Normal Steady shear flow

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

In steady shear flow, the viscosity is independent of shear rate — for all y. This property alone represents a serious qualitative failure of the conventional bead-spring models. The normal stress functions are (108) ... 

From the additional group of results of Lodge s theory, only the expressions for the normal stresses in steady shear flow will be used in the following. These expressions read ... 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

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. 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Steady shear flow measnrements, however, can measure only viscosity and the first normal stress difference, and it is difficult to derive information abont fluid structure from such measurements. Instead, dynamic oscillatory rheological measurements are nsed to characterize both enhanced oil recovery polymer solutions and polymer crosslinker gel systems (Prud Homme et al., 1983 Knoll and Pmd Homme, 1987). Dynamic oscillatory measurements differ from steady shear viscosity measnrements in that a sinusoidal movement is imposed on the fluid system rather than a continnons, nnidirectional movement. In other words, the following displacement is imposed ... 

Steady-state shear rheology typically involves characterizing the polymer s response to steady shearing flows in terms of the steady shear viscosity (tj), which is defined by the ratio of shear stress (a) to shearing rate y ). The steady shear viscosity is thus a measure of resistance to steady shearing deformation. Other characteristics such as normal stresses (Ai and N2) and yield stresses (ffy) are discussed in further detail in Chapter 3. 

We next turn our attention to the relaxation of shear and normal stresses after a steady shearing flow is suddenly stopped. Before time t=0 the fluid is in a state of steady shear flow with a velocity profile vx = K0y, vy = 0, v2 = 0. After time t = 0 the fluid is completely motionless, but the shear and normal stresses decrease gradually to zero. This flow was analyzed for rigid dumbbells by Giesekus (12) and later by Bird, Warner, and Ramakka (10). Schremp, Ferry, and Evans (69), Benbow and Howells (2), and Huppler et al. (36) have measured the time-decay of the stresses in an experiment which closely approximates the above-described idealized problem. 

In stress relaxation after cessation of steady shear flow, the elastic dumbbells give no dependence of the relaxation process on the steady-state shear rate, but the rigid dumbbells do. In addition the elastic dumbbells show the shear and normal stresses relaxing with exactly the same... 

In this section we investigate some of the properties of mixtures undergoing steady shearing flow. Specifically we consider the viscosity and normal stress functions for suspensions of rigid dumbbells of various lengths which have the same zero shear rate viscosity as a solution containing dumbbells of length L only. 

The steady-state flow behavior is not only nonlinear but also characterized by the development of normal stresses, which are completely absent in a simple Newtonian fluid. Thus, a steady shear flow in the x-y plane,, not only leads to a (nonlinear) shearing stress, X but also leads to normal stresses, Ojjx vv zz Associatea with these normal stresses are many distinctive phenomena exhibited by polymeric fluids, such as the Weissenberg effect, where a polymer being stirred by a turning shaft tends to climb up the shaft instead of being thrown outward by centrifugal action (22). 

Parallel-plate rheometers are often more useful for studying rheology of filled polymers or composite materials, particularly when the size of the fillers is comparable fo fhe disfance between the truncated cone and the surface of the plate. Again, the torque M and the normal force N tending to separate the two plates are measured. In steady-shear flow, the shear rate and the shear stress at the edge of the disks located atr = R are given by... 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

Fig. 1. Influence of silane coupling agent on steady shear flow properties of GF/PP composites (a) viscosity as a function of shear rate, (b) the first normal stress difference as a function of shear rate...

Fig. 6. Relationship between relative values of steady shear flow functions and concentration of silane coupling agnet for GF/PP and GF/mPP/PP composites (a) relative viscosity, (b) relative first normal stress difference...

Fig. 19. Normal stress relaxation following steady shear flow for 60 mole % PHB/PET. 

For a one-dimensional steady shear flow of a fluid between two planes, the velocities of an infinitesimal element of fluid in the y- and z-directions are zero. The velocity in the x-direction is a function of y only. Note that in addition to the shear stress Tyx (refer to t subsequently), there are three normal stresses denoted by Txx, Tyy, within the sheared fluid. Weissenberg in 1947 [6] was the first to observe that the shearing motion of a viscoelastic fluid gives rise to tmequal normal stresses, known as Weissenberg effects. Since the pressure in a non-Newtonian fluid cannot be deflned, and as the normal stress differences... 

A more fundamental difference is observed in the normal stress difference in steady shear flow ... 

Polymers usually show unequal normal stress components in steady shear flow (for a Newtonian fluid in steady shear flow, the three normal stress components are always equal). With the three normal stress components normal stress differences ... 

The subscript 1 denotes the direction of flow, the subscript 2 denotes the direction perpendicular to the flow (i.e. the direction along the velocity gradient) and the subscript 3 denotes the neutral direction. The various stress components are shown on a representative cubic volume of the fluid in Figure 2.2. All the components are not shown in the figure in order to maintain clarity. Note that in steady shearing flow, the stress components Ti3.T23, Tsi.Tjz, are identically equal to zero. Ti2 = T21 is called the shear stress and Th, T22, T33, are called normal stresses. 

The viscosity function rj (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients ij/, and respectively, are the three viscometric functions which completely determine the state of stress in any rheologically steady shear flow. They are defined as follows ... 

---