Extensional Thickening Models

A problem with this model is that it gives effects in uniaxial extension only. In planar extension we also expect thickening, but here 11 ho = 0, as shown in Exercise 2.8.1. 

Typically nsUho) is a shear thinning function of the Cross or Carreau form and tjuilho) is an extensionally thickening function of the same form (but n 1). The ratio Wni/S = 1 in simple shear and 0 in pure extension. 

The approaches above can describe only steady state, time-independent viscosities. In Chapter 4 we will show that for time-dependent viscoelastic models, like Maxwell s, extensional thickening arises naturally. 

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In fact we have lost something over Chapter 1. The Finger tensor B is able to give us normal stresses and extensional thickening. We will have to wait until Chapter 4 to get these factors back into our models. But in the next chapter we will see how to bring in the phenomenon of time dependence, which is so important for polymeric systems. We should note that for concentrated suspensions, especially flocculated systems, there is little elastic recovery, and time dependence is often either very short or extremely long. Thus the viscous models of this chapter are often quite adequate (recall Figures 2.5.3 and 2.5.4 and look ahead to Chapter 10). 

If shear thinning is the main phenomenon to be described, the simplest model is the general viscous fluid. Section 2.4. It has no time dependence, nor can it predict any normal stresses or extensional thickening (however, recaU eq. 2.4.24). Nevertheless, it should generally be the next step after a Newtonian solution to a complex process flow. The power law. Cross or Carreau-type models are available on all large-scale fluid mechanics computation codes. As discussed in Section 2.7, they accurately predict pressure drops in flow through channels, forces on rollers and blades, and torques on mixing blades. 

Figure 10.19 shows the extensional viscosity data of Laun and Munstedt [187] for several low-density polyethylenes made in tubular reactor. We note that these branched polymers are Newtonian at low strain rates, become extension thickening at higher strain rates, and finally exhibit extension thinning. This is similar to the behavior observed by Kurzbeck ef a. [ 184] for a crossKnked polypropylene. In Chapter 11, it will be shown that the pom-pom model predicts this type of extensional viscosity curve. 

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