Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method Vivek Ganvir a,b , Ashish Lele c , Rochish Thaokar b , B.P. Gautham a a Tata Research Development and Design Centre, Pune, India b Department of Chemical Engineering, Indian Institute of Technology, Mumbai, India c Polymer Science and Engineering Division, National Chemical Laboratory, Pune, India Keywords: Arbitrary Lagrangian Eulerian Free surface simulations Extrudate (die) swell abstract Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown signif- icant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite ele- ment formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of mate- rial motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations. 1. Introduction Extrudate swell prediction of a polymer melt is important for appropriate die design in processes such as profile extrusion. Pre- diction of extrudate swell is a challenging task due to simulation of the free surface, which requires special techniques in the tra- ditionally used Eulerian framework. The degree of swell depends on material functions such as the first normal stress difference coefficient and the coupling between the shear and extensional viscosities, which are modeled by complex constitutive equations having convective derivatives of stress tensor. This makes the simu- lation of polymer flow in an Eulerian framework even more difficult. A better understanding of the flow behavior and resulting swell will ultimately lead to improvements in the understanding of the extrusion process, for optimization of both die design and process- ing parameters. This aspect has attracted a great deal of attention from researchers working on experimental and numerical studies of extrudate swell of molten polymer. In the 1980s, several experimental studies on extrudate swell of polymer melts were performed, [1,2] and semi-empirical cor- relations were established by relating the swell ratio to the first normal stress difference (N 1 ). However, with the development of numerical techniques for viscoelastic flow, several detailed investi- gations on the extrudate swell phenomenon have been carried out using simulations. Many of these early numerical simulations were undertaken using the upper-convected Maxwell (UCM) constitu- tive equation [3,4]. Later, more realistic constitutive equations such as the integral K-BKZ model [5–13] and the differential PTT model