# Anisotropic Extrudate Swell from a Slit Die: A Velocity-Centre Hypothesis and Numerical Verification

## TL;DR

This paper introduces a new theoretical model to understand and predict anisotropic extrudate swell in polymer processing using a velocity-centre hypothesis and numerical simulations.

## Contribution

The paper proposes a novel semi-empirical framework to isolate geometric and kinematic effects in extrudate swell using a multi-velocity-centre hypothesis.

## Key findings

- Analytical predictions align well with CFD data (error < 5%) in the core zone of high-aspect-ratio dies.
- Velocity decay near die edges causes local deviations due to 3D flow kinematics.
- The model provides a mathematical basis for inverse die-profile design and shape compensation.

## Abstract

While anisotropic extrudate swell in polymer processing is fundamentally driven by physical viscoelastic recovery, this paper proposes a theoretical framework to explicitly isolate and map the purely geometric and kinematic components of this phenomenon. Serving as a mathematical proof-of-concept, a multi-velocity-centre hypothesis is proposed. By introducing a semi-empirical, lumped material-flow calibration parameter, the macroscopic diameter swell ratio is mathematically extended to the discrete local flow field of a rectangular slit die. To evaluate its validity, the analytical framework is subjected to a numerical test for kinematic consistency utilizing isothermal, inelastic power-law fluid CFD simulations, thereby separating geometric mapping from complex viscoelastic stress relaxation. Results indicate that analytical predictions show good agreement with CFD data (error < 5%) strictly within the core zone of high-aspect-ratio dies. However, due to the infinite-slit assumption, 3D flow kinematics near die edges induce velocity decay, leading to local deviations that require future empirical corrections. Although comprehensive physical extrusion experiments and non-isothermal viscoelastic coupling are required for industrial deployment, this semi-empirical kinematic mapping provides a foundational mathematical basis that could potentially inform future inverse die-profile design and shape distortion compensation.