Diffusion phase space

In terms of the fluid mass seen by the particle f, conservation of mass at the mesoscale leads to the following mesoscale models in the limit of zero particle Stokes number (i.e. u = U = Uf)  

The positive-definite matrix Bfp will be determined by the structure of the fluid flow around the particle, and will more than likely be significantly anisotropic (Tenneti et al, 2012). Note that we have not included the fluid-velocity-fluctuation-dissipation model in Eq. (4.104) when writing the GPBE. Here, for clarity, we will focus exclusively on the interdependence of the fluid and particle velocities, for which it suffices to consider a ID velocity phase space for Vp and Vf. The GPBE for this case is given by 

if the initial variances are null, then [/p[/f] is completely determined by 17j I. At steady state, the velocity fluctuations are related to Bfp by 

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In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

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) ... 

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... 

Class and sectional methods The phase-space diffusion term... 

For the KE, phase-space diffusion is usually associated with Brownian motion caused by random collisions with an external phase. In comparison, C is due to self-collisions between particles of the same type. The representation of Brownian motion as a diffusion term implies that collisions with the external phase occurs on a time scale that is much smaller than that for self-collisions. 

Moment equations with phase-space diffusion... 

This PBE is written in a general form and contains the terms representing accumulation, real-space advection, phase-space advection, phase-space diffusion, and second-, first-, and zeroth-order point processes. (See Chapter 5 for more details on these processes.) Let us... 

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