Inhomogeneous KE

In this section, we look at a kinetic equation for the velocity NDF n t, x, v), where v = ( , v) is a two-component velocity vector (i.e. the velocity phase space is two-dimensional). In order to show the dynamics for different amounts of particle-particle collisions, we will use the BGK collision model. (See Chapter 6 for more details on collision models.) The inhomogeneous kinetic equation for this case is 

Here we have used the statistical symmetry between the second and third directions in velocity phase space to express the granular temperature in terms of the bivariate moments mj k- The NDF n is the equilibrium (Maxwellian) distribution with the conservation properties m QQ = mo,o, tnl = mi,o, wJq i = Physically, these equalities result 

The transport equation for the bivariate moments found from Eq. (8.95) is dmjj, dmj+i k 

However, by symmetry, the four moments mo,i, mi,i, mo,3, and mi,3 are null if the initial NDE is Gaussian, in which case only nine moments are required in order to solve Eq. (8.96). Nevertheless, it is convenient to solve for all 13 moments (referred to hereafter as the transported moment set) and use the known zero moments to check for the numerical accuracy of the algorithm. 

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In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

Akhmatskaya E, Todd BD, Davis PJ, Evans DJ, Gubbins KE, Pozhar LA. (1997) A study of viscosity inhomogeneity in porous media. J Chem Phys 106 4684 695. 

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