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... 

Here we have three parameters r/o the zero-shear-rate viscosity, Ai the relaxation time and A2 the retardation time. In the case of A2 = 0 the model reduces to the convected Maxwell model, for Ai = 0 the model simplifies to a second-order fluid with a vanishing second normal stress coefficient [6], and for Ai = A2 the model reduces to a Newtonian fluid with viscosity r/o. If we impose a shear flow,... 

Steady shearfree flow for the White-Metzner model. This model is a nonlinear model which modifies the convected Maxwell model by including the dependence on 7 in the viscosity, i.e.,... 

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 ... 

It can be shown [8] that Hadamard instabilities are possible for admissible motions if a is in the interval (—1,1), e.g., in extensional flows. On the other hand, restrictions on the eigenvalues of r prevent Hadamard instabilities for a = 1. This is immediately seen from the integral forms qf (4)-(5) for the upper- and lower-convected Maxwell models, which imply constraints on the eigenvalues of the Cauchy-Green tensors. (See, for instance, [12].)... 

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. 

Equation (6.52) has the exact form of a convected Maxwell model. Therefore, many results from continuum mechanics can be applied to Eq. (6.52) directly. The relevant aspects of continuum mechanics are discussed in Appendix 6.A. The rheological tensors in Eq. (6.52) are all evaluated by following a particular fluid particle at any current moment t. Since a particular fluid point is followed, t may be replaced by a past time t. ... 

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. 

SO that for Marshall and Mentzner s definition a" = 0.5. Marshall and Men tzner (1964), using the contra variant convected Maxwell model to obtain a relaxation time for their polymer solutions, noticed the onset of viscoelastic behaviour at a Deborah number between 0.1 and 1.0 for flow through sintered bronze. Some of their results for various flexible coil polymers are shown in Figure 6.10. 

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). 

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