Flow of incompressible non-Newtonian fluids in pipes

Among other characteristics, non-Newtonian fluids exhibit an apparent viscosity that varies with shear rate. Consequently, the determination of the shear stress-shear rate curve must be an initial consideration. Although the apparent viscosity of a thixotropic or a rheopectic fluid changes with the duration of shearing, meaningful measurements may be made if the change is relatively slow. Viscoelastic fluids also exhibit behaviour that is a function of time but their apparent viscosities can be measured provided conditions of steady shearing are obtained. 

There are two main types of viscometer rotary instruments and tubular, often capillary, viscometers. When dealing with non-Newtonian fluids, it is desirable to use a viscometer that subjects the whole of the sample to the same shear rate and two such devices, the cone and plate viscometer and the narrow gap coaxial cylinders viscometer, will be considered first. With other instruments, which impose a non-uniform shear rate, the proper analysis of the measurements is more complicated. 

With any viscometer the flow generated should ideally have only one non-zero velocity component, causing shearing in only one direction. The purpose of a viscometer is simultaneously to measure (or control) both the shear stress and the shear rate. Not only must the flow be laminar but viscous forces must be dominant, that is, inertial effects must be negligible. 

The tangential velocity component vg varies linearly from zero at the lower plate to the speed of the cone at the cone s surface. At a radial distance r, the cone s tangential speed is Hr where ft is in radians per second. At this location the height of the gap is or where a is the angle of the gap in radians. Thus, the shear rate y is given by 

The couple acting on either the cone or the plate may be measured as they are equal but act in opposite directions. Thus tm in equation 3.3 is strictly the magnitude of the shear stress. Dividing equation 3.3 by equation 3.1 gives an expression for the apparent viscosity  

---