Phan-Thien And Tanner Research Articles

Effect of die exit flow conditions on air-gap film dynamics in two-dimensional film casting processes: Short Communication

The impact of the inlet velocity profile at the die exit on the film dynamics within the air-gap region was investigated using numerical simulations for both Newtonian and viscoelastic Phan-Thien and Tanner (PTT) fluids in isothermal two-dimensional (2-D) film casting processes. In an industrial context, intentional adjustments were made to reduce the inlet velocities at the edge of the casting die, effectively mitigating the edge-beads characterized by a higher edge thickness than the center thickness of the final films. By varying the inlet velocity conditions with decreasing edge velocities, the steady film dynamics were correlated with the onsets of draw resonance instability and frequency responses to a disturbance, which were determined using the transfer function data obtained under tension-controlled conditions. The results revealed that decreasing the inlet velocity at the edge improved not only the formation of films but also the process stability for both Newtonian and viscoelastic fluids. In addition, the sensitivity or frequency response to a disturbance was effectively reduced by decreasing the inlet velocity at the edge. This observed impact of the inlet velocity was closely related to the increased tension levels at take-up and a greater portion of the neck-like deformation type in the air-gap region.

Lubrication approximation of pressure-driven viscoelastic flow in a hyperbolic channel

We investigate theoretically the steady incompressible viscoelastic flow in a hyperbolic contracting channel. The fluid viscoelasticity is modeled using the upper convected Maxwell (UCM), Oldroyd-B, Phan-Thien and Tanner (PTT), Giesekus, and the finite elasticity non-linear elastic dumbbell with the Peterlin approximation (FENE-P) models. We first develop the general governing equations for flow within a non-deformable channel whose cross section varies with the distance from the inlet. We then exploit the classic lubrication approximation, assuming a small aspect ratio of the channel to simplify the original governing equations. The final equations, which we formulate in terms of the stream unction, are then solved analytically using a high-order asymptotic scheme in terms of the Deborah number, De, and the formulas for the average pressure drop are derived up to eight orders in De. The accuracy of the original perturbation solution is enhanced and extended over a wide range of parameters by implementing a convergence acceleration method for truncated series. Furthermore, convergence of the transformed solutions for the average pressure drop is demonstrated. The validity and accuracy of the theoretical results is independently confirmed through comparison with numerical results from simulations performed using high-order finite differences and pseudospectral methods. The results reveal the decrease in the average pressure drop with increasing the Deborah number, the polymer viscosity ratio, and the ratio of the inlet to the outlet height. We also show that the fundamental UCM and Oldroyd-B models can predict the major viscoelastic phenomena for this type of internal and confined lubrication flows, while the effect of the rheological parameters of the PTT, Giesekus, and FENE-P models on the results is minor.

New insights into the extended and generalised PTT constitutive differential equations: weak flows

This work presents a comparison between the PTT-X (extended Phan-Thien and Tanner (PTT)) and the generalised PTT (gPTT) viscoelastic models. The PTT-X model was derived based on a combination of reptation and network theories, allowing in this way a microstructural justification for the kernel function. The gPTT model is based on the network theory, with an empirical kernel function for the rate of destruction of junctions, that proved to be effective fitting experimental rheological data for polymer melts and solutions. A review on the background of both models is provided and the two models are then compared considering simple flows. This comparison allows one to attribute in some way a microstructural nature to the parameters involved in the gPTT model. Also, a new analytical solution is derived for the Poiseuille flow of the PTT-X model.

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.

A study on the boundary of linear viscoelasticity in simple shear flows: model calculations

Linear viscoelastic region is important to investigate material properties. We investigated boundary of linear viscoelasticity for simple shear flows with various time functions of shear strain. We used Phan-Thien and Tanner (PTT) model and Giesekus model in order to extract the linearity criterion. We determined critical strain (γc) as a function of dimensionless number such as De or Wi and the nonlinear parameter of the models. We found a superposition of the critical strain at start-up shear flow and oscillatory shear flow. Replacing relaxation time by mean relaxation time ( $$\left({\overline {\rm{\lambda }} = {J_{\rm{e}}}{{\rm{\eta }}_{\rm{o}}}} \right)$$ ), we checked the validity of the equation with experimental data.

A generalised Phan–Thien—Tanner model

In this work we propose a novel generalised form of the Phan–Thien and Tanner (PTT) model by considering a new functional form of the nonlinear expression characterizing the destruction of physical network junctions and entanglements. This new function of the trace of the stress tensor is given by the generalized Mittag–Leffler function, and contains the familiar exponential form of the original Phan–Thien and Tanner model as a limiting case, but affords additional fitting flexibility through the inclusion of one or two additional fitting constants. We perform fits to experimental data in shear and extension and show that this generalized expression allows a better description of the rheological responses for a range of complex materials such as polymer melts and semidilute polymer solutions. By using an appropriate information criterion, we also demonstrate that the resulting generelized model remains parsimonious but provides improved fits of real world data.

The Phan-Thien and Tanner Model Applied to the Lubrication of Knee Prostheses.

This work aims to provide a contribution to determine a proper model for the study of fluid film lubrication for the reduction of knee prostheses failure due to polyethylene wear. The Phan-Thien and Tanner (PTT) rheological law and the elastic deformation of the articular surfaces were considered in this modeling. The governing equations were solved numerically for different geometries and different Weissenberg numbers. The lubrication approximation applied to the PTT rheological law leads to an expression for the apparent viscosity similar to the Cross model. The results attest the importance of considering the non-Newtonian behavior of the synovial fluid, the elastic deformation, and the geometrical features of the prostheses to obtain quantitative information.

Operability windows in viscoelastic slot coating flows using a simplified viscoelastic-capillary model

The viscoelastic-capillary model to predict approximately coating windows for the stable operations of viscoelastic coating liquids is derived using a lubrication approximation in slot coating processes. Pressure distributions and velocity profiles for viscoelastic liquids based on the Oldroyd-B and Phan-Thien and Tanner (PTT) models are solved in the coating bead region considering the Couette-Poiseuille flow feature and the pressure jumps at upstream and downstream menisci. Practical operating limits for the uniform coating of rheologically different liquids that are free from leaking and bead break-up defects are constructed under various conditions, incorporating the position of the upstream meniscus as an important indicator while determining limits. The shift of the uniform operating range shows different patterns for the Oldroyd-B liquid with a constant shear viscosity and the PTT liquid with a shear-thinning nature in comparison with the Newtonian case. The windows predicted by the simplified model are corroborated with experimental observations for one Newtonian and two viscoelastic liquids.

Steady viscoelastic film flow over 2D Topography: II. The effect of capillarity, inertia and substrate geometry

We examine the two-dimensional, steady flow of a viscoelastic film under the action of gravity over a substrate with periodic topographical features. We account for the rheology of the viscoelastic material using the exponential Phan–Thien and Tanner (PTT) constitutive model. The conservation equations are solved via the mixed finite element method combined with a quasi-elliptic grid generation scheme, while the viscoelastic stresses are discretized using the EVSS-G/SUPG method. Our scheme allows the computation of accurate steady-state solutions up to high values of Deborah, Reynolds and capillary numbers. We perform a thorough parametric analysis to investigate the effect of the elastic, capillary and inertia forces on the flow characteristics. Our results indicate that surface tension and elasticity affect the film closer to the location with abrupt changes of the substrate topography; the sizes of the capillary ridge before a step down and of the depression before a step up are increased and move upstream as fluid elasticity or interfacial tension increase. It is shown that under creeping flow conditions the length scale of the capillary ridge increases with De following a power law of ¼, which can also be predicted by simple scaling arguments. Inertia has a more global effect on the film affecting larger portions of it, while in its presence the length scale of the capillary features is not affected significantly by the material elasticity. Moreover, it is shown that similarly to the case of Newtonian liquids, high inertia causes the formation of a ridge just after the step up. We also explore the effect of the geometrical characteristics of the substrate as well as its inclination angle and it is shown that the interface shape becomes more deformed as the topography appears wider, deeper or it approaches the vertical plane.

The Inverse Prediction for Profile Extrusion Die Based on the Finite Element Method

Considering the extrudate swell, the polymer extrusion process was calculated by the inversed simulation based on the visco-elastic ecology theory. The fluid characteristics of the polymer melt were described by the Phan-Thien and Tanner (PTT) model. The Finite Element Method was used. Based on the simulation data, the extrusion die lips were analyzed. So it is feasible to design the polymer extrusion die lips using inversed simulation method.

Predictions of flow behaviors and entrance pressure drop characteristics of a rubber compound in a capillary die using various rheological models

Rubber compounds have highly viscoelastic properties. The viscoelastic behaviors that have been exhibited during die extrusion include die swell and vortices in regions of sudden contraction. In this study, the application of rheological models to the capillary die extrusion process is investigated. Experiments and simulations were conducted using a fluidity tester and finite element analysis, respectively. The velocity distributions, velocity profiles, pressure drops, and vortices at the capillary die entrance were analyzed through computer simulations for various viscoelastic models [i.e., Phan‐Thien and Tanner (PTT), Giesekus, POMPOM, simplified viscoelastic, and generalized Newtonian models]. Different models exhibited different pressure drops and different velocity profiles in the capillary die. Only the full viscoelastic models (PTT, Giesekus, and POMPOM) predicted the vortex at the corner of the reservoir that is the capillary die entrance. However, the simplified viscoelastic and generalized Newtonian models did not predict the vortex. All the viscoelastic models studied in this article predicted the die swells in various ways, and these were compared with the experimental results. The PTT and simplified viscoelastic models exhibited good agreement with the experimental results of the die swells. POLYM. ENG. SCI., 54:2441–2448, 2014. © 2013 Society of Plastics Engineers

Numerical investigation of the thermally and flow induced crystallization behavior of semi-crystalline polymers by using finite element–finite difference method

The thermally and flow induced crystallization behavior of semi-crystalline polymer in processing can significantly influence the quality of final products. The investigation of its mechanism has both scientific and industrial interest. A mathematical model in three dimensions for thermally and flow induced crystallization of polymer melts obeying differential Phan-Thien and Tanner (PTT) constitutive model has been developed and solved by using the finite element–finite difference method. A penalty method is introduced to solve the nonlinear governing equations with a decoupled algorithm. The corresponding finite element–finite difference model is derived by using the discrete elastic viscous split stress algorithm incorporating the streamline upwind scheme. A modified Schneider's approach is employed to discriminate the relative roles of the thermal state and the flow state on the crystallization phenomenon. The thermally and flow induced crystallization characteristics of polypropylene is investigated based on the proposed mathematical model and numerical scheme. The half crystallization time of polypropylene in a cooled couette flow configuration obtained by simulation are compared with Koscher's experimental results, which show that they agree well with each other. Two reasons to cause crystallization of polypropylene in pipe extrusion process including the thermal state and the flow state are investigated. Both the crystalline distribution and crystalline size of polypropylene are obtained by using the finite element–finite difference simulation of three-dimensional thermally and flow induced crystallization. The effects of processing conditions including the volume flow rate and the temperature boundary on the crystallization kinetics process are further discussed.

Modeling the melt blowing of viscoelastic materials

Several outstanding issues concerning modeling of melt blowing of viscoelastic materials are addressed in this work. Using a slender-jet model for the melt-blown fiber, we probe the effects of rheology by considering Newtonian, upper-convected Maxwell, Phan-Thien and Tanner (PTT), and Giesekus constitutive equations. The effects of a non-uniform shear stress along the fiber length and heat transfer are also addressed. Our results suggest that by combining the slender-jet approach with a Giesekus (or PTT) constitutive equation, useful engineering predictions can be made concerning the final fiber diameter, even when assuming a constant shear stress and neglecting heat transfer. Finally, questions related to linear stability, nonlinear dynamics, and sensitivity are explored. Steady-state fiber profiles are found to be linearly stable, and numerical simulations indicate that the predictions from linear theory can be carried over into the nonlinear regime. Sensitivity analysis reveals that disturbances are likely to become especially amplified at particular frequencies, with elasticity reducing the magnitude of the amplification but broadening the spectrum of frequencies susceptible to large amplification. This suggests an explanation for the narrower fiber diameter distributions that are observed experimentally.

Numerical Investigation of Viscoelastic Flow and Swell Behaviors of Polymer Melts in the Hollow Profile Extrusion Process

The viscoelastic flow and swell behaviors of polymer melts in the profile extrusion process can significantly influence the performance and dimension of the final products. In the study, the viscoelastic flow pattern of a commercial low density polyethylene (LDPE) extruded through out of the hollow profiled extrusion die is investigated by means of finite element simulation. The mathematical model of three-dimensional viscoelastic flow and swell of polymer melts is established with a differential Phan-Thien and Tanner (PTT) constitutive model. A penalty method is employed to solve the non-linear problem with a decoupled algorithm. The computation stability is improved by using the discrete elastic-viscous split stress (DEVSS) algorithm with the inconsistent streamline-upwind (SU) scheme. A streamface-streamline method is introduced to adjust the swelling free surface of the extrudate. The essential viscoelastic flow characteristics of LDPE flowing through out of the hollow profile extrusion die is investigated based on the proposed numerical scheme. Both the redistribution of flow velocity and the release of stress are found to be the reasons for the swell phenomenon.

On the stick-slip flow from slit and cylindrical dies of a Phan-Thien and Tanner fluid model. I. Steady state

The steady planar and cylindrical stick-slip flows for a viscoelastic fluid are computed using the Phan-Thien and Tanner (PTT) constitutive model. The mixed finite element method is used in combination with the elastic-viscous stress-splitting technique and the streamline upwind Petrov–Galerkin discretization for the constitutive equation. This combination of methods when applied to the PTT constitutive model allows us to compute steady state solutions up to high Weissenberg numbers; practically without an upper limit. Equally important, the global Jacobian matrix is generated in order to be able to perform a linear stability analysis of the computed steady state. The dependence of the steady solutions on all the problem parameters is examined. In the limit of a Newtonian fluid, the expansion coefficients near the singularity are computed with comparable accuracy to those from previous analytical and numerical studies, which include the singular finite element method. In the case of a viscoelastic liquid, it is shown that the computed solutions converge quadratically with mesh refinement even at the exit plane of the die and also locally very close to the singularity. The form of the converged solution near the singularity is examined as well as its dependence on various rheological parameters. It is shown that the singularity at the die exit is a logarithmic one and always integrable. Under such conditions our calculations can be extended to determine the linear stability of the herein computed steady states.

Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, \(\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0\), where \(\widetilde{\eta}_p\) and \(\widetilde{\eta}_s\) are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and \(\xi \ne 0\); (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, \(\tilde{R}\), and flow rate on the velocity profile, the stress components, and the film thickness, \(\tilde{H}\), is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, \(De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}\), and Stokes, \(St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}\), numbers, depend on \(\tilde{H}\) and the average film velocity, \(\widetilde{U}\). This makes necessary a trial and error procedure to obtain \(\tilde{H}\)a posteriori. We find that increasing De, ξ, or the extensibility parameter e increases shear thinning resulting in a smaller St. The Stokes number decreases as \({\tilde{R}}/{\tilde{H}}\) decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when \({\tilde{R}}/{\tilde{H}}\) decreases down to unity for a film on the inner surface. When \(\xi \ne 0\), an upper limit in De exists above which a solution cannot be computed. This critical value increases with e and decreases with ξ.

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.