Steady shear flow results

FIG. 16.33 Steady shear flow results for 12% PBLG solution in m-cresol at 293K (O) viscosity (A) positive N, (A) negative Nv Reproduced with permission from Mewis J and Moldenaers P (1987) Mol Cryst Liq Cryst 153, 291. Copyright Taylor and Francis Ltd., http //www.informaworld.com. 

Figures 3.66 and 3.67 show typical steady shear flow results for polymer systems.

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. 

At this point an important result of Lodge s theory should be quoted. It can easily be derived from eq. (2.11). It concerns stress-relation after steady shear flow. As has been pointed out by Tobolsky and Murakami (51), the investigation of relaxation of shear stress after steady shear flow is a convenient method to evaluate the longest relaxation... 

This statement suffices for the present purpose. In fact, a look on eqs. (2.14) and (2.15) which hold for the interesting moment of the memory function, makes the expectation acceptable that only a restricted number of the longest relaxation times will actually be of influence on the final results for slow steady shear flow, provided the g3 s are not too different. For a further discussion of the validity of the stress-optical law see Chapter 5. 

For steady shear flow the strain is reduced to 7yx(t, t ) the constant shear rate. This results in [6]... 

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

In the limit as m tends to zero, the expression for the shear stress must simplify (for the special case of k° real) to the steady-state flow result with velocity gradient nm + k° from Eq. (6.7) this should be... 

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]. 

For Newtonian lipid-based food systems, it is sufficient to measure the ratio of shearing stress to the rate of shear, from which the viscosity can be calculated. Such a simple shear flow forms the basis for many rheological measurement techniques. The rheological properties resulting from steady shear flow for variety of food systems have been studied by many laboratories (Charm, 1960 Holdsworth, 1971 Middleman, 1975 Elson, 1977 Harris, 1977 Birkett, 1983 Princen, 1983 Shoemaker and Figoni, 1984 Hermansson, 1994 Kokini et al., 1994, 1995 Morrison, 1994 Pinthus and Saguy, 1994 and Meissner, 1997). 

The shear reversal experiment was repeated but in this case reversal was carried out directly from steady shear flow rather than after allowing the stresses to relax. Results from this experiment are presented in Figure 6. The stress growth curve is significantly... 

Guido, S. and Greco, F. (2001) Drop shape under slow steady shear flow and during relaxation. Experimental results and comparison with theory. Rhed. Acta,... 

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... 

Fig. 20 Onset of steady shear flow for the unfilled LDPE melt and a series of highly filled LDPE/LDH nanocomposites at yo = 0.3 s h experimental data (points) and predicted results (lines)...

Most flows of polymeric fluids are not slow enough for the second-order-fluid equation or any of the equations of the retarded-motion expansion to apply to them. Of course eq 4.3.1 can be made accurate for any steady shearing flow merely by replacing the constants rjo, 1,0, and 2,0 by the shear rate dependent coefficients l(y). and V 2(i )- Although the resulting equation, called... 

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