Normal stress differences illustrations

In shear flows, polymer melt experiences significant elastic deformations, and the normal stress difference reflects its resistance to the shear stress. As illustrated in Fig. 7.13, the direction of flow is labeled as 1 , the direction of velocity gradient is labeled as 2 , and the direction of equal velocity (normal to the book) is labeled as 3 . Since the deformation along direction 1 is large, part of the normal stress initially along direction 2 will be added onto direction 1, resulting in the order of normal stresses as... 

Clearly, there are other mechanisms by which interfacial distortion occurs. Dooley and Hughes [43] performed careful experiments to illustrate the extent of interface distortion in coextrusion of polymers with the same flow characteristics. In their analysis they attribute the interface distortion to normal stress differences within the polymer melt, and they used a finite element program capable of handling viscoelastic fluids to predict the distortion within the fluid. This issue is discussed further in Section 12.4.2 see Figs. 12.19 to 12.21. 

Svabik, Samsonkova, and Perdikoulias [45] proposed another explanation for the interface distortion in coextrusion of fluids with equal viscosity. They performed three-dimensional flow analysis of coextrusion flow and found that even in coextrusion with Newtonian fluids with equal viscosity layer distortion takes place. Obviously, this type of distortion cannot be caused by normal stress differences since these do not occur in Newtonian fluids. Also, with the viscosities being equal the distortion cannot be caused by viscosity differences. The predicted layer distortion is schematically illustrated in Fig. 9.43. The authors call the distortion resulting from purely viscous flow geometrical encapsulation. 

The experimental studies of the influence of fillers on the rheological properties of polymer melts by White et al. [29] best illustrate the effect of filler type. The steady shear elastic data were generated in terms of the first normal stress difference using the cone and plate arrangement of the Rheometrics Mechanical Spectrometer at a fixed temperature of 180°C. 

It should be noted that Figures 7.1 and 7.2 are plotted as Ni vs. rather than Nj vs. y. Where the latter representation was used then the data would appear as shown in Figure 7.3(a). Orxly curves for particulate fillers - titarxium dioxide and carbon black - are shown for illustration purposes. It can be seen that both these fillers appear to increase normal stress differences of the polymer melt at the same shear rate. Thus from Figure 7.3(a) it can be concluded that small particulate fillers like carbon black and titanium dioxide increase the elasticity of polymer melts. This is contrary to the trend in Figure 7.1 for the same filled systems using the same data but different representation wherein... 

The viscoelastic forces that produce the remarkable manifestations illustrated above are properly characterised in shear flow by the so-called first and second normal-stress differences, Ni and N2, which occur in addition to the shear stress CT (with which we are already familiar) —note that occasionally Ni and N2 are called the primary and secondary normal stress differences. The complete stress distribution in a flowing viscoelastic Hquid may be written down formally as follows,... 

Particular examples of the first normal-stress difference Ni or its coefficient are shown in figures 12 - 29, where a comprehensive collection of examples is given for polymer solutions and melts, as well as an emulsion. These are compared with either the equivalent shear stress or the viscosity. All the different possible combinations of shear and normal-stress difference, and viscosity and normal-stress coefficient are displayed to show the way that results are presented in the rheological literature. The figures are set out to illustrate overall behaviour, with especial emphasis on low shear-rate and mid-range (i.e. power-law) behaviour. 

We now introduce the major rheological material functions, with illustrations provided by typical experimental results. Figure 7.15 depicts data obtained for low density polyethylene under steady shear flow conditions, employing a cone-and-plate rheometer. Curves display both the shear rate dependence of the viscosity, with similar results as in Fig. 7.1, and the shear rate dependence of the first normal stress difference. The stresses arising for simple shear flows may be generally expressed by the following set of equations... 

Figure 11.2 Plots illustrating the typical form of the rheology in a suspension of Brownian particles, plotted as a function of the Peclet number, Pe = (67tr ya )/kr (a) shear viscosity with increasing cj) and (b) first and second normal stress differences. (Adapted from Frank, M. et al., /. Fluid Mech., 493, 363, 2003, following forms observed by Foss, D.R. and Brady, J.F., /. Fluid Mech., 407, 167, 2000 and references therein.)...

The finite control-volume dimensions as illustrated in Fig. 2.13 may be a potential source of confusion. While the stress tensor represents the stress state at a point, it is only when the differential control volume is shrunk to vanishingly small dimensions that it represents a point. Nevertheless, the control volume is central to our understanding of how the stress acts on the fluid and in establishing sign conventions for the stress state. For example, consider the normal stress xrr, which can be seen on the r + dr face in the left-hand panel and on the r face in the right-hand panel. Both are labeled rrr, although their values are only equal when the control volume has shrunk to a point. Since the stress state varies continuously and smoothly throughout the flow, the stress state is in fact a little different at the centers of the six control-volume faces as illustrated in Fig. 2.13 where the... 

Rheological properties have also been studied. It has been observed for some concentrations that the first difference of normal stress becomes negative for a certain range of shear stress. This effect had already been observed in the case of HPC (2fl). Figure 5 illustrates the case of another cellulosic derivative ethylcellulose. 

The effect of shear displacement under different normal stress conditions on fracture cross-flow for RWS, YBS and LC is illustrated by the TCSFT results in Figs. 12-14. The (current) bulk permeability Kf) at any given shear displacement has been normalized with the initial bulk permeability prior to... 

The t s are the normal stresses in the coordinate directions indicated. Figure 3-11 illustrates both shear (two different letter subscripts) and normal (same letter subscripts). The shear stresses act along a plane, whereas the normal... 

Figure 12. Numerical results of fmite element analysis for the stress distribution profile (normal stresses. Ox) of the specimens with relative notch depths of oo/W 0.1, 0.5, and 0.9 at the respective critical loads Pc, illustrating a significant difference in the tensile/compressive distribution profiles in the locations away of the cracked region.

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

Here we apply the summation convention, i.e. if the same index appears twice on the same side of an equation then summation over this index is implicitly assumed (unless explicitly stated otherwise). The relation between indices, force components, and the faces of the cubic volume element is depicted in Fig. 1.3. Upper and lower sketches illustrate the shear and the normal contribution to the force component fa acting on the volume element in ct-direction. Notice that fa can be written as the sum over two shear stress and one normal stress contribution. The latter are stress differences between adjacent faces of the cubic volume element. Note also that the unit of Oa is force per area. 

The zone at risk is defined differently from the infarct area and from viability as the area of reduced flow reserve by dipyridamole stress perfusion imaging rather than by rest perfusion imaging. The physiologic basis for this approach is the well documented observation that resting coronary flow may be normal with up to 85% stenosis consequently, rest imaging may not define the correct size of zone at risk, as illustrated by the example below. 

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