Vorticity volume-averaged

This is a semiheuristic volume-averaged treatment of the flow field. The experimental observations of Dybbs and Edwards [27] show that the macroscopic viscous shear stress diffusion and the flow development (convection) are significant only over a length scale of i from the vorticity generating boundary and the entrance boundary, respectively. However, Eq. 9.22 predicts these effects to be confined to distances of the order oi Km and KuDN, respectively. We note that Km is smaller than d. Then Eq. 9.22 predicts a macroscopic boundary-layer thickness, which is not only smaller than the representative elementary volume i when i d, but even smaller than the particle size. However, Eq. 9.22 allows estimation of these macroscopic length scales and shows that for most practical cases, the Darcy law (or the Ergun extension) is sufficient. 

For solid-body rotation at constant angular velocity, the vorticity vector, defined by I (V X v), is equivalent to the angular velocity vector of the solid. For two-dimensional flow in cylindrical coordinates, with Vr(r,0) and V0 r,9), the volume-averaged vorticity vector. 

The second integral of (8-201) vanishes, due to the periodicity of at 0 = 0 and 2ji. Hence, the volume-averaged vorticity... 

---