The concentric cylinder geometry

Working equations relating the measured torque to the requisite shear stress, and angular velocity (of the cup) to the requisite shear rate, are widely available 

Several designs have been described which overcome end-effects due to the shear flow at the bottom of the concentric cylinder geometry. These include the recessed bottom system which usually entails trapping a bubble of air (or a low viscosity liquid such as merciuy) beneath the inner cylinder of the geometry. Alternatively the Mooney-Ewart design, which features a conical bottom may, with suitable choice of cone angle, cause the shear rate in the bottom to match that in the narrow gap between the sides of the cylinders, see Figme 2.4. 

To minimise end-effects the lower end of the inner cyhnder is a truncated cone. The shear rate in this region is equal to that between the cylinders if the cone angle, a, is related to the cylinder radii by  

The main sources of error in the concentric cylinder type measuring geometry arise from end effects (see above), wall shp, inertia and secondary flows, viscous heating effects and eccentricities due to misahgnment of the geometry [Macosko, 1994], 

Secondary flows are of particular concern in the controlled stress instruments which usually employ a rotating inner cylinder, in which case inertial forces cause a small axisymmetric cellular secondary motion ( Taylor vortices). The dissipation of energy by these vortices leads to overestimation of the torque. The stability criterion for a Newtonian fluid in a narrow gap is 

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While the concentric cylinder geometry is relatively easy to use in rheological studies, some of its limitations should be recognized as shown in Figure 3-44. 

Fluid inertia effects have been found to be very small for the cone-and-plate geometries typically supplied with these instruments. While inertial corrections are foimd to be imimportant for the parallel plate geometries, for shearing gaps of the order of 2 mm or less (except possibly for very thin fluids), they must be taken into accoimt in the concentric cylinder geometry (especially for high-density, mobile fluids). Evaluation methods are available for p, in the case of cylindrical and plane Couette flow, taking into account fluid inertia [Aschoff and Schummer, 1993]. 

Liquid convection baths are best suited for the concentric cylinder geometry and for temperatures between —10 and 1S0°C. Temperature gradients with cone and plate and parallel disk geometries can be a problem because of heat loss from the upper fixture. Insulated covers and thin samples help. For other geometries and wider temperature ranges, gas convection is preferred. 

The concentric cylinder geometry was used in the first studies of flow birefringence." The shear field is inhomogeneous within the gap, and as narrow a gap as possible is needed to reduce this effect. A very small gap gives alignment difficulties and reduces the ability to view through the gap for birefringence measurements. 

Figure 2.4 A commercial instrument, the Brookfield Digital Viscometer, based on the geometry of the concentric cylinder viscometer. (Photo courtesy of Brookfield Engineering Laboratories, Inc., Stoughton, Mass. 02072.)...

Fig.4.5.7 (a) Chemical shift imaging pulse sequence and (b) schematic drawing of CSI data for a given pixel of an oil-in-water emulsion inside the horizontal concentric cylinders geometry. 

Flow effects on non-neutrally buoyant emulsions and suspensions can be studied in various geometries. For example, flow in rotating cylinder and narrow gap concentric cylinder geometries in both horizontal and vertical orientations can be studied. Flow instabilities in settling suspensions in a horizontal rotating cylinder have recently been reported [84], Measurements of velocity fields have not been reported in the literature, but can be performed by using the methods presented in this work. 

As already mentioned in Chapter 1, there are mainly three geometries suitable for the measurement of flow birefringence, viz. those of the concentric cylinder apparatus, the adapted cone-and-plate apparatus and the slit-capillary with a rectangular cross-section. The general principles of the pertinent techniques have been described in the same chapter. The purpose of the present chapter is to give details of the design and construction in order to enable the reader to form a judgement as to the efficiency of the proposed methods, i.e. the relation between information and experimental effort. 

The vane viscometer is yet another form of the concentric cylinder instrument, in which the bob is replaced by a rotor with four blades or vanes each attached by one edge to a vertical shaft, at 90° intervals around the shaft (Figure 22.7). This geometry, which can be used either with a cup or in the infinite sample mode, is particularly useful for measuring yield stress, and can also be used to measure the rheological properties of non-Newtonian liquids. Its advantages are described by Gunasekaran and Ak (2002). 

In a concentric cylinder geometry, the shear stress can be determined from the total torque (A/) ... 

In contrast to a cone and plate geometry to be discussed next, the shear rate of non-Newtonian foods cannot be determined from a simple expression involving the angular velocity and often one must use a suitable relationship between rotational speed and shear stress to correct for non-Newtonian behavior. More complex equations are needed to describe the flow of non-Newtonian fluids in concentric cylinder geometry. For example, for fluids that can be described by the power law model, an expression presented by Krieger and Elrod (Van Wazer et al., 1963) has been used extensively in the literature ... 

Other expressions for concentric cylinder geometry include that for Bingham plastic fluids where the yield stress must be taken into account which leads to the Reiner-Riwlin equation ... 

Figure 3-3 Schematic of a Concentric Cylinder Geometry Air Bubble to Minimize Shear at the Bottom of a Rotating Cylinder.

The apex of a cone is brought into close proximity, but not in to contact, of a horizontal plate (Figure 3-7). Often, the apex is truncated slightly to eliminate a sharp point. The minimum gap between the cone and plate is usually of the order of 50 xm so that this geometry may not be suitable for dispersions containing larger diameter solids. The test fluid fills the gap between the cone and the plate, and because the gap is small, only a small volume (typically, 1-5 mL) of fluid is needed. The cone is rotated and the torque is measured at various speeds of rotation. A cone and plate viscometer can be used to obtain shear stress-shear rate curves and shear-stress versus time at constant shear rate curves as described above for concentric cylinder geometry. The... 

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