Phan-Thien And Tanner Model Research Articles

Numerical study on oblique stretching of viscoelastic polymer film

• Numerical format for oblique stretching of viscoelastic film is established. • The application scenarios of the membrane hypothesis are expanded. • Detailed numerical procedure and a complete numerical case are provided. • The influence of the y -direction stretching on thickness uniformity is investigated. Oblique stretching is a special approach for the fabrication of polymer film whose product is mainly used in the optical display field, such as compensation films, retardation films, et al. Nonetheless, the previous research reports on oblique stretching processing are still lacking, and the related numerical research is almost blank. This contribution uses the FEM (finite element method) to simulate the oblique stretching process of viscoelastic polymer film for the first time. The differential viscoelastic PTT (Phan-Thien and Tanner) model, one of the most realistic constitutive models is chosen. Furthermore, the polycarbonate melt was rheologically characterized and used as input material parameters in our simulations. Based on the membrane hypothesis and stabilization algorithms like DEVSS (discrete elastic viscous stress splitting), the specific numerical scheme is derived and the algorithm implementation is summarized. Finally, a complete numerical example of oblique stretching is demonstrated and compared with symmetric stretching. Based on the simulation results, the evolution of thickness and the influence of the y -direction stretching on thickness uniformity are further investigated.

Application of the PTT Model for Capillary Extrusion of Rubber Compounds

Abstract Rubber compounds have high viscoelastic property. One of the viscoelastic behaviors shown in profile extrusion is an extrudate swell and circulation flow at the corner of inside of die. Application of viscoelastic model to a capillary extrusion has been investigated in this study. Experiments and simulations have been performed using Fluidity Tester and commercial computational fluid dynamics (CFD) code, Polyflow respectively. Die swell of rubber compounds in a capillary die were predicted using a non-linear differential viscoelastic model, Phan-Thien and Tanner (PTT) model for various relaxation times and relaxation modes. As relaxation time and number of relaxation mode increase, die swell increases. The results of simulations were compared with the experiment. Pressure and velocity distributions, and circulation flows at the corner of reservoir have been analyzed through computer simulation. Two and three relaxation modes with large range of relaxation time examined in this study showed good agreement with experimental results of die swell and well represented circulation flow at the corner of reservoir in the capillary die.

Steady extrusion of viscoelastic materials from an annular die

The steady extrusion of viscoelastic materials from a straight, annular die is studied theoretically. The viscoelastic behavior is modelled using the affine Phan-Thien and Tanner (PTT) constitutive equation of the exponential form. For the numerical solution of the governing equations the mixed finite element method is combined with a quasi-elliptic mesh generation scheme in order to capture the large deformations of the two free surfaces of the extrudate. The elastic-viscous stress splitting technique (EVSS-G) is used to separate the elastic and viscous contributions to the polymeric part of the stress tensor together with a streamline upwind Petrov-Galerkin (SUPG) weighting for the discretization of the constitutive equation. This combination of solution methods and constitutive model allows us (i) to compute accurate steady-state solutions up to very high Weissenberg numbers resulting in very high deformations of the free surfaces (ii) construct and store the Jacobian matrix, which is necessary to conduct linear stability analysis for this flow. First, results for the fully developed flow of a PTT liquid inside an annular die are presented. They reveal a complex interplay between material elasticity, shear thinning and solvent viscosity. Next, a complete parametric analysis of annular extrusion is performed. Such a complete study using the PTT model has not been reported before, even at much lower Wi numbers. It is found that swelling of the material increases sharply up to moderate Weissenberg numbers, whereas its rate of increase is reduced for higher values of Wi, as shear thinning becomes increasingly important. The latter generally plays a crucial role, in addition to elasticity, on the swelling of the extrudate. Moreover, as the contribution of the solvent viscosity increases, the contribution of elastic stresses decreases causing a decrease in the swelling of the material which approaches the Newtonian limit. The predicted swelling ratios, which characterize the geometry of the extrudate, are in satisfactory agreement with earlier experimental and theoretical data for three particular HDPE resins.

The Phan-Thien and Tanner Model Applied to Thin Film Spherical Coordinates: Applications for Lubrication of Hip Joint Replacement

The Phan-Thien and Tanner (PTT) model is one of the most widely used rheological models. It can properly describe the common characteristics of viscoelastic non-Newtonian fluids. There is evidence that synovial fluid in human joints, which also lubricates artificial joints, is viscoelastic. Modeling the geometry of the total hip replacement, the PTT model is applied in spherical coordinates for a thin confined fluid film. A modified Reynolds equation is developed for this geometry. Several simplified illustrative problems are solved. The effect of the edge boundary condition on load-carrying normal stress is discussed. Solutions are also obtained for a simple squeezing flow. The effect of both the relaxation time and the PTT shear parameter is to reduce the load relative to a Newtonian fluid with the same viscosity. This implies that the Newtonian model is not conservative and may overpredict the load capacity. The PTT theory is a good candidate model to use for joint replacement lubrication. It is well regarded and derivable from molecular considerations. The most important non-Newtonian characteristics can be described with only three primary material parameters.

Modelling human hip joint lubrication subject to walking cycle

AbstractThis paper deals with the study of a hip joint prosthesis subject to a walking cycle. Human joints are lubricated by synovial fluid (SF). To properly model joint performance, one must take into account the behaviour of this fluid, which is highly viscoelastic as confirmed in early experimental work. The Phan‐Thien and Tanner (PTT) model is one of the most widely used rheological models. It can properly describe the common characteristics of viscoelastic non‐Newtonian fluids. Based on a microstructural description of the fluid, it is possible to establish laws of rheological behaviour to reproduce shock or the sudden inception of shear, taking into account large deformation and viscoelastic effects. Modelling the geometry of the total hip replacement, the PTT model is applied in spherical coordinates to a thin confined fluid film. Using a modified Reynolds equation developed for this specific geometry, we present a numerical simulation of an artificial hip joint subject to a walking cycle. The equilibrium position depends on the load magnitude and also depends strongly on the load direction. The non‐linear response of the SF leads to some unanticipated complexities. In our case, the dominant effect is the squeeze action between the surfaces. The contact behaviour depends not only on the geometry and kinematics but also on the fluid properties. The minimum film thickness remains higher than likely roughness height during these ‘normal’ conditions. When considering the design of hip joint prostheses, attention should be paid to the rheological properties of SF. Copyright © 2008 John Wiley & Sons, Ltd.

Pressure profiles along an abrupt 4:1 planar contraction

AbstractThe ability to accurately predict the pressure profile along an abrupt 4:1 planar contraction is investigated. Predicted pressure profiles obtained using the Phan Thien and Tanner (PTT) and generalized Newtonian fluid (GNF) models are compared to experimental measurements for low‐density polyethylene (LDPE) and linear low‐density polyethylene (LLDPE) polymer melts. The results obtained for the extensional strain hardening, LDPE resin show that numerical predictions are consistently less than the experimental values by 10–15%. Furthermore, no significant difference between the PTT and GNF predictions were observed over the range of convergent solutions. On the other hand, the numerical predictions for the LLDPE melt were within 1.5–7% of the experimental values over the entire range of stable flow rates investigated. Again, the differences in accuracy of the PTT and GNF models were small. In fact, the GNF model was generally more accurate than the PTT model for the case of LLDPE. Closer investigation of the predicted streamlines patterns near the contraction clearly shows larger, more intense vortices for the extensional strain hardening LDPE than for the LLDPE melt, which is in agreement with experimental observations. However, analysis of the plane of symmetry shows that the magnitude of the planar extensional stress is mitigated by the fluid relaxation behavior as the melt briefly passes through the contraction region. Although it appears that a viscoelastic constitutive equation is not necessary for predicting the pressure profile along an abrupt 4:1 planar contraction, this is a fortuitous result of the small extensional deformations observed in the contraction region over the limited range of convergent solutions.

Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model

A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm–Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge–Kutta method. The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter ϵ) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.

Rheology of acrylonitrile–butadiene–styrene polymer melts and viscoelastic constitutive models

Experimental shear and tensile stress growth coefficients for startup of steady shear and uniaxial elongation for a commercial grade acrylonitrile–butadiene–styrene (ABS) polymer melt are presented. One differential [Phan-Thien and Tanner (PTT)] and two Kaye–Bernstein–Kearsley–Zapas (K-BKZ) type single integral [Wagner and Papanastasiou–Scriven–Macosko (PSM)] nonlinear viscoelastic constitutive models were fit to shear and extensional experimental data. A comparison of the fit quality was performed, and the PTT model was found to result in the best overall quality fit, although deviations from linear viscoelastic behavior at small extensional strains could not be adequately described by the linear viscoelasticity theory without a time correction. Experimental data for stress relaxation after cessation of both steady shear and uniaxial elongation were compared with three constitutive model predictions in order to evaluate their anticipated accuracy in the prediction of residual stresses. The PTT model provided better predictions of ABS relaxational behavior than K-BKZ models except for shear stress relaxation in the nonlinear regime. Qualitative differences between experimental data and model predictions were found to be present for all models tested at large shear and extensional strains.

The confined flow of polyethylene melts past a cylinder in a planar channel

This paper is concerned with the comparison of the results of numerical simulation of confined flow past a cylinder to birefringence data for two polymer melts. The Phan-Thien and Tanner (PTT) constitutive equation and the Rivlin-Sawyers (RS) constitutive equation with the Papanastasiou, Scriven, and Macosko (PSM) damping function were each fit to the shear viscosity and extensional viscosity data of both linear low-density polyethylene (LLDPE) and low-density polyethylene (LDPE) melts to determine the values of the model parameters. Finite element calculations were carried out using the 4 × 4SUPG and 4 × 4SU methods for the PTT model and the method developed by Dupont, Marchal, and Crochet for the RS model. Isochromatic birefringence patterns calculated from the predicted stress field and the stress-optic law were compared to birefringence data. Good agreement was found between the birefringence data and the numerical predictions, except in the immediate vicinity of the cylinder surface. Large extensional stresses were observed and predicted along the centerline downstream of the cylinder for LDPE. This behavior was not observed or predicted for LLDPE. Stress fields obtained from birefringence measurements for LDPE flowing past three cylinders in a channel indicate an effect of deformation history on the flow behavior of LDPE. It is shown that the PTT model does not correctly predict the rheological behavior of LDPE as a function of shear history because the time scale of structural recovery is much longer than the relaxation time associated with viscoelasticity.

Viscoelastic flow past a confined cylinder of a polyisobutylene solution

Viscoelastic constitutive equations are evaluated using the benchmark problem of the planar flow past a confined cylinder for a well-characterized solution of 5%(w/w) polyisobutylene in tetradecane. The ratio of channel height to cylinder diameter is equal to two. We compare finite element simulations with point-wise measured velocities and stresses obtained by means of laser Doppler anemometry and a flow-induced birefringence technique, respectively. The Deborah number (De) ranges from 0.25 to 2.32. In the case of the geometry with a symmetrically confined cylinder, computations were made with a generalized Newtonian model and with both a single- and a four-mode Phan-Thien and Tanner (PTT) model. All model parameters were determined in simple shear flow. A similar analysis is presented in case of an asymmetrically confined cylinder (with De=1.87). Impressively good agreement was found between the predictions of the four-mode PTT model and the measured velocities and stresses. The agreement was even excellent in the geometry with the asymmetrically confined cylinder.

The use of large transient deformations to evaluate rheological models for molten polymers

The prediction of models proposed by Acierno et al. (ALMRT) and by Phan-Thien and Tanner (PTT) are compared with data for a low density polyethylene film resin and a high density blow molding resin. The deformations involved are interrupted shear, reduction in shear rate, large amplitude oscillatory shear and uniaxial extension. Both models predict characteristic times for ”reentanglement“ that are significantly larger than stress relaxation times, but that are still much smaller than observed values. The origin of this effect is shown to be quite different in the two models. The ALMRT model fails to predict the observed nonlinearity in the response to large amplitude oscillatory shear. The PTT model predicts nonlinearity and gives the correct value of the maximum stress, but the predicted shape of the stress curve is incorrect. Extensional stress growth data were not in agreement with the predictions of the ALMRT model except in the linear region. It was not possible to fit the PTT model to these data.