White-Metzner model

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

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.,... 

Develop expressions for the steady shear viscometric functions for the White-Metzner model. 

Figure 9.41 presents the predicted secondary flow patterns that result from the vicoelastic flow effects. The Giesekus model with one relaxation time was used for the solution presented in the figure. For the simulation, a relaxation time, A, of 0.06 seconds was used along with a viscosity, r], of 8,000 Pa-s and a constant a of 0.80. Similar results were achieved using the Phan-Thien Tanner-1 model. As expected, when the White-Metzner model was used, a flow without secondary patterns was predicted. This is due to the fact that the White-Metzner model has a second normal stress difference, N2 of zero. 

Hakim [25] has shown that Theorem 3.1 is also true for White-Metzner models, where T = satisfies... 

A similar result has been proven by Hakim [39] for a cleiss of White-Metzner models (still with e < 1) under suitable assumptions on the constitutive functions A(II) and t (I1). These assumptions are satisfied in particular by the Carreau smd the Gaidos-Darby laws [40]. 

A similar result for White-Metzner models is proven in [25] under the same hypotheses on the constitutive functions as for the local existence result mentioned earlier, plus the hypothesis x) < M, for all x in R+. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

Example 3.1. Shear Flow Predictions for the White-Metzner Model... 

Calculate the material functions for the White-Metzner model in simple shear flow (i.e., = Yxy(t)y, Vj, = =... 

Finally, one other approach is to use the White-Metzner model (see Table 3.1), as at least the viscosity function can be fit to the viscosity data (parameters for the Carreau viscosity model are found using nonlinear regression and are rio = 23,000 Pa-s, n = 0.587, and A = 19.7 s). Values of N cannot... 

TABLE 3.6 Predicted Values ofiVi,iVi/2Tj, t, and Thickness SweU Hp/H ) for LDPE (NPE 953) Using the White-Metzner Model... 

The constitutive equation used by Denn and co-workers (Denn et al., 1975 Fisher and Denn, 1976) was the White-Metzner model (see also Section 3.2.1 Eq. 3.42), which was thought to be applicable to high Deborah number processes such as melt spinning. The zz and rr components of the constitutive equation in cylindrical coordinates are... 

Vijay, R., Deshpande, A.P., and Varughese, S. 2008. Nonlinear rheological modeling of asphalt using White-Metzner model with structural parameter variation based asphaltene structural build-up and breakage. Appl. Rheol. 18 23214—23228. 

The vortex development of a low-density polyethylene in different flat dies under various proeessing conditions has been analyzed by the Finite Element Method enploying the modified White-Metzner model as constitutive equation. The theoretical results are conpared with the velocity distributions measured by Laser-Doppler Velocimetry (LDV). 

Table 1 Modified White Metzner model parameters for LDPE Lupolen 1840H melt at 150 °C.

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