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Figure 5-t. Schematic illustration of laminar flow of a Newtonian liquid.

Figure 5-3a. Schematic representation of laminar flow of a Newtonian fluid in round pipe. Fluid velocity increases from 0 at the wall to a maximum at the axis of the pipe.

Figure 5-3b. Velocity profile of the fluid in Figure 5-3a. The shear rate at any point is the slope of the profile at that point.

The unit of viscosity in the metric system is the poise, defined as the stress in dynes per square centimeter required to produce a difference in velocity of one centimeter per second between two layers one centimeter apart. The centipoise is one hundredth of a poise. The American unit of viscosity, the reyn, is similar, but expressed in poundals and feet.

Laminar flow of a Newtonian fluid in a round pipe may be visualized as a telescope of concentric cylinders, as shown in Figure 5 -3a. The velocity of the cylinders increases from zero at the pipe wall to a maximum at the axis of the pipe, resulting in a parabolic velocity profile (Figure. 5- 3b). The shear rale at any point on the radius is given by the slope of the profile at that point. Note that it is maximum at the pipe wall, and zero at the axis.

The relationship between pressure and flow rate is derived as follows: If a fluid flows in a pipe of length L, and radius R, the force on the end of a cylinder of radius r will be the pressure difference between the ends of the pipe, P multiplied by the cross sectional area of the cylinder, and the shear stress will be:

Substituting for t in Equation 5 1 gives:

dv rP

0 0