Parallel plate flow both plates moving

COUETTE FLOW. In one form of layer flow, illustrated in Fig. 5.17, the fluid is bounded between two very large, flat, parallel plates separated by distance B. The lower plate is stationary, and the upper plate is moving to the right at a constant velocity Uq. For a newtonian fluid the velocity profile is linear, and the velocity u is zero at y = 0 and equals Uo t y = B, where y is the vertical distance measured from the lower plate. The velocity gradient is constant and equals uq/B. Considering an area A of both plates, the shear force needed to maintain the motion of the top plate is, from Eqs. (3.3) and (3.4),... 

The theory developed up to this point is based on a model where the screw is stationary and the barrel rotates around the screw. It is assumed that the flow that results is the same as when the barrel is stationary and the screw rotates in the opposite direction. This assumption was considered valid for over fifty years until several workers challenged this assumption, first in the early 1990s [272-276], and then more recently [323]. Because of the importance of this issue we will critically analyze this assumption to determine to what extent the assumption is correct. Flow will be analyzed using the parallel plate assumption with either the barrel or the screw considered moving. Flow will also be analyzed without the parallel plate assumption, using a cyhndrical coordinate system, again considering both cases. This analysis is based on a study by Osswald et al. [281]. 

Let us first consider a simple shear flow. When a fluid is confined between two parallel plates, both perpendicular to the y-axis (Figure 7.2a), and one is moved with a constant velocity in the direction of the x-axis relative to the other, the velocity field of the fluid is given by... 

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