Collision terms for arbitrary moments

For an arbitrary moment // the collision term is defined by (Fox Vedula, 2010) 

As was done in Section 6.1.2, we can now use a Taylor-series expansions about the collision contact point 

This expression is further simplified by introducing the pair correlation function gcp and using a truncated Taylor-series expansion around the point x up to first order in dap (see the details given in Section 6.1.2)  

Note that we have assumed that gap may depend on xi2 when a p, and will discuss a possible dependence below. We have also introduced the velocity distribution functions for each particle type fa and fp), which need not be the same (unless a = p). 

In general, gap depends on the volume fractions of each particle type and on the particle diameters. However, it can also depend on other moments of the velocity distribution function. For example, if the mean particle velocities Uq. and Vp are very different, one could expect that the collision frequency would be higher on the upstream side of the slower particle type. The unit vector Xi2 denotes the relative positions of the particle centers at collision. If we then consider the direction relative to the mean velocity difference, (Uq, - U ) xi2, we can model the dependence of the pair correlation function on the mean velocity difference as ° 

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