**Comparison between different viscosity models**

Viscosity measurements are performed with a high pressure capillary rheometer at discrete shear rates. In transition zones of the real process, or especially in flow simulation, a mathematical function of the slope of viscosity, as a function of shear rate, is necessary to calculate points in-between the measured data. Therefore, models have be developed to describe the mathematical connection of the measured viscosity data, as a function of shear rate. The models have different strengths and weaknesses and most are only suited for specific materials. The following list of models is restricted to those that are integrated in our standard Capillary Rheometer Software.

**Ostwald – de Waele (power law)**

Especially elastomers can be described quite well by this simple model, which describes the slope of the shear thinning area. For thermoplastic materials it is less suitable, because of the Newtonian plateau and transition zone. It must only be applied in the pseudo plastic (shear thinning) range, because the curvature of the viscosity in the transient area is not described. If drawn with double logarithmic axis, the model is a straight line.

**Carreau – Winter**

The 3–Parameter expression also covers the so-called Newtonian Plateau, transition and shear thinning area. The viscose flow behavior is described in a physically meaningful way over a wide shear rate region, especially in the region of larger shear rates.

**Yasuda (modified Carreau – Model)**

Yasuda is an extended Carreau model, which describes the viscosity of branched thermoplastic materials. These kinds of materials show a typical curvature of the viscosity in the transient area between zero-shear viscosity and pseudo plastic (shear thinning) area, which can be described with the additional exponent. Typically, this model also fits quite well at low and high shear rates. As the model fits well over the whole measured shear rate, extrapolation is also possible with a high level of confidence.

**Muenstedt**

This 4-Degree polynomial expression is a mathematic approximation of the viscosity function. The function typically matches the experimental data very well, but because of its purely mathematic nature, data extrapolation is not advised.

**Cross**

The cross model is similar to the Carreau-Winter Model. The exponent is applied differently, which leads to a slightly different form of the curve.

**Comparison of models with HDPE**

Figure shows the Ostwald model calculated from measurement data with HDPE, drawn with log-log scaling. The graph shows, why it makes little sense to use the Ostwald model over the whole shear rate range of this thermoplastic HDPE test. The typical curvature at the onset of the shear-thinning region cannot be modelled accurately.

The Carreau model shown in Figure 3 fits the data more closely and resembles the curve quite well. Still, there are some smaller discrepancies between the measured data and the model. The Cross model also fits the measured data quite well, with smaller discrepancies especially at higher shear rates.

In a direct comparison the Cross model is more accurate at low shear rates, while the Carreau model is more accurate for high shear rates. The curvature of the models are slightly different, because of the different position of the exponent in the denominator of the equations.

Figure 4 shows the Muenstedt and Yasuda model, calculated from measurement data with HDPE. Both models perfectly fit the data. While Muenstedt is a purely mathematical approximation of the data, Yasuda’s equation is physically plausible.

**Extrapolation of data**

Figure shows the extrapolation of data using the Carreau and Cross models. The results with Carreau seem to be a bit more plausible at high shear rates. At low shear rates, the Cross model results seem to be more accurate.

Compared to Cross and Carreau, Muenstedt and Yasuda fit the data more closely and are therefore better suited for data extrapolation, at least with this material. Compared to Muenstedt, Yasuda calculates slightly lower viscosities at low shear rates, and slightly higher ones, at high shear rates. For extrapolation Yasuda seems to be the most suited model because of its accurate data replication and physically plausible equation.

The easiest way to select the best model is to compare the correlation coefficient, also given with the model coefficients. The ideal fit is given by a correlation coefficient of 1.