Newtonian viscosity, zero shear rate

Figure 8.5. Apparent viscosity-shear rate curves for dilatant fluid, a Newtonian fluid and pseudoplastic fluid which have the same apparent viscosity at zero shear rate...

The viscosities of most real shear-thinning fluids approach constant values both at very low shear rates and at very high shear rates that is, they tend to show Newtonian properties at the extremes of shear rates. The limiting viscosity at low shear rates mq is referred to as the lower-Newtonian (or zero-shear /x0) viscosity (see lines AB in Figures 3.28 and 3.29), and that at high shear rates Mo0 is the upper-Newtonian (or infinite-shear) viscosity (see lines EF in Figures 3.28 and 3.29). 

First the temperature dependence of the limiting zero shear rate viscosity (Newtonian) is calculated at a shear rate of 0.01 1/s using the data in Table 3.7 ... 

Mendelson (169) studied the effect of LCB on the flow properties of polyethylene melts, using two LDPE samples of closely similar M and Mw plus two blends of these. Both zero-shear viscosity and melt elasticity (elastic storage modulus and recoverable shear strain) decreased with increasing LCB, in this series. Non-Newtonian behaviour was studied and the shear rate at which the viscosity falls to 95% of the zero shear-rate value is given this increases with LCB from 0.3 sec"1 for the least branched to 20 sec"1 for the most branched (the text says that shear sensitivity increases with branching, but the numerical data show that it is this shear rate that increases). This comparison, unlike that made by Guillet, is at constant Mw, not at constant low shear-rate viscosity. 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

Here we have three parameters r/o the zero-shear-rate viscosity, Ai the relaxation time and A2 the retardation time. In the case of A2 = 0 the model reduces to the convected Maxwell model, for Ai = 0 the model simplifies to a second-order fluid with a vanishing second normal stress coefficient [6], and for Ai = A2 the model reduces to a Newtonian fluid with viscosity r/o. If we impose a shear flow,... 

In this case, p is an arbitrary constant, chosen as the zero shear rate viscosity. The expression for the non-Newtonian viscosity is a constitutive equation for a generalized Newtonian fluid, like the power law or Ostwald-de-Waele model [6]... 

The zero shear rate viscosity of a narrow molecular weight distribution fraction of polystyrene is 6.5 x 10 poise at 160°C. If the molecular weight between entanglements in this polymer is about 18,000 g/mol, make a rough estimate of the shear rate k, above which this fraction will display non-Newtonian behavior. 

The 17, 170, A, and n are all parameters that are used to fit data, taken here as 17 = 0.05, 170 = 0.492, A = 0.1, and n = 0.4. A plot of the viscosity vs shear rate is given in Figure 10.8. For small shear rates, the viscosity is essentially constant, as it is for a Newtonian fluid. For extremely large shear rates, the same is true. For moderate shear rates, though, the viscosity changes with shear rate. In pipe flow, or channel flow, the shear rate is zero at the centerline and reaches a maximum at the wall. Thus, the viscosity varies greatly from the centerline to the wall. This complication is easily handled in FEMLAB. 

Eq. (6.7) may be rearranged in order to display the results in terms of the intrinsic non-Newtonian viscosity M and the corresponding result at zero shear rate Mo ... 

An example is shown in Figure 6.6, lower curve. At very low shear rate, the solution shows Newtonian behavior (no dependence of r] on shear rate), and this is also the case at very high shear rate, but in the intermediate range a marked strain rate thinning is observed. The viscosity is thus an apparent one (/ ,), depending on shear rate (or shear stress). It is common practice to give the (extrapolated) intrinsic viscosity at zero shear rate, hence the symbol [ri]0 in Eq. (6.6). The dependence of t] on shear rate may have two causes. 

For colloidal particles, the dimensionless parameters are generally small and non-Newtonian effects dominate. Considering the same example as above, but with particles of radius a = 1 /xm, the parameters take on the values Pe = y, N y = 10 y, and N = 10 y so that for shear rates of 0.1 s or less they are all small compared to unity. The limit where the values of the dimensionless forces groups are very small compared to unity is termed the low shear limit. Here the applied shear forces are unimportant and the structure of the suspension results from a competition between viscous forces. Brownian forces, and interparticle surface forces (Russel et al. 1989). If only equilibrium viscous forces and Brownian forces are important, then there is well defined stationary asymptotic limit. In this case, there is an analogue between suspensions and polymers which is similar to that for the high shear limit, wherein the low shear limit for suspensions is analogous to the zero-shear-rate viscosity limit for polymers. 

The flow curve of a broad MWD polyethylene is more non-Newtonian than that of a narrow MWD polyethylene (Fig. 5.7). These polymers have the same Mw, so, by Eq. (3.8), have the same zero-shear rate viscosity. The elastic stresses at low shear rates are influenced by the high molecular weight tail of the MWD. When the tensile stress difference is small, it can be described by... 

For a viscoelastic material, the viscosity in the limit of zero-shear rate, the Newtonian viscosity, can be obtained from the integral of the stress relaxation modulus ... 

We studied the dependence of spin modes on viscosity in the HDDA/persulfate system. Determining the viscosity is complicated by the shear-thinning behavior of silica gel suspensions. Figure 6 shows the apparent viscosity vs. shear rate for different percentages of silica gel in HDDA. The linear stability analysis assumes Newtonian behavior so we need to estimate the viscosity at zero shear, which is something we can not reliably estimate with our viscometer. We used the viscosities at the lowest shear rate we could measure and recognize that we are underestimating the true value. Fortunately, this does not affect the qualitative trends. 

It is also essential to take into account delayed elastic effects, specially in the neighborhood of Tg. In this case, one has to wait for the relaxation of elastic response before the glass enters into Newtonian behavior. For polymers, which are not Newtonian, one has to determine the viscosity as a function of shear rate. In some instances, the zero shear viscosity is reported, where one extrapolates the viscosity value to zero shear rate. 

Fig. lA, Reduced plot of the non-Newtonian viscosity shown in Fig. 1.3, where is the viscosity at zero shear rate, and the time constant is chosen empirically for each solution. Reproduced from W. W. Graessley Adv. Pofym. 

Limiting viscosity at zero shear rate, i.e., at the upper Newtonian plateau... 

For Newtonian behaviour r = ri where r is the shear stress, 77 is the viscosity and y is the strain rate. Kaolinite dispersions are cohesive sediments and show shear thinning behaviour ie rj decreases as y increases. The extrapolated value of r at zero shear rate is called the Bingham yield stress, Tg. For a Hookean solid r = G y, where G is the modulus of rigidity. Most substances are neither purely elastic (solid-like) nor purely viscous (liquidlike) but are viscoelastic. Kaolinite is in this category. For viscoelastic behaviour... 

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