Tubes, Bingham plastic flowing

Derive the E t) curve for a Bingham plastic flowing in a cylindrical tube. [2nd ed. PI3-16]... 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

Neglect the entrance and exit effects in the tube as well as the changes in kinetic energy. Assume laminar flow in the tube and (i) power-law behaviour, and (ii) Bingham plastic behaviour. 

A Bingham plastic is placed in a cylindrical tube and the pressure difference between the ends is gradually increased from zero. Show that flow first occurs at the wall of the tube, causing the material to flow as a plug along the tube. 

C. G. Lawson, Heat Transfer to Bingham Plastics, ThOo Slurries Flowing Turbidently in Tubes -In Exploratory Study, US.VEC Report CF-56-9-132, Oak Ridge National Laboratory, 1956. 

From the heat-transfer relation.ships for Bingham plastic slurries, described in Article 4-4.5, it is evident that for optimum design of steam generators the flow of slurry through the tubes should be turbulent. 

A similar problem on a nonisothermal rectilinear flow of a viscoplastic Shvedov-Bingham fluid in a circular tube for the case in which the yield stress and the plastic viscosity are inversely proportional to temperature was studied in [298],... 

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