Coupling between active and passive internal coordinates

3 Coupling between active and passive internal coordinates 

In the GPBE the internal coordinates include both active and passive variables. For example, consider the ID NDF for the velocity (v) and surface area (5) of a particle n(t, X, V, s), where the two internal coordinates are = (v,. y). The free-transport term for the NDF is the same as in Eq. (8.3). However, we now have bivariate moments defined by fnk,i = f dv d.y, so that the moment transport equation is 

As before, the exact solution to Eq. (8.3) is n(t, x, v, s) = n(0, x - vt, v, s), from which we can observe that. y is indeed passive because it is simply carried along with velocity v. Or, in other words, the free-transport term in the GPBE can be solved separately for each value of. y. However, the moment-transport equations are coupled because they depend on both k and 1. Nevertheless, we can observe that if we consider a set of moments with k fixed, but I free, it is possible to use QBMM to represent the unclosed moments. For example, with A = 2 the moment-transport equations are 

For example, if a 2D quadrature with A = 4 nodes is constructed, the 12 mixed moments in Table 3.5 must be transported. The free-transport term will correspond to Eq. (8.10) with I = 0,1,2,3. However, for I = 2,3, only the four moments with k = 0,1 are transported (i.e. mo,2, mi,2, mo,3, and mi,3), and hence we will not be able to find two weights and 

The conditional moments (v ) for A = 1,2,3 are found by solving the linear system 

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