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dr which leads to Poiseulle's equation for laminar flow of Newtonian liquids in round pipes:

gnR4P

HL/i

where Q is the volumetric flow rate, and R is the radius of the pipe.

It is often more convenient to express Poiseulle's equation in terms of V, the average velocity, and the diameter of the pipe, D. Since Q — V it R2, Equations 5-4 becomes:

J2V,L

For flow in concentric annuli of inner and outer diameters /), and D respectively. Equation 5 3 may be written:

(D2 — D{)/4 is called the mean hydraulic radius, and may be substituted for Dl4 in many hydraulic equations. Poiseulle's equation then becomes:

48 VpL

The viscosity of a Newtonian fluid is determined in a capillary viscometer by measuring the time of discharge of a standard volume under a standard gravity head. The viscosity may be calculated from Equation 5-4, or by calibration with liquids of known viscosity, or from a constant supplied by the manufacturer of the viscometer. Various capillary sizes are available, so that a wide range of viscosities may be conveniently measured.

### The Bingham Plastic Flow Model

Plastic fluids were first recognized by Bingham,1 and are therefore referred to as Bingham plastics, or Bingham bodies. They are distinguished from New Ionian fluids in that they require a finite stress to initiate flow. Figure 5 4a shows the consistency curve for an ideal Bingham plastic, the equation for which is dv . „

Figure<5-4a. Consistency curve of an ideal Bingham plastic. Note: The effective viscosity at shear rate 2 is much greater than at shear rate 1.

wherer0 is the stress required to initiate flow, and pp is the plastic viscosity, which is defined as the shear stress in excess of the yield stress that will induce unit rate of shear, Thus

The total resistance to shear of a Bingham plastic may be expressed in terms of an effective viscosity, at a specified rate of shear. Effective viscosity is defined as the viscosity of a Newtonian fluid that exhibits the same shear stress at the same rate of shear. Figure 5-4a shows that effective viscosity at shear rate y, is given by

Figure 5-4b. Observed consistency curve of a Bingham plastic. P0 is the actual yield point, neglecting creep. 4/3 P0 is the apparent yield point.

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