Mesoscale models in the GPBE

As derived from the microscale model in Chapter 4, the GPBE for a polydisperse multiphase flow has the following form  

In general, the phase-space advection terms are modeled as the sum of contributions due to pure advection and to phase-space diffusion (Gardiner, 2004)  

With pure advection processes, we refer to continuous phenomena that cause continuous changes in the external and internal coordinates. Continuous changes of the particle s position in real space are quantified by the real-space advection (or free-transport) term  

The phase-space diffusion terms in Eq. (5.2) generate a very large number of terms in the GPBE (many of which are zero). For example, considering only the fluid-particle interaction term in the limiting case in which particle-velocity fluctuations are due to microscale fluid turbulence (i.e. Bp = 0, Bp, = 0) yields the diffusion terms in velocity phase space 

The diffusion matrices Bpy, and Bfy, are symmetric and, most importantly, conservation of momentum at the microscale will require that they be dependent and, hence, at most only six diffusion coefficients need be determined from the microscale model (see the discussion leading to Eq. (5.17)). The simplest case occurs when the diffusion matrices are isotropic  

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