Concentric cylinders shear stress

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

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. 

There are two well-accepted models for stress transfer. In the Cox model [94] the composite is considered as a pair of concentric cylinders (Fig. 19). The central cylinder represents the fiber and the outer region as the matrix. The ratio of diameters r/R) is adjusted to the required Vf. Both fiber and matrix are assumed to be elastic and the cylindrical bond between them is considered to be perfect. It is also assumed that there is no stress transfer across the ends of the fiber. If the fiber is much stiffer than the matrix, an axial load applied to the system will tend to induce more strain in the matrix than in the fiber and leads to the development of shear stresses along the cylindrical interface. Cox used the following expression for the tensile stress in the fiber (cT/ ) and shear stress at the interface (t) ... 

A fluid with a finite yield. stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion was 0.01 N m. Calculate the force needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/ni2. 

Fig. 15. Effects of turbulent shear stress level and exposure time on cell viability measured by trypan blue staining. Cells were sheared in a concentric cylinder viscometer [1]...

This equation will only apply 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. This can be allowed for by introducing a stress concentration factor, in a similar manner to the method used for torispherical heads,... 

As the name implies, the cup-and-bob viscometer consists of two concentric cylinders, the outer cup and the inner bob, with the test fluid in the annular gap (see Fig. 3-2). One cylinder (preferably the cup) is rotated at a fixed angular velocity ( 2). The force is transmitted to the sample, causing it to deform, and is then transferred by the fluid to the other cylinder (i.e., the bob). This force results in a torque (I) that can be measured by a torsion spring, for example. Thus, the known quantities are the radii of the inner bob (R ) and the outer cup (Ra), the length of surface in contact with the sample (L), and the measured angular velocity ( 2) and torque (I). From these quantities, we must determine the corresponding shear stress and shear rate to find the fluid viscosity. The shear stress is determined by a balance of moments on a cylindrical surface within the sample (at a distance r from the center), and the torsion spring ... 

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

This section describes common steps designed to measure the viscosity of non-Newtonian materials using rotational rheometers. The rheometer fixture that holds the sample is referred to as a geometry. The geometries of shear are the cone and plate, parallel plate, or concentric cylinders (Figure HI. 1.1). The viscosity may be measured as a function of shear stress or shear rate depending upon the type of rheometer used. 

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 first of these techniques an approximation to uniform rate of shear throughout the sample is achieved by shearing a thin film of the liquid between concentric cylinders. The outer cylinder can be rotated (or oscillated) at a constant rate and the shear stress measured in terms of the deflection of the inner cylinder, which is suspended by a torsion wire (Figure 9.2) or the inner cylinder can be rotated (or oscillated) with the outer cylinder stationary and the resistance offered to the motor measured. 

To measure elastic and viscous properties which are characteristic of the material under consideration and independent of the nature of the apparatus employed, the applied stress and the resulting deformation must be uniform throughout the sample. Concentric cylinder and cone and plate methods approximate these requirements. For materials which are self-supporting, measurements on, for example, the shearing of rectangular samples are ideal. 

The Couette rheometer. Another rheometer commonly used in industry is the concentric cylinder or Couette flow rheometer schematically depicted in Fig. 2.48. The torque, T, and rotational speed, 0, can easily be measured. The torque is related to the shear stress that acts on the inner cylinder wall and the rate of deformation in that region is related to the rotational speed. The type of flow present in a Couette device is analyzed in detail in Chapter 5. 

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. 

Hydrodynamic Analysis of an Idealized Internal Mixer Consider the idealized mixer shown in the accompanying figure, consisting of two concentric cylinders in relative motion with a step change in clearance. Calculate the mean shear stress in the narrow gap. 

In rotational instruments, one member (e.g., the cup in a concentric cylinder viscometer) rotates while the other (e.g., the bob) remains stationary. The sample is held, and sheared, in the gap between the two. In a controlled shear rate measurement, the rotational speed is constant, and the torque on one member caused by the viscous resistance to flow exerted by the sample is measured. In a controlled stress measurement, a constant torque is applied to one member and its speed of rotation measured. Controlled stress instruments are particularly useful for measuring yield stress, the minimum stress causing flow of a plastic material. 

In general, shear stress at one location (e.g., the bob surface in a concentric cylinder viscometer) is calculated from the dimensions of the sample gap and the measured or applied torque. Shear rate is calculated at the same location from sample gap dimensions and rotational speed. By making experimental measurements over a range of speeds or torques, the flow curve (shear stress versus shear rate) of the sample can be established. Suitable mathematical treatment of the flow curve data yields the sample s constitutive equation and rheological properties. 

In a rotational viscometer the solution is filled in the annulus between two concentric cylinders of which either the external (Couette-Hatschek type) or the internal (Searle type) cylinder rotates and the other, which is connected to a torsion-measuring device, is kept in position. Let R, and Ro be the radii of the inner and outer cylinders, h the height of the cylinder which is immersed in the solution or its equivalent height if end effects are present, a> the angular velocity of the rotating cylinder, and T the torque (or moment of force) required to keep the velocity constant against the viscous resistance of the solution. It can be shown that the shearing stress (see, for example, Reiner, 1960) ... 

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

---