Spatial transport with scalar-conditioned velocity

4 Spatial transport with scalar-conditioned velocity 

In this section, we consider the intermediate case in which the particle velocity is given by Eq. (8.107) with flow-dependent parameters o, u, and 2, or by the EQMOM representation given in Eq. (8.112). As a starting point, we will take the following GPBE for the joint velocity-size NDE n t, x, v, )  

Note that both the spatial-flux terms and the drag terms require moment closures. These closures will depend on whether the NDF is represented by Eq. (8.108) or Eq. (8.112). When the NDF is represented by Eq. (8.108), the transport equation for mi,3 is not used. 

As an example, we will employ a two-node beta EQMOM to represent n f) and, thus, the mixed moments can be expressed in closed form as 

The realizable scheme for solving Eq. (8.125) is the same as in Eq. (8.89) (without diffusive fluxes), but with a slightly different definition for the numerical fluxes  

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The bio-heat transfer equation with both of these assumptions has been solved for various tissue geometries and initial and boundary conditions (Shitzer and Eberhart, 1985). Because of scalar treatment of the convective heat transport by blood, the use of the bio-heat transfer equation has been questioned repeatedly (Charny, 1992). Considering tissue as porous media, Wulff (1974, 1980) introduced the blood velocity vector ub, in the bio-heat transfer equation. Unfortunately, the complex nature of the system defies any attempt to specify the circulation vector at the microscopic level. As non-invasive technologies (e.g., MRI) provide improved spatial resolution, it may be possible to incorporate such data numerically (Dutton et al., 1992). 

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