Material Functions for Steady-State Shear Flow

1 Material Functions for Steady-State Shear Flow 

In the preceding sections, we have presented the material functions derived from various constitutive equations for steady-state simple shear flow. During the past three decades, numerous research groups have reported on measurements of the steady-state shear flow properties of flexible polymer solutions and melts. There are too many papers to cite them all here. The monographs by Bird et al. (1987) and Larson (1988) have presented many experimental results for steady-state shear flow of polymer solutions and melt. In this section we present some experimental results merely to show the shape of the material functions for steady-state shear flow of linear, flexible viscoelastic molten polymers and, also, the materials functions for steady-state shear flow predicted from some of the constitutive equations presented in the preceding sections. 

Meister Bird-Carreau Coleman-Noll second-order 0 (e 0) (P + 2v)y2 

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Appendix 12B Derivation of Three Material Functions for Steady-State Shear Flow from Equation (12.30)... 

It is seen that Eq. (3.35) becomes identical to Eq. (3.15) only when the symmetric part of tensor x (x - 2i Qd) disappears in other words, when x - 2i Qd = 0. The Giesekus model is very versatile in that it has an additional parameter, the so-called mobility parameter a, which allows for more accurate predictions of the material functions in steady-state shear flow as compared with the ZFD model. This will become clear later in this chapter when we present logarithmic plots of r versus y, and logarithmic plots of Alj versus y based on predictions from both the Giesekus and ZFD models. In Chapter 6 of Volume 2 we will use a modified version of the Giesekus model to simulate the high-speed melt spinning process. 

For steady-state shear flow, three material functions, t] y), N y), and be expressed by... 

In Chapter 3, we show that the contravariant and covariant components, respectively, of the convected derivative of the stress tensor give rise to different expressions for the material functions in steady-state simple shear flow. When compared with experimental data, it turns out that the material functions predicted from the contfavariant components of the convected derivative of the stress tensor give rise to a correct trend, while the material functions predicted from the covariant components of the convected derivative of the stress tensor do not. 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l < n < 2) as y increases further above a certain critical value. 

Note that since m(s) and a( i, 2) are functions only of time y, then t]q, y3, and v are constants. A material that can be represented by the constitutive equation given in Eq. (3.76) is called a Coleman-Noll second-order fluid (Coleman and Markovitz 1964 Truesdell and Noll 1965). For steady-state simple shear flow, Eq. (3.76) yields... 

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10. 

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