0
Research Papers: Flows in Complex Systems

Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics

[+] Author and Article Information
Mehmet Yildiz

Faculty of Engineering and Natural Sciences,
Advanced Composites & Polymer
Processing Laboratory,
Sabanci University,
34956 Tuzla, Istanbul, Turkey

Manuscript received September 2, 2011; final manuscript received January 5, 2013; published online April 8, 2013. Editor: Malcolm J. Andrews.

J. Fluids Eng 135(5), 051103 (Apr 08, 2013) (18 pages) Paper No: FE-11-1354; doi: 10.1115/1.4023645 History: Received September 02, 2011; Revised January 05, 2013

This work presents the development of a weakly compressible smoothed particle hydrodynamics (WCSPH) model for simulating two-dimensional transient viscoelastic free surface flow which has extensive applications in polymer processing industries. As an illustration for the capability of the model, the extrudate or die swell behaviors of second-order and Olyroyd-B polymeric fluids are studied. A systematic study has been carried out to compare constitutive models for second-order fluids available in literature in terms of their ability to capture the physics behind the swelling phenomenon. The effects of various process and rheological parameters on the die swell such as the extrusion velocity, normal stress coefficients, and Reynolds and Deborah numbers have also been investigated. The models developed here can predict both swelling and contraction of the extrudate successfully. The die swell of a second-order fluid was solved for a wide range of Deborah numbers and for two different Reynolds numbers. The numerical approach was validated through the solution of fully developed Newtonian and non-Newtonian viscoelastic flows in a two-dimensional channel as well as modeling the die swell of a Newtonian fluid. The results of these three benchmark problems were compared with analytic solutions and numerical results in literature when pertinent, and good agreements were obtained.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

The simulation domain in particle representation

Grahic Jump Location
Fig. 2

The particle distribution on the die channel whose length and width are L = 0.2 m and H = 0.01 m along the x- and y-directions, respectively

Grahic Jump Location
Fig. 3

A schematic for enforcing zero pressure condition on the free surface

Grahic Jump Location
Fig. 4

Velocity profile at the middle of the channel

Grahic Jump Location
Fig. 5

Fully developed non-Newtonian second-order periodic flow

Grahic Jump Location
Fig. 6

The die swell of Newtonian fluid where color indicates the velocity magnitude

Grahic Jump Location
Fig. 7

SPH particle distribution for eight possible forms of momentum balance formulations. The results on the left column are obtained by using the Rivlin–Ericksen constitutive equation while those on the right column are obtained by employing the CEF constitutive equation, where (a) covariant convected derivative, (b) contravariant convected derivative, (c) mixed covariant-contravariant convected derivative, and (d) corotational (Jaumann) derivative. Note that in each figure, the particle distributions are colored in accordance with the values of the first normal stress difference N1, and the magnitude of the centerline velocity v is given on each subfigure. The results are obtained using the solution procedure (case 3) expounded below.

Grahic Jump Location
Fig. 8

Simulation results for case 3 (left column) and case 4 (right column), where (a) the magnitude of the velocity (m/s), (b) the normal component Txx of the extra stress tensor (Pa), (c) the normal component Tyy of the extra stress tensor (Pa), (d) the shear component Txy of the extra stress tensor (Pa), and (e) the first normal stress difference N1 (Pa)

Grahic Jump Location
Fig. 9

(a) The magnitude of the velocity versus the axial distance, (b) the magnitude of the velocity as a function of channel width, (c) the shear component Txy of the extra stress tensor, (d) the normal component Txx of the extra stress tensor, (e) the normal component Tyy of the extra stress tensor, and (f) the first normal stress difference N1. Results are shown for x = 0.185 m.

Grahic Jump Location
Fig. 10

Simulation results for the extrudate contraction (left column) and swelling (right column), where (a) the magnitude of the velocity (m/s), (b) the normal component Txx of the extra stress tensor (Pa), (c) the normal component Tyy of the extra stress tensor (Pa), (d) the shear component Txy of the extra stress tensor (Pa), and (e) the first normal stress difference N1 (Pa)

Grahic Jump Location
Fig. 11

(a) The magnitude of the velocity versus the axial distance, (b) the magnitude of the velocity as a function of channel width, (c) the shear component Txy of the extra stress tensor, (d) the normal component Txx of the extra stress tensor, (e) the normal component Tyy of the extra stress tensor, and (f) the first normal stress difference N1. Results are shown for x = 0.185 m.

Grahic Jump Location
Fig. 12

The results of the extrudate simulation with fine particle numbers for convergence analysis, (a) the shape of the extrudate with colors denoting the velocity magnitude, m/s, and (b) the magnitude of the velocity as a function of channel width, (c) the shear component Txy of the extra stress tensor. Results are shown for x = 0.185 m.

Grahic Jump Location
Fig. 13

Summary of the die swell simulation with the centerline velocity of 2 m/s, where (a) the magnitude of the velocity (m/s), (b) the normal component Txx of the extra stress tensor (Pa), (c) the normal component Tyy of the extra stress tensor (Pa), (d) the shear component Txy of the extra stress tensor (Pa), and (e) the first normal stress difference N1 (Pa)

Grahic Jump Location
Fig. 14

(a) The magnitude of the velocity versus the axial distance, (b) the magnitude of the velocity as a function of channel width, (c) the shear component Txy of the extra stress tensor, (d) the normal component Txx of the extra stress tensor, (e) the normal component Tyy of the extra stress tensor, and (f) the first normal stress difference N1. Results are shown for x = 0.185 m.

Grahic Jump Location
Fig. 15

Particle distributions with colors denoting values of the first normal stress difference N1 for Re numbers of 5 and 10, and for various De numbers. The left and right columns for subfigures from (a) to (e) correspond to the centerline velocity of v = 5 m/s and v = 10 m/s, respectively. On the other hand, the subfigures (f) and (g) have the centerline velocity of v = 10 m/s. Here, (a) De = 0, (b) De = 0.02, (c) De = 0.1, (d) De = 0.2, (e) De = 0.3, (f) De = 0.4, and (g) De = 0.5.

Grahic Jump Location
Fig. 16

The swelling ratios at different axial positions x = 0.21, 0.23, and 0.25 m for different centerline velocities, (a) v ≅ 5 m/s, (b) v ≅ 10 m/s, respectively, (c) averaged swelling ratio for v ≅ 5 m/s and 10 m/s

Grahic Jump Location
Fig. 17

The left and right columns, respectively, correspond to the centerline velocities of 5 and 10 m/s, where (a) the shear component Txy of the extra stress tensor, (b) the normal component Txx of the extra stress tensor, (c) the normal component Tyy of the extra stress tensor, and (d) the first normal stress difference N1. Results are shown for x = 0.185 m.

Grahic Jump Location
Fig. 18

The summary of the WCSPH simulation for Weeffect=0.1 in terms of (a) the velocity magnitude, (b) the first normal stress difference, and (c) the shear component of the extra stress tensor stress. Graphical results are given for x = 0.180 m.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In