Shear rate, dependence

With several springs, which function as torque gauges, and a number of spindles, viscosities can be measured up to 10 mPa-s with the Brookfield viscometer. The shear rates depend on the model and the sensor system they are ca 0.1 100 for the disk spindles, <132 for concentric cylinders, and <1500 for the cone—plate forlow viscosity samples. Viscosities at very low (ca 10 — 1 )) shear rates can be measured with the concentric... 

Shear rate dependence of apparent viscosity for Newtonian and non-Newtonian fluids plotted on linear co-ordinates... 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

For polymer melts or solutions, Graessley [40-42] has shown that for a random coil molecule with a Gaussian segment distribution and a uniform number of segments per unit volume, a shear rate dependent viscosity arises. This effect is attributed to shear-induced entanglement scission. 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

Qualitatively, the same behaviour is observed for the flow curves at a fixed polymer concentration with various molar masses (Fig. 10). The shear rate depend-... 

For a precise analysis of the shear rate dependent viscosity it is necessary to know at which critical rate of deformation shear-induced disturbance can no longer be leveled out by the recoil of the polymers. 

It is possible to approximate the shear rate dependent viscosity at any rate of deformation (y> ycrit), to such an extent that degradation may be neglected. 

In Eq. (34) K is a constant that approximates to unity. Substitution of (t 0-r s) by the r 0-Mw-c relationship, by the A0-Mw-c relationship and the exponent n, as mentioned above, leads to the results that r ( j) is solely a function of the overlap parameter, the term c1+1/a and the rate of deformation, i.e. the shear rate dependence of T) can be satisfactorily described by application of a three-parameter approach. 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

A similar variety of samples was tested for thermal stability by capillary rheometry and TGA. Figure 6.3 shows the viscosity-shear rate dependence for PCTFE homopolymers and one copolymer (Alcon 3000). All materials, save one, showed virtually identical viscosity relationships despite large changes in inherent viscosity. Only the polymers from runs initiated by fluorochemical peroxides (FCP) showed a dependence of molecular weight (as measured by inherent viscosity) upon melt viscosity. 

There is a relationship that is used to cross between time and shear rate dependence regimes and that is the Cox-Merz rule.5 The dynamic viscosity when plotted as a function of frequency, has a similar... 

Now for a power iaw fluid the viscosity n is repiaced by the shear rate dependent viscosity q. 

The presence of fillers in viscous polymer melts not only increases their viscosity but also infiuences their shear rate dependency, especially with non-spherical particles (fibrous or fiake-like) which become oriented in the fiow field. As Fig. 6 shows, particle orientation increases the non-Newtonian behaviour which commences at a lower rate of shear than for unfilled melt. 

In the in situ consolidation model of Liu [26], the Lee-Springer intimate contact model was modified to account for the effects of shear rate-dependent viscosity of the non-Newtonian matrix resin and included a contact model to estimate the size of the contact area between the roller and the composite. The authors also considered lateral expansion of the composite tow, which can lead to gaps and/or laps between adjacent tows. For constant temperature and loading conditions, their analysis can be integrated exactly to give the expression developed by Wang and Gutowski [27]. In fact, the expression for lateral expansion was used to fit tow compression data to determine the temperature dependent non-Newtonian viscosity and the power law exponent of the fiber-matrix mixture. 

Fig. 28a, b. Shear rate dependence of the primary and secondary normal stress diffemeces (oN1, alN2) a Magda et al. s experimental results [148] for a liquid crsytal solution of PBLG with M, = 23.5 x 10 (N = 0.54) at 25 °C b Larson s theoretical results [154]... 

This theory seems not to have been extended to branched polymers in the non-Newtonian range so far as the extended theory does not agree very well with experiment in the Newtonian range, a satisfactory extension to predict the shear-rate dependence of the viscosities of branched polymers is likely to be difficult. The experimental evidence available is discussed in later sections. 

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