Phan-Thien

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

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. 

Phan-Thien N, Atkinson JD, Tanner RI (1978) J Non-Newt Fluid Mech 3 309... 

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. 

N. Phan-Thien. A nonlinear network viscoelastic model. J. Rheol., 31(8) 259-283, 1978. 

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,... 

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. 

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. 

A study was made of the ability of viscoelastic models to describe the measured material functions of unplasticised PVC during extrusion and to determine whether it was possible to reproduce the elastic properties of the large entrance pressure drop and small extrudate swell during the extrusion of PVC using a capillary rheometer. Models used were the Phan-Thien and Tanner model and the K-BKZ-Wagner model with a single exponential damping... 

Wu X, Phan-Thien N, Fan X, Ng TY. (2003) A molecular dynamics study of drop spreading on a solid surface. Phys Fluids 15 1357-1362. 

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 ... 

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. 

The first kind of modification to the UCM model that may be conceivable is that of the convected derivative. This leads one to consider that the motion of the network junctions is no more that of the continuum and thus, the afiine assumption of the Lodge model is removed. Among the various possibilities, Phan Thien and Tanner suggested the use of the (Jordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative ... 

Provided that a suitable Y function is chosen, this is claimed to give a correct picture of many phenomena displayed in simple shear and uniaxial elongational flows. One should note that the original model of Phan-Thien and Tanner uses the non-af ne derivative together with nonlinear stress term. 

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. 

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. 

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. 

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. 

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. 

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