Phan-Thien/Tanner 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. 

Considering these previous remarks, two network models, thought to be representative of each class of equation, have been investigated, namely the Wagner model and the Phan-Thien Tanner model,... 

At least, it is worth noticing that the Phan Thien Tanner model is, in its mathematical form, a non-separable equation. However, it has been pointed out that, for some special forms of the relaxation spectrum, apparent separability may be displayed [61]. 

Two different constitutive equations, namely the Wagner model and the Phan Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in numerical calculation in complex flows. Additionally, advantages and problems related to the use of these equations have been reviewed. 

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

Theorem 6.1 is still valid for models with several relaxation times, as well as for models with a quadratic in r, such as the Giesekus and the linearized Phan-Thien-Tanner models. (See [110])... 

In pioneering work, Keunings and Crochet [18] used the Phan-Thien Tanner model, with an additional purely viscous retardation term to correct the unrealistic rapid viscosity decrease predicted by this model in simple shear flow. The improvement of convergence with a re ed mesh was found to be opposite to that observed in previous munerical studies. This improved convergence mi t be due to the Phan-Thien Tanner model itself or to the adcbtional retardation term, which appears to stabilize the solution. 

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]. 

Two equations have been selected for the viscoelastic extra-stress component a generalized Oldroyd-B model (GOB) and a multimode Phan-Thien Tanner model (mPTT). The veilues of the corresponding parameters are given in sub-section 3-2... 

A fixed point algorithm permits to adjust, at each iteration, the value of ext rpjjg same kind of method is also used for the multimode Phan-Thien Tanner model. 

The streamline patterns are quite identical for both constitutive equations. However, the vortex is more pronounced for the multimode Phan-Thien Tanner model, whereas the swelling is greater for the generalized Oldroyd-B model... 

The velocity aloi the symmetry axis is significantly different an overshoot at the contraction as well as an undershoot downstream of the die exit are observed with the generalized Oldroyd-B model, but only a smooth overshoot is indicated by the multimode Phan-Thien Tanner model. The final value of the velocity after swelling is more important for the PTT, which is consistent with the lower value of swelling observed in Fig 20. 

Figure 9.7. Steady shear and normal stress data at 170 °C for two polyethylenes (a) low-density Resin 10 and (b) high-density Resin 22. The lines are the fit to the Phan-Thien/Tanner model, discussed below. Reprinted with permission from Tsang and Dealy, /. Non-Newtonian Fluid Mech., 9, 203 (1981). Copyright Elsevier.

Figure 9.13. Comparison of the Phan-Thien/Tanner model to the extensional data of Meissner. Reprinted with permission from Phan-Thien and Tanner, J. Non-Newtonian Fluid Mech.,...

Table 10.1. Phan-Thien/Tanner model parameters used by Verbeeten and co-workers for DSM Stamylan LD 2008 XC43 LDPE M = 1.55 X 10, = 11.9) cUT = 110 °C...

Best overall fit of the Larson model (left), and the Phan Thien-Tanner model (right) to data for a high density polyethylene (circles) in step shear (a), in steady elongation (b), and in step biaxial extension (c). Adapted from Khan and Larson (1987). 

Differential models obtained by replacing the ordinary time derivative in Eq. 10.21 by one that can describe large, rapid deformations are able to describe some nonlinear viscoelastic phenomena, but only qualitatively. To improve on such models, it is necessary to introduce additional nonlinearity into the equation. In the popular Phan-Thien/Tanner model, the Gordon-Schowalter convected derivative is used, and nonlinearity is introduced by multiplying the stress term by a function of the trace of the stress tensor. The Giesekus and Leonov models are other examples of nonlinear differential models. All of the models mentioned above are described in the monograph by Larson [7j. 

FIGURE 3.13 Comparison of the predictions for viscosity for the White-Metzner, upper convected Maxwell, and Phan-Thien-Tanner models with experimental data for polyst5 rene (Styron 678, Dow Chemical Company) at 190 °C. (Data from Gotsis, 1987.)... 

TABLE 3.4 Predicted Values of Ni, NiHxyx, and Extrudate Swell (Hp/Ho) for LDPE (NPE 953) at 170 °C Using the Phan-Thien-Tanner Model... 

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