Multi-mode Constitutive Equation Research Articles

Non-isothermal analysis of extrusion film casting using multi-mode Phan-Thien Tanner constitutive equation and comparison with experiments

Extrusion film casting (EFC) is an industrially important process which produces thousands of tons of polymer films, sheets, and coating used for various industrial as well as household applications. In this paper, we focus on an instability which occurs during certain polymer processing operations operating under predominantly elongational flow, such as extrusion film casting and fiber spinning. This instability, called the draw resonance, occurs in the form of sustained periodic fluctuations in the film dimensions. It appears when the process goes beyond the critical line speed of the EFC process. In this work, a conventional linear stability analysis is carried out for nonisothermal EFC process to determine the onset of the draw resonance. The polymer rheology is modeled by the Phan-Thien Tanner (PTT) multi-mode constitutive equation. For the implementation, a conventional shooting method approach is used. Extrusion film casting experiments were also carried out using a conventional linear low-density polyethylene (LLDPE) by varying process parameters such as draw ratio and aspect ratio, to observe the effect on the stability of the process. Linear stability analysis results under non-isothermal conditions are compared and validated with existing results from literature and with our own experimental data. This work displays the effect of multiple relaxation modes as well as the temperature influence on the stability of EFC process. Finally, results also indicate that the temperature highly affects the stability of the EFC process and cannot be ignored from modeling of EFC process.

Nonisothermal analysis of extrusion film casting process using molecular constitutive equations

Extrusion film casting (EFC) is a commercially important process that is used to produce several thousand tons of polymer films and coatings. In a recent work, we demonstrated the influence of polymer chain architecture on the extent of necking in an isothermal film casting operation (Pol et al., J Rheol 57:559–583, 2013). In the present research, we have explored experimentally and theoretically the effects of long-chain branching on the extent of necking during nonisothermal film casting conditions. Polyethylenes of linear and long-chain branched architectures were used for experimental studies. The EFC process was analyzed using the 1-D flow model of Silagy et al. (Polym Eng Sci 36:2614–2625, 1996) in which the energy equation was introduced to model nonisothermal effects, and two multimode constitutive equations, namely the “extended pom-pom” (XPP, for long-chain branched polymer melts) equation and the “Rolie-Poly stretch version” (RP-S, for linear polymer melts) equation, were incorporated to account for the effects of polymer chain architecture. We show that the model does a better job of capturing the qualitative features of the experimental data, thereby elucidating the role of chain architecture and nonisothermal conditions on the extent of necking.

On the discrete EVSS method

This paper is a completion of a previous paper written by two of the authors [M. Fortin, R Guénette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] concerning the DEVSS method, which is a discrete variant of the popular elastic-viscous-split-stress (EVSS) method introduced by [Rajagopalan, R.A. Brown, R.C. Armstrong, Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity, J. Non-Newtonian Fluid Mech. 36 (1990) 159–199] within the context of viscoelastic fluids. The method uses the rate of deformation tensor as an additional variable. In this paper, it is shown that a simple modification of the theoretical framework introduced in the first paper [M. Fortin, R Guénette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] allows the treatment of multi-mode constitutive equations of viscoelastic flows as well as that of the velocity gradient introduced as the additional variable instead of the rate of deformation tensor. Moreover, this general framework based on a generalized Brezzi–Babuska theory, allows different discretizations for the stress and the additional variable. The paper presents a general criteria insuring the stability of the method. Three particular mixed finite elements satisfying this criteria are studied. An error analysis is performed on Stokes and viscoelastic flows.

Comparison of computational efficiency of flow simulations with multimode constitutive equations: integral and differential models

A decoupled finite-element method (INT/FEM) is presented for calculation of two-dimensional viscoelastic flows with integral constitutive models. The momentum and continuity equations are solved by Galerkin's method with the viscoelastic stress treated as a fixed body force. The viscoelastic stress is computed by using the stream function to track fluid particles upstream, integrating a system of ordinary differential equations that govern the displacement-gradient tensor, and evaluating the integral constitutive equation by numerical quadrature. The quasi-linear upper-convected Maxwell and Oldroyd-B models, as well as the nonlinear model of Papanastasiou, Scriven and Macosko (PSM), are used in the simulations. The efficiency of the integral method is compared to that of the recently developed finite-element method (EVSS/FEM) for differential constitutive models. Convergence and accuracy of the INT/FEM are shown by calculations for flow between eccentric cylinders. The upper limit in De attainable by using the INT/FEM is comparable to values for the EVSS/FEM only for constitutive models with a shear-thinning ratio of the first normal stress difference to the shear stress and a large solvent contribution to the solution viscosity. The INT/FEM becomes the more efficient technique for simulation with this type of constitutive equation when three or more relaxation modes are included in the memory function.