The Couette Viscometer

In the analysis which gives Eqns 4-23 and 4-24 the following assumptions are made (1) incompressible liquid (2) non-turbulent flow (3) streamline gradient in horizontal plane perpendicular to axis of rotation (4) steady-state motion (5) no slip at wall of either cylinder (6) motion invariant in direction of axis of rotation. Assumption (3) neglects the effect of centrifugal forces, and this assumption as well as assumption (2) implies small values of R. Assumption (6) means the end effect is neglected. 

To comply with assumption (6) in an operating instrument, the upper and lower surfaces of the stator are protected by guard rings or liquid locks, as shown in Fig. 4-lOb, so that the liquid is interacting with only the cylindrical surfaces. Or the viscometer can be constructed with the depth of immersion variable, as shown schematically in Fig, 4-1 la. Equation 4-24 may then be written as 

Going back to Eqn 4-23, we see that regarded as an instrument constant and since M 

In practice most coaxial viscometers operate with very thin films of liquid x..e. ( 2 1. In such case it is easy to show that dw/ 

A correctly designed and constructed rotational viscometer is a convenient instrument for detecting departure from Newtonian viscosity behavior, since the rate of shear is readily computed by Eqn 4-27 from easily measured quantities such as Q, and R2. Barber, Muenger and Villforth [10] discuss the design and construction of an instrument which has the desired attributes and in addition operates so that the temperature gradient in the sheared film of liquid is only one or two degrees Kelvin. 

Another common device for measuring viscous properties is the cup-and-bob, or Couette viscometer, a diagram of which is shown in Fig 16.6. The fluid is confined in the gap between two concentric cylinders, one of which rotates relative to the other at a known angular velocity while the torque on one is measured. This is another classic example of a viscometric flow. In cylindrical coordinates, we assume only a tangential velocity component, so the 1 coordinate is the tangential or 6 direction and the 2 coordinate is the radial direction. 

Example 5. a. Neglecting end eSects, determine the shear stress as function of radius in terms of the measured torque on the stationary inner cylinder (bob), M(Ri), and the geometry of the apparatus as the outer cylinder (cup) is rotated with an angular velocity ta (radians/s). 

Solution, a. In a rotating system at equilibrium, S torques = 0, or there would be angular acceleration. Consider a ring of fluid with inner radius Ri and outer radius r M(r) = M Ri) 

Where the geometric approximations in Example 5b above are not applicable, Kreiger and Maron have presented an analysis similar to the Rabinowitsch development for flow in tubes. It involves differentiation of the M vs. (o data, but unfortimately, is in the form of an infinite series. If RoIRi 1.2, a closed-form approximation is given. 

Obviously, the analysis above is not valid in the area beneath the bob at the bottom of the viscometer. This is best taken into account by making measurements with two fluid depths, the lower being well above the bottom of the bob, and using the differences between the torques and depths in (16,28), thereby subtracting out the effects of non-Couette flow. Another approach is illustrated in Example 7. 

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By taking into account that v = cor, the velocity profile of the fluid layers in the Couette viscometer with rotating inner cylinder is given in Figure 13.18. Equation (13.70) in combination with the first boundary condition in Eq. (13.69) gives... 

The apparatus with a cylinder rotating in liquid in a fixed cylinder is usually called the Couette viscometer in the Pochettino viscometer6 the two cylinders rotate in opposite directions. In Searle s7 apparatus (Fig. 6.VIIIE) a brass cylinder C of radius a is supported on a vertical spindle A pivoted at its ends, and dips to a depth / in a liquid in a cylinder B of radius b. The cylinder C is rotated by weights mg in scale pans which pull on cords passing round a drum of diameter d attached to the spindle, on the top of which is a disc a used with an index b to measure the period of rotation, t0. Then ... 

A nearly identical flow pattern exists in the annular space between two concentric cylinders, one rotating and the other stationary, provided the width of the annulus, B, is small compared to the diameters of the cylinders. A device that makes use of this is the Couette viscometer. By measuring the torque required to rotate one cylinder at a known speed, the viscosity may be readily calculated from Eq. (5.70). 

The cone-and-plate viscometer is one of the rotational methods of measuring the polymer viscosity. It consists of a fiat horizontal plate and a cone with an obtuse angle. The cone touches the plate at its tip and rotates at a constant speed. The melt is charged into the gap forming between the horizontal plate and the cone. The rotational velocity determines shear rate and the torque applied gives shear stress. Shear rate is constant across the gap, thus it eliminates the need for non-Newtonian behavior of the melt. In a plate-plate viscometer, the cone is replaced by a second flat plate. The Couette viscometer is comprised of two concentric cylinders where one can be rotated at a constant speed. 

Let us now take the planes y, and — yj to be the walls of the Couette viscometer so that yi + y2 = and U is the relative velocity of the walls. When the viscometer is filled with pure liquid, the shear stress t is given by the subscript 0 referring to the particle-free state. However, with particles in the fluid the planes move with the new relative velocity... 

These facts indicate that, from the chemical point of view, there is only a single actomyosin complex of composition three parts of myosin to one of actin and that mixtures in any other proportion contain one component in excess. If this is so, the maximum in true viscosity should be at the 3 1 ratio, a point which should be tested in the Couette viscometer. Since artificial actomyosin solutions contain several very rapidly sedimenting fractions (H. H. Weher, 1947 Snellman and Gelotte, 1950 Johnson and Landolt, 1950), and since their viscosities are very variable (Jaisle, 1951 see Section III, 4d), it would have to be assumed that the same 3 1 complex forms threads of very vaiiable length and thickness. This may be so, but the problem requires further elucidation. It is possible that actomyosins with different physical properties represent sharply defined stages in the interaction of the two components. 

Numerous methods for measuring fluid viscosity exist, for example, capillary tube flow methods (Ostwald viscometer), Zahn cup method, falling sphere methods, vibrational methods, and rotational methods. Rotational viscometers measure the torque required to turn an object immersed or in contact with a fluid this torque is related to the fluid s viscosity. A well-known example of this type of system is the Couette viscometer. However, it should be noted that as some CMP slurries may be non-Newtonian fluids, the viscosity may be a function of the rotation rate (shear rate). An example of this is the dilatant behavior (increasing viscosity unda increasing shear) of precipitated slurries that have symmetrical particles [33]. Furthermore, the CMP polisher can be thought of as a large rotational plate viscometer where shear rates can exceed 10 s and possibly affect changes to the apparoit viscosity. The reader can refer to the comprehensive review of viscosity measurement techniques in the book by Viswanath et aL [34]. 

The equations for deriving the rj/y relationship for a polymer solution are shown in Figure 3.14 for the three most common viscometer geometries discussed above. The fundamental measurements for a capillary viscometer are a flow rate Q and a pressure drop AP along the capillary. For both the cone and plate and the Couette viscometers, the basic measurements are a torque on the central spindle for a given rotation rate, which, as the equations... 

A model used to demonstrate laminar shear mixing is Couette flow [16, 17], the principle used in the Couette Viscometer ... 

Movement of the screw surface shears the viscous polymer melt in a similar manner to the Couette viscometer. Using the scheme described by McKelvey [16], a striation A B Cj Dj will be sheared as the screw rotates to position A B C2 D2 which can be defined by the angle 0. 

Green and Weltman themselves also apply the CouETTE viscometer to characterize thixotropy but they let their systems gel during different periods, then break down the structure partially by ihear of different strength and durations and determine the residual rheological properties at a lower rate of shear. 

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