Phan-Thien/Tanner equation

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

The original Phan Thien Tanner equation was written using simultaneously both modifications Gordon Schowalter derivative and segment kinetics term. The segment kinetics term (exponential form) enables a more... 

The ability of the Phan Thien Tanner equation and related models for the prediction of data in shear and elongation has been investigated. Attention has been focused on special simplified cases of the original equation which enable the imderstanding of the influence of each parameter. 

At least, using the complete Phan Thien Tanner equation, with non-affine motion and modified kinetics enables a correct description of the data in shear and in elongation. However, the parameters that can be determined for this model are bound to be some compromise. This is necessary in order to minimize the deviation to the Lodge-Meissner rule, due to the use of the Gordon-Schowalter derivative. This is also required to give adequate description of both the shear and uniaxial elongational behaviour. Additional undesirable phenomena in some flows have also been pointed out such as oscillations in transient flows. 

Though the Wagner and Phan Thien Tanner equations seem to give adequate description of the observed behaviour either in shear or in uniaxial elongation, it is worth mentioning some peculiarities and key points that should keep the attention of the user to avoid misleading conclusions. 

In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the... 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

In this approach the contribution to the extra stress of the matrix and the rod-matrix interaction is captured using a multi-mode viscoelastic constitutive relation. For the model predictions in the paper, we chose to use the Phan-Thien Tanner equation (PTT) [13]. The ability to indirectly capture the rod-matrix interaction is based on the presence of the fiber retarding the long relaxation... 

For each r, a constitutive equation must be selected. Dooley and Dietsche [5] evaluated the White-Metzner, the Phan-Thien Tanner-1, and the Giesekus models, given by,... 

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

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,... 

A differential constitutive equation the Phan-Thien Tanner mod ... 

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. 

One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay bounded—as in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-Thien-Tanner constitutive equation [50]. 

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... 

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... 

Table 9.4. Components of the Phan-Thien/Tanner Fluid, Equation 9.23, for two-dimensional flows...

The Phan-Thien/Tanner constitutive equation does not represent the state of the art in modeling melt flow at the time of this writing, but it is adequate to illustrate the response of melts of flexible polymers in complex flows and it has a mathematical structure that does not differ substantively from other equations with a firmer basis in molecular theory. Furthermore, it has been widely used in simulation studies to date. Hence, we will use it for illustrative purposes in this text, recognizing that it is likely to be replaced as the preferred constitutive equation for applications. The minimum rheological information required for simulations is thus the temperature-dependent linear viscoelastic spectrum and the temperature-dependent viscosity as a function of shear rate. Extensional data should be used, but they are often unavailable when the PTT equation is employed it is therefore common to select a reasonable value of to describe the extensional response. 

Here we have removed the overbars from the averaged quantities and made use of the fact that trr = ree. (This equality does not hold for hoUow-fiber spinning.) The steady-state equations for each mode of a Phan-Thien/Tanner fluid (Table 9.4) are as follows ... 

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. 

The shear viscosity for the PVDF compoimd has been measured and fitted by a 3-mode Phan Thien-Tanner (PTT) viscoelastic model [6], as shown in Figure 2. The constitutive equation of the PTT model has the form ... 

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

The various changes that may be carried out can be either on the convected derivative or in the right term of equation (35) or both these imply the removal of some assmnptions of the initial model. Such a possible modification, that was claimed to give a correct description of the essential phenomena of the nonUnear viscoelastic behaviour of polymer melts, is that proposed by Phan Thien and Tanner [44-46] involving the use of a special convected derivative and special kinetics of the junction. 

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