Upper convected Maxwell model

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

For the upper-convected Maxwell model, the full equations for reads... 

Remark 4.4 No result such as Theorem 4.2 seems to be known for Maxwell models. We however have to mention the result [44], where the upper-convected Maxwell model in the whole space R is considered. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

An upper-convected Maxwell model has been used with the full relaxation spectrum for the calculation of the stress, but for calculating the birefringence this spectrum has been restricted to long relaxation times as shown in Fig. 12. The model predictions for the data of the Fig. 9 are shown in Fig. 13. The deviations from the linear stress-optical nole are well accounted for by this very simple model. However, the model does not describe the stress data in simple elongation, and in particular the initial stress values at temperatures close to the Tg. 

Bagley (1992a) measured the apparent biaxial elongational viscosity of wheat flour dough. The upper convected Maxwell model was considered to be adequate in explaining both the effect of crosshead speed and sample... 

Bagley, E. G., Christianson, D. D., and Martindale, J. A. (1988). Uniaxial compression of a hard wheat flour dough Data analysis using the upper convected Maxwell model. J. Text. Stud. 19, 289-305. 

Apelian, M. R., e. a. (1988). hnpad of the constitutive equation and singularity on the calculation of stick-slip flow The modified upper-convected maxwell model, /. Non-Newtonian Fluid Mech. 27 299-321. 

Christenson and McKinley [ 19] evaluated a generalized linear Maxwell model as well as the upper convected Maxwell model and the Giesekus model. These authors worked with the tensorial forms of these functions which are capable of correctly treating large strain deformations. 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

The rheological predictions that derive from this simple molecular model are very similar to the upper-convected Maxwell model see example 4.3.3. Recall that t = Tj + tp = r , 2D + Xp. We obtain Steady shear viscosity... 

This model, often referred to as the upper convective Maxwell model, is weakly non-linear in that it predicts a first normal stress, but no shear thinning effects, i.e, the shear stress increases linearly with shear rate so that the viscosity is independent of shear rate. Combining Eqs. 2, 4, 5 and 6, we see that the tube model predicts the viscosity to be. 

At low shear rates i.e., yr < 1, the second term on the left hand side of Eq. 92 can be neglected, and the expression becomes an exact differential version of the upper convected Maxwell model, hi steady state shear flow Eq. 92 gives each component first normal stress as. 

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. 

It is seen from Eq. (3.50) that the Lodge model predicts virtually the same form for material functions as the upper convected Maxwell model does (see Eq. (3.6)). 

Let us consider the upper convected Maxwell model given by Eq. (3.4). Since we are only interested in small-amplitude oscillations with Uj = Vi(t,x2), all nonlinear terms appearing in the convected derivative of stress tensor a (see Eq. (2.107)) can be neglected and thus Eq. (3.4) reduces to the classical Maxwell equation, Eq. (3.3). Applying Eq. (3.79) to (3.3) we obtain ... 

Figure 3.11 gives plots of n /irjo versus A.jC that are predicted from two constitutive equations (1) the upper convected Maxwell model, and (2) the Oldroyd three-constant model. It is seen in Figure 3.11 that both models predict values of increasing very rapidly without bound as e increases, in contrast to the experimental results given in Figure 3.10. As a matter of fact, all the expressions summarized in Table 3.3 predict similar elongational behavior, which is considered to be physically unrealistic.

Giesekus (1982) summarized nicely a series of his papers on the formulation of a new class of constitutive equations. The origin of Eq. (3.23) comes from a modification of the upper convected Maxwell model as applied to a dilute polymer solution, namely... 

A) 100, and ( ) 10. The values of X, defined by Eq. (3.14), were determined by curve fitting the experimentally obtained log versus log y plot to the theoretical prediction of the modified upper convected Maxwell model, Eq. (3.12). 

This is also the integral form of the differential constitutive equation called the upper convected Maxwell model , which is given in the next section. 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

Koscher and Fulchiron [140] assumed to be proportional to the first normal stress difference Ni, which they calculated with an upper-convected Maxwell model. Although a larger HMW tail leads to longer time scales in the relaxation spectrum, and therefore enhances the development of the first normal stress difference, this approach does not emphasize the importance of the longest molecules as much as when the nucleation rate... 

The major effect of polymer cooling is that it retards stress relaxation and, as mentioned previously in the chapter, some of the stress remains frozen-in even after the molding has completely solidified. This stress relaxation cannot be predicted using an inelastic constitutive equation (why ) the simplest equation that we can use for the purpose is the upper-convected Maxwell model. In the absence of flow, the use of this model yields... 

Thus far, we have limited our discussion to the melt spinning of Newtonian liquids. For materials that are more elastic than PET, this restriction needs to be relaxed. The simplest way to do this is by using the upper-convected Maxwell model introduced in Chapter 14. In this case, the equation equivalent to Eq. (15.4.13) is... 

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