Bingham plastic and yield-pseudoplastic fluids

A fluid with a yield stress will flow only if the applied stress (proportional to pressure gradient) exceeds the yield stress. There will be a solid plug-like 

In the annular area Rp r R, the velocity will gradually decrease from the constant plug velocity to zero at the pipe wall. The expression for this velocity distribution will now be derived. 

For the region Rp r R, the value of shear stress will be greater flian the yield stress of the fluid, and the Bingham fluid model for pipe flow is given by (equation (1.16) in Chapter 1)  

Now combining equations (3.2) and (3.10) followed by integration yields the following expression for the velocity distribution 

At the walls of the pipe (i.e. when r = R the velocity must be zero to satisfy the condition of no-slip. Substituting the value = 0, when r = R  

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