Transformed frame

Substituting Eq. 7.18 into Eq. 7.3 and solving Eqs. 7.1 and 7.3 for V, 14, and Vp, the solution for the transformed boundary condition problem Is obtained, and the equations are shown by Eqs. 7.21, 7.23, and 7.26. These equations physically represent the flow due to rotation and pressure in the transformed frame of reference in Fig. 7.10. Equation 7.21 is the velocity equation for the x-direction recirculatory cross-channel flow for the observer attached to the screw, and Eq. 7.23 is the apparent velocity in the z direction for the observer attached to the moving screw. 

The down-channel velocity in the transformed frame for/ /PFless than 0.1 is shown by Eq. 7.26. This equation is the same as the z-direction flow in the literature. 

Figure 7.11 Qualitative predicted particle trajectories in the transformed frame (Lagrangian) a) down-channel flows induced by the drag motion of the moving barrel, and b) recirculation flow in thex-y plane...

The solution for screw rotation is obtained by moving Eq. A7.51 from the transformed frame back to the laboratory (Eulerian) frame ... 

Applying the same transformation to the inhomogeneous part of Eq. (21.68) yields (a denotes the density matrix in the transformed frame, i.e. a = S2S1P) ... 

Proofs Transformation (frame change) of objective scalar a (3.55) with (3.28) gives the objectivity of scalar material derivative... 

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