Pressure drop for Bingham plastics in laminar flow

8 Pressure drop for Bingham plastics in laminar flow 

As the shear stress for flow in a pipe varies from zero at the centre-line to a maximum at the wall, genuine flow, ie deformation, of a Bingham plastic occurs only in that part of the cross section where the shear stress is greater than the yield stress ry. In the part where r ry the material remains as a solid plug and is transported by the genuinely flowing outer material. 

As part of the Rabinowitsch-Mooney analysis, it was shown that the volumetric flow rate can be written in terms of the shear stress distribution  

For a Bingham plastic, there is a change in the flow behaviour at r= ry and the range of integration must be split into two parts  

The first integral vanishes because y = 0 for 0 r s ry. Substituting for y from equation 1.73, equation 3.53 becomes 

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