Rotation between coaxial cylinders

The use of different torque drums and cylinder dimensions permits measurements of materials over a wide range of consistencies. Good thermal isolation is achieved by a glass section in the torque rod. 

In this geometry, the strain and rate of strain are not homogeneous, but diminish from the inner cylinder outwards. For Newtonian liquids, they are proportional to 1 jr, where r is the distance from the center of rotation for non-Newtonian liquids, the distribution is more complicated. Corrections must be made for end effects of the cylinders and can be checked by filling the apparatus to different levels or varying the end clearance. Filling is difficult with liquids of very high viscosity. 

Since in equation 6 the ratio 721 (0/ 2i is the creep compliance Jit), the latter is obtained directly by substituting in this equation the time-dependent angular displacement a(t). 

The torques must be kept small to approach linear viscoelastic behavior. A large viscous deformation does not necessarily imply departure from linear behavior if the associated elastic deformation (as measured by the recoverable strain after removal of the torque) is small as a rule of thumb, the product a2 J° should perhaps be less than 0.1. The maximum value of T2 (at the inner cylinder wall) can be calculated from the torque formulas for all geometries are given in Appendix C. 

Another serious limitation on the magnitude of the torque is avoidance of the Weissenberg effect in which, owing to normal stresses (Section C below), the liquid climbs up the inner cylinder during rotation. This alters the geometry and falsifies the form factor b calculated from equation 7. Observation of the sample is therefore highly desirable to make sure that the torque is sufficiently small to render this effect negligible. 

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FIG. 5-1. Geometries, coordinates, and dimensions for investigations of viscoelastic liquids, (a) parallel plate simple shear (b) annular pumping (c) rotation between coaxial cylinders (d) torsion between cone and plate (e) torsion between parallel plates (/) axial motion between coaxial cylinders. 

PIG. S-S. Identification of normal stresses in different experimental geometries, (a), rotation between coaxial cylinders (b), torsion between cone and plate or parallel plates (c) annular axial flow between coaxial cylinders. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

In this case, G is called the gradient of the flow rate or the shear rate. The Couette flow occurs between two parallel moving planes or in the gap between coaxial cylinders rotating at different... 

As a third example, consider viscous flow between coaxial cylinders, as shown in Fig. 5.5. The outer cylinder is rotating with a linear velocity wb, while the inner cylinder is at rest. The particles in the fluid move in concentric circles around the axis of rotation. In polar coordinates, only the tangential component of fluid velocity, u, is non-zero. Using the elastic analogy, from Eq. (4.26) one can write... 

It is known, however, that settling is enhanced or initiated when the slurry is subjected to an externally imposed strain rate. Early experiments were performed by Highgate and Warlow [3] using spheres in a pseudoplastic fluid that was sheared in the space between coaxial cylinders. As the fluid had no yield stress, the particles settled slowly in the quiescent case. When shear was applied by rotating the outer cylinder, the settling velocity increased reaching five times the initial value at high strain rates. 

Flow birefringence of polymer solutions is, in general, measured with the aid of an apparatus of the Couette type, containing two coaxial cylinders. One of these cylinders is rotated at constant speed, the other is kept in a fixed position. The light beam for the birefringence measurement is directed through the annular gap between these cylinders, in a direction parallel with the axis of the apparatus. In this way, the difference of principal refractive indices An is measured just in the above defined plane of flow (1—2 plane). 

Figure 6-4. Rotation plastimeter geometry, (a), (b) and (c) Coaxial cylinder types (d) and (e) concentric disc types ((d) is the Mooney geometry). A is usually the stator and B the rotor, C is the rotating shaft and r is the cylinder radius (much larger than the clearance between A and B). xy indicates the mid-plane along which the chamber can be opened for filling.

Typical of this class of viscometer is the coaxial or Couette type of instrument described in Volume l, Section 3.7.4. The sample fluid is contained within the annular space between two coaxial cylinders, either of which may be rotated by a motor with the remaining cylinder suspended elastically in such a way that the torsional couple exerted on the latter can be measured. If the outer cylinder of radius r2 rotates with an angular velocity cou and the inner cylinder of radius r, is stationary, and the torque (or viscous drag) per unit length of cylinder exerted on the inner cylinder is T, then, for a Newtonian fluid(49) ... 

Fig. 1 a, b. Schematic diagram of a flow of fluid under combined shear conditions a — between flatly parallel plates under the action of pressure difference AP = P -P2 (the upper plane moves in the direction transverse to the main flow) b — between two coaxial cylinders rotating towards one another at angular velocities flj and fi2... 

Taylor (1923) first observed the instability of fluid when the inner cylinder exceeds a critical speed between two coaxial cylinders in his established work. The laminar flow confined within the annulus region between two coaxial cylinders with the inner one differentially rotating with respect to the outer suffers centrifugal instability depending on the geometry and rotation rates. Taylor (1923) showed that an inviscid rotating flow to be... 

The Kanavets plastometer represents in principle two coaxial cylinders, into the gap between them is placed the mass under pressure. The inner cylinder (in the form of a bar) is set in rotation. On the inner surface of the matrix and at the outside surface of the bar grooves are made preventing the melt from slipping along the wall. 

Fluid motion between rotating cylinders. The problem about the transient motion of an initially stagnant fluid in the gap between two coaxial cylinders of radii b and a (b < a) that suddenly begin to rotate at angular velocities uib and u>a is considered in [40]. The only nonzero velocity component Vv satisfies Eq. (1.9.29) with the initial condition (1.9.30) and the following boundary conditions ... 

Consider the flow between two coaxial cylinders in relative rotation (Fig. 5P-7). Write a dimensionless relation between torque and angular frequency. 

Figure 4-10a shows the basic schematic diagram for the operating parts of the Couette concentric cylinder viscometer. The liquid to be investigated is in a thin layer between two coaxial cylinders, the outer one with a radius R2 rotating with angular velocity n and the inner one... 

So far, the general point of departure for the nonlinear analyses attempted has been not the set of equations of motion in their complete form, but rather the set that results by neglecting the temperature dependence of all fluid properties except density, and also the dissipation of kinetic energy. This is known as the Boussinesq approximation. Using the Boussinesq equations, Woronetz (Wl) was able in 1934 to obtain expressions for the velocity perturbations in a fluid confined between coaxial rotating cylinders or spheres which were maintained at different temperatures. Volkovisky (V3) used the... 

Figure 5.5 Linear viscous flow between rotating coaxial cylinders.

Several high-precision viscometers are based on the concentric-cylinder method. The liquid under test is contained in the annulus between two vertical coaxial cylinders one cylinder can be made to rotate at a constant speed, and the couple required to prevent the other cylinder rotating can be measured. For more detailed information on practical viscometry reference should be made to specialized publications (Dinsdale and Moore 1962, BS 188, 1993). 

The polymer solutions under study were poured between two coaxial cylinders made of organic glass- The diameter of the external rotating cylinder amounted to 40 mm, the diameter of the internal one did to 36 mm. The height of the internal cy-... 

Tangential laminar flow of a Newtonian fluid with constant density is occurring between two vertical coaxial cylinders in which the outer one is rotating (S4) with an angular velocity of w as shown in Fig. 3.8-5. It can be assumed that end effects can be neglected. Determine the velocity and the shear stress distributions for this flow. 

Flow between Two Rotating Coaxial Cylinders. The geometry of two coaxial... 

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