Concentric cylinders normal stress

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

Chapter 3. In-plant measurement of flow behavior of fluid Foods. Using a vane-in-a-cup as a concentric cylinder system. The vane yield stress test can be used to obtain data at small- and large-deformations. Critical stress/strain from the non-linear range of a dynamic test. Relationships among rheological parameters. First normal stress difference and its prediction. 

Equations (P17.6.11) and (PI7.6.12) indicate that the shear stress at any point within the cross section of an elliptical cylinder under torsion is tangent to an ellipse passing through this point with the same axis ratio, a/b, as that of the boundary ellipse. In other words, the lines of shear stress are concentric ellipses. Consequently, the inner boundary also coincides with a line of shear stress. Therefore the shear stress acting normally on the internal surface parallel to the z axis in null. Moreover, if a concentric cylinder is removed from the rod, the stress distribution in the remaining portion will be the same as in the solid cylinder. For this reason, the stress function will be given by... 

In some ways a simpler approach to obtaining normal stresses from the concentric cylinder system is to use the rod climbing phenomenon quantitatively. Because the flow is complex, analysis requires the assumption of some constitutive relation. Joseph and Fosdick (1973) have done this using the second-order fluid, which should be an exact representation of elastic liquids in the limit of slow flows (see Section 4.3). They derive a power series for the... 

In the foregoing derivations we neglected the inertia term in eq. 5.4.1. We saw with concentric cylinders that inertia causes a depression around the inner cylinder rather than the climb due to viscoelastic normal stresses (Figure 5.3.3). In cone and plate... 

For polymer solutions, the rheologist should start with the cone and plate, unless the concentric cylinder sensitivity is needed. Normal stress data can be collected simultaneously, and the entire range of strain fiom linear to nonlinear is possible. Temperature control is typically available over a wide range, but solvent evaporation at the edge can cause problems. 

Bird, et al. enumerate a considerable series of effects driven by nonzero normal stress differences(2). For example, for laminar flow in the space between two concentric cylinders, the flow being parallel to the cylinder axes, for a Newtonian fluid the pressures at the two walls at the same distance along the pipes must be equal for a polymer solution the fluid pressure along a radial coordinate may differ between... 

Besides being used as a calibration device, concentric cylinders can be used to find the first normal stress difference (c e —O to investigate orientation effects in shearing and oscillatory ffow. The measurements provide shear orientation information for both the rigid and flexible types of microstructure. 

The above problem is difficult to avoid when it occurs in filled systems [pastes] because a certain minimum gap is needed to avoid artifticts due to the size of the particles. In general, the g should be a minimum of 10 x particle size, and the remedy for this problem is to reduce the gap to improve the stabilizing effect of surftice tension. However, it is also possible to change the geometry type to a concentric cylinder, with a sufficiently large radius ratio to accommodate the particles. Since the fi ee edge is at the top, and the cause is due to the second normal stress difference in a direction perpendicular to the rotation axis, this effect is almost non-existent. 

A considerable reduction in stress concentration could be achieved by using a cross-bore which is eUiptical in cross-section, provided the major axis of the eUipse is normal to the axis of the main cylinder. A more practical method of achieving the same effect is to have an offset radial hole whose axis is parallel to a radius but not coincident with it (97,98). Whenever possible the sharp edges at the intersection of the main bore with the cross bore are removed and smooth rounded corners produced so as to reduce the stress raising effects. 

This equation will apply only at points away from the cone to cylinder junction. Bending and shear stresses will be caused by the different dilation of the conical and cylindrical sections. A formed section would normally be used for the transition between a cylindrical section and conical section, except for vessels operating at low pressures or under hydrostatic pressure only. The transition section would be made thicker than the conical or cylindrical section and formed with a knuckle radius to reduce the stress concentration at the transition see Figure 13.11. The thickness for the conical section away from the transition can be calculated from equation 13.48. 

Socrate et al. (2000) considered an axially symmetric problem, with a rubber sphere in the centre of a short cylinder of matrix the spheres are in a row, aligned with the tensile stress axis. The potential positions of crazes were predetermined, initially running radially from the material interface, then becoming normal to the tensile stress along the cylinder. The initial stress concentration is greatest in the polymer near the equator of the sphere (Fig. 4.11a). The model, for a 20% volume fraction of rubber, predicts a yield point in the tensile stress-strain curve at an average strain of 1%, and 24 MPa stress, when the first craze propagates across the section. However, this relieves the stress in the polystyrene, and a tensile stress concentration... 

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