1. A COMPUTATIONAL STUDY OF THE EFFECT OF EXIT GEOMETRY ON SHARKSKIN INSTABILITY IN THE EXTRUSION OF A NON-NEWTONIAN VISCOELASTIC POLYMER Amanda Pascoe (amanda.pascoe@furman.edu) and Jill Roth (jill.r.roth@vanderbilt.edu) Research Mentor: Mark Sussman (sussman@math.fsu.edu) Research Experience for Undergraduates in Computational Math, CSIT, Florida State University, Tallahassee, Florida 32306-4120

2. This study investigates sharkskin instability in the extrusion of a non- Newtonian, viscoelastic polymer. Our goal is to determine if the geometry at the die exit causes sharkskin. We alter the inflow rate and relaxation time of the polymer. Our numerical approach is the coupled level set volume-of-fluid (CLSVOF) method. ABSTRACT

3. BACKGROUND INFORMATION During the extrusion of polymers, a series of instabilities occurs that affects its manufacturing. One must choose between reducing the extrusion rate or sacrificing the quality of the extrudate. Sharkskin, the first instability, is of particular interest as it occurs at relatively low extrusion rates. Characterized by surface roughness, sharkskin consists of a regular pattern of fractures.

4. During extrusion, polymeric materials often expand to a diameter larger than that of the die. At sufficiently high flow rates, this swelling can lead to sharkskin in the extrudate. Although the source of sharkskin is still unknown, there is general agreement, supported by both experimental and numerical analysis, that the location of the initiation of sharkskin is at the die exit. Migler identifies a cohesive failure at the die exit, where the polymer splits into a surface layer and a core layer. Then the surface layer bulges and pinches off from the core layer. Studies have identified a stress singularity at the corner which we attempt to eliminate by altering the exit geometry. BACKGROUND INFORMATION (continued)

5. Sharkskin Instability http://polymers.msel.nist.gov/highlights/image/sharkskinfig1.jpg

6. APPROACH The coupled level set volume-of-fluid (CLSVOF) method combines the advantages of both the level set (LS) method and the volume-of-fluid (VOF) method. These methods track the solid/liquid and liquid/gas interfaces. The VOF function measures the fraction of gas and liquid in each cell of a grid. The LS function represents the distance from any cell center to a free surface. Our study uses two LS functions: Φ measures to the die geometry and Ψ measures to the surface of the advecting liquid.

7. LEVEL SET FUNCTIONS Ψ<0 Ψ>0 Φ>0 Φ<0

8. ADAPTIVE MESH REFINEMENT

9. APPROACH (continued) Rather than using a body fitted grid method as in previous studies, our simulations utilize a Cartesian grid that cuts through the geometry. An adaptive mesh refinement (AMR) algorithm magnifies the die exit region through a hierarchical grid system. The levels of refinement range from coarsest at the lowest layer to most refined at the top.

10. GOVERNING EQUATIONS Given that gravity is ignored, the Navier-Stokes equations appear as the following, Here L is the diameter of the nozzle, and U is the inflow velocity. After a change of variables, we have

11. PARAMETERS This study uses the physical properties of polyethylene to model a non- Newtonian, viscoelastic fluid.

12. GEOMETRY We alter the geometry of the die exit from a sharp corner to a rounded corner whose curvature is controlled by the length of a radius. Sharp Corner Round Corner

13. SIMULATIONS During the simulations, the inflow velocity is varied. It has been shown that sharkskin is present at sufficiently high inflow rates. Altering the inflow rate effects the Deborah number (De) and the capillary number (Ca).

14. 2D SIMULATIONS η p = 12636 (g/cm s) Square η p =12636 (g/cm s) Inflow rate = 2mm3/s Step 45600 Square η p =12636 (g/cm s) Inflow rate = 10 mm3/s Step 84300 Round η p =12636 (g/cm s) Inflow rate = 10 mm3/s Step 84300

15. 2D SIMULATIONS η p = 17755 (g/cm s) Square Inflow rate = 2mm3/s Step 26700 Round Inflow rate = 2mm3/s Step 26700 Square Inflow rate = 10 mm3/s Step 26700 Round Inflow rate = 10mm3/s Step 26700

16. 3D GEOMETRY