Viscosity/shear rate profile

Low-shear-rate solution viscosity was measured on a Couette-type rheometer (Contraves LS 30) with a No. 1 bob and cup. The viscosity-shear rate profile was determined fi om 10" to 10 s" at 25 BC. The system was allowed to reach steady state at each shear rate before the measured viscosity was recorded. 

The Ubbelohde-type capillary viscometer gives viscosity at a low shear rate. To obtain the full viscosity-shear rate profile, a coaxial cylinder viscometer can be employed. The shear rate in a coaxial cylinder viscometer is calculated using the following equations ... 

Figure 1 Apparent viscosity-shear rate profiles of linear and branched PVDF samples at 210°C (L/d=30).

Figure 7.7 Viscosity verses shear rate profiles (polymer concentration, 2.25 kg m 3). Adapted from as Figure 7.5.

The relationship between steady-shear viscosity and dynamic-shear viscosity is also a common fundamental rheological relationship to be examined. The Cox-Merz empirical rule (Cox, 1958) showed for most materials that the steady-shear-viscosity-shear-rate relationship was numerically identical to the dynamic-viscosity-frequency profile, or r] y ) = r] m). Subsequently, modified Cox-Merz rules have been developed for more complex systems (Gleissle and Hochstein, 2003, Doraiswamy et al., 1991). For example Doriswamy et al. (1991) have shown that a modified Cox-Merz relationship holds for filled polymer systems for which r](y ) = t] (yco), where y is the strain amplitude in dynamic shear. 

In looking at the new size tank, estimates should be made of the shear rate profile around the system, and then using the relationship that viscosity is a function of shear rate, and the fact that it is shear stress... 

The MRI-based viscosity measurement relies on local calculations of the velocity gradient based on using MRI velocity profiles to provide a wide range of shear viscosity-shear rate data. Two observations are used to characterize fully developed and steady laminar flow in a MRI-based viscometer the velocity profile and the pressure drop per unit tube length measurements. Viscosity measurements were acquired on strawberry milk samples using H P LG tubing for the sample channel. 

The melt viscosity data were obtained in a conventional manner using an Instron capillary rheometer over the temperature range of 180-240°C. The capillary had a 90° entry angle, and it was 5 cm long and 0.125 cm in diameter. The well known equations were used to calculate apparent viscosity. Shear rates at the wall, calculated assuming a Newtonian fluid, were corrected for nonparabolic velocity profile using the Rabinowitch equation. No correction was made for entrance effect because of the length-to-diameter ratio of the capillary used. 

When solutions of fully water-soluble Type 1 polymers are investigated using a flow curve experiment, it is possible to see a classic double plateau profile in the viscosity versus shear rate profile. An upper Newtonian viscosity plateau exists at very low shear rates. This is... 

Figure 1.8. (a) Viscosity of the colloidal dispersion versus shear rate profiles that can... 

Equation 16C.2 is called the Power Law and Equarion 16C.3 is called the Cross Model. Plots of these two models are shown in Figure 16C.2. In general, the Cross Model does a better job of describing the actual viscosity versus shear rate data observed on commercial polymers. Now we have a mathematical model that describes the actual viscosity versus shear rate profile of mbber materials. 

When processing dispersions, a particular viscosity is often desired both at low and at high shear rates, and these viscosities may be very different A key objective in the formulation and production of polymer dispersions is therefore to adjust the shear-rate profile to meet the demands of the particular appHcation. One way in which this can be achieved is by the addition of polymeric thickeners, which influence the viscosity at low and at high shear rates differently depending upon their stracture and molecular weight... 

Figure 2.17 Profiles for laminar flow of poly(styrene) melt at 453 K between two parallel plates separated by a distance of 10 cm. The imposed pressure gradient is —1 184 000 Pa/m, which in the absence of shear-thinning yields a centerline velocity of 10 cm/s (a) velocity profile of Newtonian (dash-dot) and shear-thinning (solid) fluids (b) linear shear-stress profile (c) shear-rate profile (d) viscosity profile.

The viscosity-shear rate curves of the samples are shown in Figure 1. The branched PVDF samples have a similar viscosity profile to those of the reference samples across the entire shear rate range. The change in the viscosity is mostly controlled by the molecular weight of the samples. The melt viscosity of the resin plays an important role in the first step of the extrusion blow molding process where the resin is processed under moderate shear rates well below the onset of melt instabilities. This is extremely important for the formation of a smooth parison and ultimately in a quality product. 

Figure 1 outlines the profiles of both velocity and shear rates of the laminates of polymers as they flow in a runner. Referencing the velocity profile present in this figure shows how the velocity at the mold wall is zero at the frozen layer (flow is stagnant) and increases dramatically until leveling off at the center of the flow channel. The sharp increase in velocity near the frozen layer contributes to the high shear rate in this area visible in the shear rate profile under the velocity profile, again present in Figure 1. This explains the dynamic decrease in viscosity present in this area.

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

Once the velocity profile has been obtained, the shear rate is calculated. This is the most difficult step. To ensure that the viscosity is determined without any bias, no assumption is made regarding the constitutive behavior of the material. Every effort is made to obtain smooth, robust values of the shear rate without any bias towards a particular model of the flow behavior. Particularly near the tube center, the velocity profiles are distorted by the discrete nature of the information. The size of a pixel is defined by the velocity and spatial resolutions. These are given by... 

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

Fig. 4.2.8 Shear viscosity versus shear rate data for a 0.6% by weight aqueous carboxymethylcellulose solution. Data from MRI were obtained from one combined measurement of a velocity profile and a pressure drop, (a) Cone and plate ( ) MRI.

A plot of Tvs. G yields a rheogram or a flow curve. Flow curves are usually plotted on a log-log scale to include the many decades of shear rate and the measured shear stress or viscosity. The higher the viscosity of a liquid, the greater the shearing stress required to produce a certain rate of shear. Dividing the shear stress by the shear rate at each point results in a viscosity curve (or a viscosity profile), which describes the relationship between the viscosity and shear rate. The... 

Fig. 3 Newtonian and non-Newtonian behaviours as a function of shear rate (a) flow profile (b) viscosity profile. (From Ref. 65.)...

Before the viscosity can be calculated from capillary data, as mentioned above, the apparent shear rate, 7 , must be corrected for the effect of the pseudoplastic nature of the polymer on the velocity profile. The calculation can be made only after a model has been adopted that relates shear stress and shear rate for this concept of a pseudoplastic shear-thinning material. The model choice is a philosophical question [11] after rheologlsts tried numerous models, there are in general two simple models that have withstood substantial testing when the predictions are compared with experimental data [1]. The first Is ... 

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