Elementary viscometry

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 ve 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 fir where Cl is in radians per second. At this location the height of the gap is ar 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 r20 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  

Note that a must be in radians and fl in radians per second. If the rotational speed is measured as N revolutions per minute (rpm), then the required conversion is 

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