Newtonian region

For the straight line in Fig. 2.5 where m = 1.0, this equation expresses direct proportionality between and y, the condition of Newtonian behavior. In the non-Newtonian region where m < 1, Eq. (2.11) may describe the data over an order of magnitude or so. Next we consider the relationship between the constant K and viscosity. If Eq. (2.11) is solved for K and the resulting expression multiplied and divided by 7 ", we obtain... 

Controlled stress viscometers are useful for determining the presence and the value of a yield stress. The stmcture can be estabUshed from creep measurements, and the elasticity from the amount of recovery after creep. The viscosity can be determined at very low shear rates, often ia a Newtonian region. This 2ero-shear viscosity, T q, is related directly to the molecular weight of polymer melts and concentrated polymer solutions. 

Moreover, a further conclusion can be drawn from Fig. 10, namely, that the evaluation of molar mass is impossible in the non-Newtonian region because samples with different molar mass all exhibit the same viscosity at a given shear rate. 

In equation 1.75, pt0 and are the values of the apparent viscosity for the lower and upper Newtonian regions respectively. The constant ym is the shear rate evaluated at the mean apparent viscosity (po + p. )/2. 

Accordingly, plots of a t)/a(0) vs t from different shear rates should superimpose. Experimentally the curves do not superimpose when the shear rate is in the non-Newtonian region, the initial rate of relaxation being increasingly more rapid for higher shear rates. The normal stress decays more slowly than shear stress, but behaves similarly with respect to the effect of previous shearing flow in the non-Newtonian region. 

When the shear rate at steady state is in the non-Newtonian region for the fluid, the recoil is smaller than J°a0 for both broad and narrow molecular weight distributions (362). 

The dynamic moduli for infinitesimal superimposed deformations parallel and transverse to the flow direction in steady shearing flow should be unaffected by flow if the shear rate is sufficiently small. According to the theory of simple fluids, the superimposed dynamic moduli for shearing flows in the non-Newtonian region must change in order to conform with the relations (370,372 ... 

The experiments show that the time-temperature dependence of viscosity in the non-Newtonian region of flow may be represented as ... 

Measurements of the zero shear viscosity (20 °C) were made with a Bohlin VOR rheometer in the viscometry mode. If a Newtonian region was not found at the lowest measurable shear rates, the samples were characterized with a Bohlin-CS constant stress rheometer, with which it was possible to obtain extremely low shear rates. This approach was especially needed for highly viscous samples exhibiting pseudoplastic behavior on the VOR rheometer. Newtonian regions were found for each sample in this manner, yielding the zero shear viscosities reported. 

If the Power Law equation is used in pressure flows, where 0 < y < y,Tl. , an error is introduced in the very low shear rate Newtonian region. In flow rate computation, however, this is not a very significant (40). 

It further appears that, with higher rates of stress, the effect of the MMD may overrule the effect of Afw on the apparent viscosity the curves for A and C intersect. From Figure 5.10 one might conclude there is a Newtonian region for A up to y = 0.1 sec-1 or to t = 103 Pa, and for B up to y = 1 sec-1 or to r = 104 Pa. In this Newtonian region the viscosity would be constant . In how far this is indeed true, is apparent from the linear plot in Figure 5.11. As happens more often, the use of a log-scale may lead to a distorted look on reality. 

FIG. 15.9 (left) Increase of shear stress after starting steady shear flow at zero time for constant shear rate in the Newtonian region, (right) Time dependent shear viscosity, rj+[t) from Newtonian to non-Newtonian behaviour. 

Finally a phenomenon should be mentioned which polymer solutions show more often than polymer melts viz. a second Newtonian region. This means that with increasing shear rate the viscosity at first decreases, but finally approaches to another constant value. As the first Newtonian viscosity is denoted by rjor the symbol // x, is generally used for the second Newtonian viscosity. Empirical equations as those presented in Chap. 15 now need an extra term r oo to account for this second Newtonian region. This leads to ... 

There is a reasonable correspondence between the experimental and calculated viscosity values at a shear rate of 10 s-1. The calculated 77 value at y 103 s-1 is too low, however, because at this shear rate the second Newtonian region is approached. To estimate rj Eq. (16.51) can be applied with... 

Fig. 11-28. Typical rheometer shear rate ranges and polymer melt flow curve. The lower shear rate region of the flow curve exhibits viscosities that appear to be independent of y. This is the lower Newtonian region.

For Newtonian fluids the viscosity is independent of time. However, for most non-Newtonian fluids the viscosity at a shear rate high enough to place the fluid in the non-Newtonian region evolves with time as schematically indicated by the lower curve of Figure 13.39. The viscosity decreases with time until steady-state conditions are reached. This phenomenon is called thixotropy. The cause of this behavior lies in the fact... 

The viscosity of some fluids (particle solutions or suspensions) measured at a fixed shear rate that places the fluid in the non-Newtonian regime increases with time as schematically shown by curve C of Figure 13.39. This behavior can be explained by assuming that in the Newtonian region the particles pack in an orderly manner, so flow can proceed with minimum interference between particles. However, high shear rates facilitate a more random arrangement for the particles, which leads to interparticle interference and thus to an increase in viscosity. Models that illustrate the thixotropic and rheopectic behavior of structural liquids can be found elsewhere (58,59). 

These equations assume Newtonian behavior, or at least apply to the Newtonian region of a flow curve, and they usually apply if the droplets are not too large and if there are no strong electrostatic interactions. A more detailed treatment of these relationships is given in Chapter 4. 

In acknowledging that the upper Newtonian region is sometimes not observed due to experimental difficulties, the Cross and the Carreau models reduce to Eqs. (5) and (6), respectively ... 

Mathematical models, which can predict the shape of a flow curve of a shear thinning material including lower and upper Newtonian regions, require at least four parameters. The Cross model is one such model ... 

Similarly, the two models could be reduced by neglecting the lower Newtonian region. This has been done by Sisko for the Cross model in order to account for the rheological properties of grease in bearings ... 

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