Steady-State Simple Shear Flow

For the case of small velocity gradients, the variables xfk and u k can be found in the form of an expansion in powers of velocity gradients. The first terms are defined by equations (7.28) and (7.39) for the case of weakly entangled systems (x X 0.1) and by equations (7.32) and (7.43) for the case of strongly entangled systems (x X 0.1). 

Further on, we shall consider the case of shear stress when one of the components of the velocity gradient tensor has been specified and is constant, namely V12 0. This situation occurs in experimental studies of polymer solutions (Ferry 1980). In order to achieve such a flow, it is necessary that the stresses applied to the system should be not only the shear stress a 12, as in the case of a linear viscous liquid, but also normal stresses, so that the stress tensor is 

The shear stress o 2 and the differences between the normal stresses o —0x2 and 022 — 033 are usually measured in the experiment (Meissner et al. 1989). The results of calculation of the stresses up to the third-order terms with respect to the velocity gradient will be demonstrated further on. For simplicity, we shall neglect the effect of anisotropy of the environment when the case of strongly entangled systems will be considered. 

In steady-state shear, when the only component of the velocity gradient tensor differs from zero is z/12, equation (9.19) is followed by 

The third-order terms in shear stress give us the expression for the effective shear viscosity 

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If the particular product is not very non-Newtonian, a single-point measurement might be sufficient. Even then however we must be sure that the flow type is the same. For instance if our experience off-line is built up from steady-state, simple-shear flow curves, then a too-fast measurement might mean the measurement is not made under steady-state conditions. Equally if the on-hne flow has an appreciable extensional component, then problems can arise for some hquids, especially pol3nner solutions. Also if a vibrational mode is used, then it is probable that some non-hnear oscillatory function is being measured. Ah these facts could mean that we end up with no simple one-to-one correlation between on-hne and off-hne. Hence the safest way is to duphcate on-hne what is done off-hne. This is possible nowadays for most situations. 

A consequence of finite deformations is the appearance of normal stresses in simple shearing deformations. Thus, even in steady-state simple shear flow (Fig. 1-16) where the rate of strain tensor (c/. equations 3 and 5) is... 

Let us now consider steady-state simple shear flow, for which we have the velocity field of the form... 

To illustrate the usefulness of the various forms of rate tensors we have introduced, let us consider steady-state simple shear flow whose velocity field is given by Eq. (2.41) and whose motion is given by Eq. (2.44). Use of Eq. (2.45) in (2.57) gives... 

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. 

In Eqs. (2.111) and (2.112) the vorticity tensor

VISCOELASTICITY OF FLEXIBLE HOMOGENEOUS POLYMERIC LIQUIDS 53 functions in steady-state simple shear flow ... 

It can easily be shown that in steady-state simple shear flow, the use of the contravariant convected derivatives of a and d in Eq. (3.9) yields... 

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 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. 

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