Zero shear rate viscosity, nonlinear

Some modifications of the melt flow behavior of thermoplastics that can be observed depending on filler concentration are a yield-like behavior (i.e., in these cases, there is no flow until a finite value of the stress is reached), a reduction in die swell, a decrease of the shear rate value where nonlinear flow takes place, and wall slip or nearwall slip flow behavior [14, 27, 46]. Other reported effects of flllers on the rheology of molten polymers are an increase of both the shear thinning behavior and the zero-shear-rate viscosity with the filler loading and a decrease in the dependence of the filler on viscosity near the glass transition temperature [18, 47-49]. 

It can be utilised in the same way as the linear response relation for the thermal conductivity, (3.8). One calculates the shear stress for a few different strain rates and extrapolates to zero strain rate. The strain rate must be large enough to prevail over the thermal fluctuations but not so large that one goes outside the linear regime. Outside the linear regime this relation can still be used to calculate nonlinear viscosities. 

Viscosity is defined as the ratio of shear stress to shear rate. The viscosity of a Newtonian fluid is a material constant that depends on temperature and pressure but is independent of the rate of shear that is, the shear stress is directly proportional to the shear rate at fixed temperature and pressure. Low molar-mass liquids and aU gases are Newtonian. Complex liquids, such as polymers and suspensions, tend to be non-Newtonian in that the shear stress is a nonlinear function of the shear rate. Some typical melt viscosities are shown in Figure 1.7. The viscosity approaches a constant value at low shear rates, known as the zero-shear viscosity and denoted... 

The rubberlike liquid model is able to predict, qualitatively, certain nonlinear viscoelastic phenomena. In particular, some effects arising from the finite orientation of chain segments are predicted, for example a nonzero first normal stress difference. However, it fails to describe many other nonlinear effects. For example, it predicts that the viscosity is constant with shear rate and the second normal stress difference is zero. In fact, all its predictions for the shear stress in simple shear are the same as those of the Boltzmann superposition principle. We can gain some insight into the origins of nonlinearity by examining the features of the rubberlike liquid model that limit its predictive ability. 

Just as the Deborah number is often used to plot data or predictions for transient tests at finit rates, the Weissenberg number (Wi) is used for a similar purpose for representing stresses that are independent of time. This is defined as the product of a time scale governing the onset of nonlinearity, let s call it /I, and the rate of strain of the experiment. We will see in the next section that a fluid that has a shear-rate dependent viscosity must have at least one material constant with units of time. For example, one might select the reciprocal of the shear rate at which the viscosity falls to 60% of its zero-shear value. This time then characterizes the nonlinearity of the behavior for this flow. Obviously, the degree to which a melt deviates from Newtonian behavior depends on how this time constant compares with the rate of the deformation. Thus, the Weissenberg number in simple shear flow is defined as ... 

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