally, for collisional dense phase suspensions of hard, smooth, and inelastic spherical particles, the transport theorem is expressed as... 

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