Steady-State Shear Flow Measurement

1) it is assumed that the stress acting on the fluid surface at the edge of the gap may be neglected. Integration of Eq. (5.1) gives 

It is of particular interest to note in Eq. (5.7) that for small values of 9 the shear stress aj2 (hereafter denoted by a) is constant across the gap between the cone and plate. 

Let us now discuss what other parameters should be measured in order to determine normal stress differences (see Chapter 2). The total stress component normal to the direction of shear (that is, the surface of the cone or plate) is 

Substitution of Eq. (5.16) into Eq. (5.3) to eliminate p, with an understanding that is independent of 6, gives 

It is of interest to note in Eq. (5.19) that measurement of the total normal stress 00 at various positions r along the stationary surface will permit one to determine N defined byEq. (5.18). 

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The steady-state shear flow properties in the low shear rate region and the dynamic functions were measured using a rotational viscometer (cone-plate type, RGM151-S, Nippen Rheology Kiki Co., Ltd., Japan). The cone radius R was 21.5mm, the gap between the central area of the cone and plate H was kept at 175p.m, and the cone angle 0 was 4°. The measurements were carried out at 200°C Steady state shear properties (shear viscosity //, and the first normal stress difference Ni) as well as dynamic functions (storage and loss moduli G, G", respectively. 

Many of the devices mentioned in the preceding paragraph can also be used to measure stress growth following the inception of shear at a constant shear rate, as the condition of steady-state shear flow is approached. 

Figure 9.18 refers to two other standard experiments. It depicts the results of stress growth experiments, conducted again on a polyethylene melt. The figure includes both measurements probing shear and tensile properties, thus facilitating a direct comparison. Curves show the building-up of shear stress upon inception of a steady state shear flow at zero time and the development of tensile stress upon inception of a steady state extensional flow. Measurements were carried out for various values of the shear rate 7 or the Hencky rate of extension ch ... 

In the preceding sections, we have presented the material functions derived from various constitutive equations for steady-state simple shear flow. During the past three decades, numerous research groups have reported on measurements of the steady-state shear flow properties of flexible polymer solutions and melts. There are too many papers to cite them all here. The monographs by Bird et al. (1987) and Larson (1988) have presented many experimental results for steady-state shear flow of polymer solutions and melt. In this section we present some experimental results merely to show the shape of the material functions for steady-state shear flow of linear, flexible viscoelastic molten polymers and, also, the materials functions for steady-state shear flow predicted from some of the constitutive equations presented in the preceding sections. 

Oscillatory shear flow properties (also referred to as dynamic viscoelastic properties) have long been used to investigate the viscoelastic properties of polymeric materials (Ferry 1980). Oscillatory shear flow measurement requires an instrument that can generate sinusoidal strain as an input to the fluid under test and record the stress resulting from the deformed fluid as an output. For such purposes, a parallel-plates flxture as well as a cone-and-plate flxture can be used the uniform shear rate in the radial direction that is necessary when conducting steady-state shear flow experiments is no longer necessary. 

We now present the theory (Davis et al. 1973 Han 1974) that allows one to determine shear stress and first normal stress difference in steady-state shear flow using wall normal stress measurements along the axis of a slit die. Consider a fluid flowing through a slit die having the height h and the width w, and assume that flow has become fully developed. Then, for steady-state fully developed flow, the equations of motion... 

Steady state shear flow behavior can be measured with rotational equipment operating at prescribed angular velocities. The cone and plate configuration is commonly used... 

The stress relaxation experiment in Figure 2(d) involves the measurement of the time evolution of the stresses after a steady-state shear flow has been suddenly stopped at time t=0. Here again it is necessary to design the experiment in such a way that inertial effects are minimized. One can then define three relaxation functions as follows for t>0... 

An extensive examination was conducted on the yielding phenomena associated with various rheological measurements with an NR and S-SBR. Carbon blacks were NllO, N326 and N990 at 0, 10, 20 and 30% by volume. Experiments carried out were (i) stress relaxation, (ii) transient and steady state shear flow, (iii) stress relaxation after steady flow, (iv) sequential shear flow history and (v) storage effect. 

Unfortunately, Fixman has not yet given a value for the reduced steady-state shear compliance. However, from a comparison of eqs. (3.60a), (3.64) and (3.66) the impression is obtained that the theory of Ptitsyn and Eizner overestimates the influence of the excluded volume on 0 and JeR. As will be shown in the experimental section of this chapter, this impression is supported by flow birefringence measurements on solutions in 0- solvents and in good solvents. 

Two liquid crystalline polybenzylglutamate solutions, adjusted to the same Newtonian viscosity, have been investigated Theologically. The steady state shear properties and the transient behaviour are measured. For the same kind of polymer, the dynamic moduli upon cessation of flow can either increase or decrease with time. This change in dynamic moduli shows a similar dependency on shear rate as the final portion of the stress relaxation but no absolute correlation exists between them. By comparing the transient stress during a stepwise increase in shear rate with that during flow reversal the flow—induced anisotropy of the material is studied. 

In regime III, the flow field is very strong and shear-induced molecular orientation becomes important. According to birefringence measurements for anisotropic HPC/H2O solutions and HPC/ m-cresol solutions, the molecular orientation is a monotoni-cally increasing function of the steady state shear rate. 

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. 

Three types of flow are mainly used in the rheological measurements steady state shearing, dynamic shearing, and elongation. The three can be classified according to the strain, y, vorticity, as well as uniformity of stress, a, and strain within the measuring space (see Table 7.1). 

The cone and plate rheometers are useful at relatively low shear rates. For higher shear rates capillary rheometers are employed. They are usually constructed from metals. The molten polymer is forced through the capillary at a constant displacement rate. Also, they may be constructed to suit various specific shear stresses encountered in commercial operation. Their big disadvantage is that shear stress in the capillary tubes varies from maximum at the walls to zero at the center. On the other hand, stable operation at much higher shear rates is possible. Determination, however, of rjo is usually not possible due to limitations of the instruments. At low shear rates, one can determine the steady-state viscosity from measurements of the volumetric flow rates, Q and the pressure drop ... 

Owing to experimental difficulties, steady-state shear measurements of Ni and 0 2 are relatively rare. Their rate of shear gradients, Ni/y,r] = 012/y usually show a similar dependence ]322]. The value of the complex viscosity ist] >t]. In the steady shear flow of a two-phase system, the stress is continuous across the interphase, but the rate of deformation is not. Thus, for polymer blends, plots of the rheological functions versus stress are more appropriate than those versus rate, that is, a Ni = Ni oi2) plot is similar to G = G (G"). 

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