Planar extensional viscosity

M. Padmanabhan and M. Bhattacharya, Planar Extensional Viscosity of Corn Meal Dough, J. Food Eng., 18 389-414 (1993). 

Fig. 3.7 (a) Uniaxial, (b) equibiaxial, and (c) planar extensional viscosities for a LDPE melt. [Data fromP. Hachmann, Ph.D. Dissertation, ETH, Zorich (1996).] Solid lines are predictions of the molecular stress function model constitutive equation by Wagner et al, (65,66) to be discussed in Section 3.4. 

By applying force balances over a cone or wedge-shaped element, Cogswell (1972) developed relationships between pressure drop and stress. For a sudden planar contraction (6 = 90° in Figure 3-29), the apparent planar extensional viscosity is (Padmanabhan and Bhattacharya, 1993) ... 

Padmanabhan, M. and Bhattacharya, M. 1993. Planar extensional viscosity of com meal dough. J. Food Eng. 18 389 11. 

Three main types of elongational flow are uniaxial, biaxial, and planar. Although resistance to flow can be referred to loosely as an elongational or extensional viscosity (which further depends upon the type of elongational flow), this parameter generally is not constant. 

In the inflation stage, the molten or thermally softened polymer is subjected to the action of the gas pressure. One analysis [13] that involved minimal strain rates showed that a planar extensional viscosity y pe (kg/ms) could be correlated with strain rate ( pe). The relationship between these quantities was... 

Extensive reviews [6-10] and a monograph [11] summarize the literature covering significant aspects of extensional flows in various commercial processes, theoretical treatment for ttie hydrod)mamics of such flows and different methods of determining material functions such as uniaxial, biaxial and planar extensional viscosities. 

If we want to find out how a fluid behaves under extension, we have to somehow grip and stretch it. Experimentally, this is much more difficult than the shear arrangement, especially if the fluid has a low viscosity. Earlier (see Section 5) we saw that it is possible to classify steady extensional flows under the categories of uniaxial, biaxial and planar flows. We will now examine uniaxial testing, since this mode is more commonly employed as a routine characterization tool. Here we encounter two approaches the first seeks to impart a uniform extensional field and back out a true material function, while the second employs a mixed flow field that is rich in its extensional component (e.g. converging flows) and use it to back out a measured property of the fluid which is somehow related to its extensional viscosity. 

Figure 5.9 Transient uniaxial extensional (A), planar extensional ( ), and shear viscosity ( ) of an 11-mode pom-pom melt in start-up compared with measurement results for LDPE (shear/ elongation rate, 0.01/s temperature, 140 °C) [5]...

The sink flow analysis, which assumes a purely extensional flow (i.e., no shear component), was presented by Metzner and Metzner (1970) to evaluate the extensional viscosity from orifice Apen measurements. For an axisymmetric contraction, the flow into the orifice is analogous to a point sink for a planar contraction flow, the analogy is with a line sink (Batchelor, 1967). In the case of axisymmetric contraction (Figure 7.8.1), the use of spherical coordinates and continuity gives the velocity components... 

Figure 6 shows a direct comparison of the experimentally determined vortex shapes and the corresponding predictions for different mass flow rates at a temperature of 180 C (the contraction ratio is 1 19 in this case). In Figure 6a, the experimental vortex shape line color corresponds to the simulated ones. The conq)arison between the measured and the calculated mass flow rate dependent vortex size is depicted in Figure 6b. It is nicely visible that the vortex size versus the mass flow rate is non-monotonic (runs through a maximum), which can be attributed to the non-monotonic behavior of the planar extensional viscosity.

The effect of the melt temperature on the vortex size development has been studied experimentally as well as theoretically. The most important results are depicted in Figme 7. It is obvious, that the vortex area primarily increases, reaches a maximum and then it decreases again with increasing temperature. This behavior can be explained by the temperature dependency of the nonmonotonic shape of the planar extensional viscosity predicted by e improved mWM model, which is depicted in Figure 8. In more detail, the planar extensional viscosity maximum moves fi om low extensional strain rates to higher ones for increasing melt temperatures. This seems to be the driving mechanism for the maximum appearance in the vortex size vs. temperature flmction. 

Viscoelastic FEM analyses of the vortices of LDPE have been performed for different temperatures, flow rates and flow geometries. The theoretical results were compared with corresponding experimental data. It has been suggested that for the LDPE melt the planar extensional viscosity can be different (slightly lower) compared to uniaxial extensional viscosity at the maximum of the steady extensional viscosity curve. It seems that the non-monotonic function of the vortex size on the temperature for a constant mass flow rate can be explained by the temperature dependency of the planar extensional viscosity curve. Finally, it has been found that the proposed modification of the mWM model significantly improves the model behavior in the planar extensional flows. 

Figure 5 Comparison of original and improved mWM model predictions of planar extensional viscosity.

Figure 8 Improved mWM model prediction of the temperature dependent planar extensional viscosity.

Figure 9.7 Photographs of droplet shapes in planar extensional flow for various viscosity ratios M of the dispersed to the continuous phase. The droplets are viewed in the plane normal to the velocity gradient direction. The critical capillary numbers Cac and droplet deformation parameters Dc at breakup are also given. The droplet fluids are silicon oils with viscosities ranging from 5 to 60,000 centistokes, while the continuous fluids are oxidized castor oils both phases are Newtonian. (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

---