## Bingham Plastics

The Bingham plastic model usually provides a good representation for the viscosity of concentrated slurries, suspensions, emulsions, foams, etc. Such materials often exhibit a yield stress that must be exceeded before the material will flow at a significant rate. Other examples include paint, shaving cream, and mayonnaise. There are also many fluids, such as blood, that may have a yield stress that is not as pronounced.

It is recalled that a ''plastic'' is really two materials. At low stresses below the critical or yield stress (xo) the material behaves as a solid, whereas for stresses above the yield stress the material behaves as a fluid. The Bingham model for this behavior is

Because the shear stress and shear rate can be either positive or negative, the plus/minus sign in Eq. (6-54) is plus in the former case and minus in the latter. For tube flow, because the shear stress and shear rate are both negative, the appropriate form of the model is

A. Laminar Flow

Because the shear stress is always zero at the centerline in pipe flow and increases linearly with distance from the center toward the wall [Eq. (6-4)], there will be a finite distance from the center over which the stress is always less than the yield stress. In this region, the material has solid-like properties and does not yield but moves as a rigid plug. The radius of this plug (ro) is, from Eq. (6-4), r0 = R-

Because the stress outside of this plug region exceeds the yield stress, the material will deform or flow as a fluid between the plug and the wall. The flow rate must thus be determined by combining the flow rate of the "plug" with that of the "fluid" region:

Evaluating the integral by parts and noting that the Qplug term cancels with nro Vplug from the lower limit, the result is

When Eq. (6-55) is used for the shear rate in terms of the shear stress and Eq. (6-4) is used for the shear stress as a function of r, the integral can be evaluated to give

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