Small-angle cone-and-plate geometry

Two of this class of flows will now be considered. The first is circular flow between closely-fitting, narrow-gap concentric cylinders, and the second is circular flow in a small-angle, cone-and-plate geometry, see figure 1. Also shown is the cone-cylinder or Mooney geometry where the shear rates in the concentric-cylinder and the cone-and-plate parts of the geometry are arranged to be the same. 

If both cylinders have a large radius, and the gap between them is small, then the shear rate is the almost the same everywhere in the liquid-filled gap. If the inner cylinder radius is a-y and an outer cylinder radius is 2, then if the inner cylinder is stationary and the outer is rotated, the shear rate y in the contained liquid is given 

The shear stress cr is given by the force per unit area on the inner cylinder. If the torque on the inner cylinder is T (otherwise known as the couple or moment, with units of N.m), the turning force F on the inner cylinder is given by T/ai. Since the surface area over which the force F acts is 2miH, where H is the cylinder height, the shear stress cris given by 

To help locate the cone relative to the plate, and to reduce wear, the cone is usually truncated to remove the tip to the extent of a few tens of microns. This makes no practical difference to the results, but does allow the use of liquids with small suspended particles. 

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