Laminar flow in a concentric annulus

The only other force acting on the fluid element in the z-direction is that arising from the shearing on both surfaces of the element. Note that, not only will the shear stress change from r to r + dr but the surface area over which 

The shear stress distribution across the gap is obtained by integration  

Because of the no-slip boimdary condition at both solid walls, i.e. air = oR and r = / , the velocity must be maximum at some intermediate point, say at r = XR. Then, for a fluid without a yield stress, the shear stress must be zero at this position and for a viscoplastic fluid, there will be a plug moving en masse. Equation (3.76) can therefore be re-written  

It is important to write the equation in this form whenever the sign of the velocity gradient changes within the flow field. In this case, (dV /dr) is 

The only unknown now remaining is X, which locates the position where the velocity is maximum. Table 3.2 presents the values of X for a range of values 

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