Third normal stress difference

J. A. Kornfield, G. G. Fuller, and D. S. Pearson, Third normal stress difference and component relaxation spectra for bidisperse melts under oscillatory shear, Macromolecules, 24, 5429 (1991). 

If X, y and z are respectively the flow, gradient and vorticity directions, birefringence measurements in the y-z and x-z planes also lead to the second and third normal stress differences, providing the material verifies the linear... 

First normal stress difference Second normal stress difference Third normal stress difference Normal stress coefficient Birefringence Poiseuille flow No-slip condition Capillary flow... 

First, second, and third normal stress difference Reynolds number Critical Reynolds number Pressure, injection pressure Scattering intramolecular interference function... 

Difference between the second and third normal-stresses (022 - 033) in simple shear flow... 

There are some interesting points to be noted. First, it seems that also for polymer melts the normal stress differences (fin — fi22) and (fin—fi33) are practically equal. (Similar results have been obtained for melts of several polyethylenes.) Second, for the investigated polystyrene a practically quadratic dependence of nn — n33 on the shear stress is found up to the point of the inset of an extrusion defect. It is noteworthy that Fig. 1.9 shows no quadratic dependence of Pjd vs Ds, as would be expected for a second order fluid. Third, the measurements in the cone-and-plate apparatus have to be stopped at a shear stress at least one... 

Furthermore, when the cone-and-plate rheometer is outfitted with pressure taps at various radial positions, the experimentally obtained pressure distribution is found to be increasing with decreasing radial distance. This, as we will see later, enables us to compute the secondary normal stress difference, namely, x22 — T33, where direction 3 is the third neutral spatial direction. 

In examples 3.3, 3.4 and 3.5 we discuss three of the models listed above the LVE, some members of the GNF family and the CEF the first because it reveals the viscoelastic nature of polymer melts the second because, in its various specific forms, it is widely used in polymer processing and the third because of its ability to predict normal stress differences... 

It appears that there are several mechanisms for interface distortion. One is distortion caused by viscosity differences (viscous encapsulation), another is caused by normal stress differences in the fluid (elastic encapsulation), and a third is caused by normal velocity differences within the fluid (geometrical encapsulation). Obviously, the distortion will increase when viscosity differences are large and when normal stress differences play a significant role. 

The symbols have a meaning similar to those in Fig. 3 and, again, three different zones can be observed. The transition from the second to the third zone is no longer smooth and continuous. A discontinuity is observed and a twofold jump in velocity is observed for this particular case. The results obtained by Rodrigue et al. (1995) seem to indicate that a discontinuity can be observed in special cases, namely, the viscoelastic fluids involving normal stress differences as well as a certain amount of impurities present in the bulk fluid, to modify the gas-liquid conditions prevailing at the interface. 

T+(f, y) (300-400% of the steady-state value) and (t,y) (approximately 200% of the steady-state value) is observed in PSHQIO, whereas very little or no overshoot in cr+(r, y) and (t, y) is observed in PS and HOPE. Second, the peak value of (t, y) is 3-4 times greater than the peak value of a+(t, y) in PSHQIO, whereas no overshoot in N t, y) occurs in both PS and HDPE. Third, the steady-state first normal stress difference (N ) is greater than the steady-state shear stress (nematic state is attributable to the reorientation of the directors along the flow direction, which were distributed randomly in the polydomain of a solvent-cast specimen before shear startup. It has been observed that the magnitude of overshoot in r+(f, y)... 

The third force component is believed to be related to the second normal stress difference N2 (17.46). 

The wall shear stress inside the die is one-third of the maximum wall shear associated with the onset of melt fracture that is, it is equal to ([1.13 x 10 ]/3) Pa = 37,667 Pa. Thus, the allowed ratio in parentheses in Eq. 9.177 (primary normal stress difference over wall shear stress) is equal to... 

The shear stress o 2 and the differences between the normal stresses o —0x2 and 022 — 033 are usually measured in the experiment (Meissner et al. 1989). The results of calculation of the stresses up to the third-order terms with respect to the velocity gradient will be demonstrated further on. For simplicity, we shall neglect the effect of anisotropy of the environment when the case of strongly entangled systems will be considered. 

The numbers 1 to 3 are associated with normal stresses and strains, 4 to 6 with the shear components. It is useful to note that the numbers run down the diagonal of the stress and strain tensors and circle back up the third column and along the top row to the starting position. This new notation also removes the difference between simple and pure shear strains discussed earlier. With the new notation, Eq. (2.50) becomes... 

The procedure described, involving the variation of the laser energy, has some advantages relative to the alternative method of using several solutions with different transmittances. First, it provides a check for multiphoton effects simply by analyzing the quality of the linear correlations obtained. It should be stressed that the excellent correlations in figure 13.7 are typical, that is, correlation factors are usually better than 0.9995. Second, the method requires considerably less sample (only one solution is needed). Third, the analysis of experimental data is also conceptually simpler, because no normalization is required. 

The third mechanism can be seen as a combination of first and second basic mechanisms (Figure 1(c)). In a fractured rock with multiple fractures, anisotropy in fluid permeability may be significant due to the different orientations of fractures and anisotropic stresses. Each fracture is under different contact normal or shear stresses mobilized through deformations and this make the directional permeability anisotropic. 

We can identify a third category called reduced. This is mainly used to indicate the stress level of syllables containing the schwa vowel. Reduced vowels occur in at least three subtly different cases. First, some words naturally, or always, have a schwa in a certain position, for example in the first syllable of machine / m ax sh iy n/. But consider the second syllable of information. We have given this a schwa in our annotation and most speakers would agree with this pronunciation. Now consider the word inform, /ih n f ao r m/, here the second syllable is a normal full vowel. We can say that information is created from inform by the application of morphological rules, and in the process, the second syllable has been reduced from a full vowel to a schwa. A third form of reduction occurs in function words, when for instance a word such as (from, / f r ah m/ is produced as /f r ax m/ in many instances. Finally, when people speak quickly, they often under-articulate vowels, which leads to a loss of distinction in their identity leaving them sounding like a schwa. This effect is also termed reduction. The number 0 is used to represent a reduced vowel, so the full transcription for information is /ih2 f axO r m ey2 sh axO n/... 

---