Couette cylinder geometry

As stated, we begin with the special problem of flow between two rotating cylinders whose axes are parallel but offset to produce the eccentric cylinder geometry shown in Fig. 5 1. In the concentric limit, this is the famous Couette flow problem, which was analyzed in Chap. 3. 

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]. 

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. 

This is most easily achieved by rotating the inner cylinder and keeping the outer fixed in the laboratory frame. Note, however, that this geometry leads to the formation of Taylor vortex motion if inertial effects become important (Reynolds number Re 1). Most rheo-NMR experiments are actually performed at low Re. In the cylindrical Couette, the natural coordinates are cylindrical polar (q, <(>, z) so the shear stress is denoted and is radially dependent as q 2. The strain rate across the gap is given by [2]... 

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

Experimentally, the dynamic shear moduli are usually measured by applying sinusoidal oscillatory shear in constant stress or constant strain rheometers. This can be in parallel plate, cone-and-plate or concentric cylinder (Couette) geometries. An excellent monograph on rheology, including its application to polymers, is provided by Macosko (1994). 

In the last decade of the nineteenth century, Maurice Couette invented the concentric cylinder viscometer. This instrument was probably the first rotating device used to measure viscosities. Besides the coaxial cylinders (Couette geometry), other rotating viscometers with cone-plate and plate-plate geometries are used. Most of the viscometers used nowadays to determine apparent viscosities and other important rheological functions as a function of the shear rate are rotating devices. 

Normal stress differences can be observed in Couette flow, cone-plate and plate-plate geometries, and capillary flow. The only nonzero components of the stress tensor in coaxial cylinders are a. e(r), cSrr(f), CTee(r), and... 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

Figure 3-5. Typical rheometer geometries (a) parallel disk, (b) concentric cylinder (Couette) geometry, (c) cone-and-plate. Either the angular velocity is set and one measures the torque required to produce this rotation rate, or the torque is set and one measures the angular velocity. We analyze the Couette device in this section.

If one needs to investigate the dependence of r] on shear rate, y, one must have access to a rheometer, an instrument that can characterize the dependence of viscosity on shear rate, thus enabling an extrapolation to the Newtonian limit. Typically, such measurements are conducted in Couette (concentric cylinder) or cone and plate geometry. In the Newtonian limit, for Couette geometry, when the inner cylinder is rotated, and provided that the gap between the inner and outer cylinder is small (i.e., RilRo < 0.99, where Ri and Ro are the radii of the inner and outer cylinders), the shear stress on the wall of the outer (resting) cylinder is... 

Used for glass, sol-gels, blood, polymers, etc. The cup-and-bob types define the volume of sample to be sheared in a test cell. The torque needed to achieve a particular rotational speed is measured. The two geometries are known as the Couette or Searle systems the difference depends on whether the cup or bob rotates. The cup can be a cylinder. 

Figure 11 Schemes of the different geometries that can be used for flow or viscoelastic rheological measurements. From left to right double cylinders or Couette geometry, cone and plate geometry, and parallel plates geometry.

Fig. 5 Schematic diagram of tangential generalized annular Couette flow, (a) Geometry for concentric cylinder model (b) typical velocity profiles under varying flow conditions.

There are many designs of rheometer for materials with different viscoelasticities. For example, liquids can be examined in the Couette geometry, which consists of vertical concentric cylinders, one of which rotates with the sample between the gap. Cone-and-plate and parallel disc cells are also widely used. For soft solids, oscillatory shearing between vertical parallel plates yields the dynamic shear moduli. 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

Secondary flows in the Couette geometry have been well studied following Taylor s classic work (1923). Vfith the inno- cylinder rotating at some speed, inertial forces cause a small axisymmet-ric cellular secondary motion known as Taylor vortices or Taylor cells. These dissipate energy and cause an increase in the measured torque. Some data from Denn and Roisum (1969) for a glycerin-water solution are shown in Figure 5.3.8. For Newtcmian fluids and narrow gaps, the criterion for stability is... 

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