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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.

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References

Liang, Y. C., Oztekin, A., and Neti, S., 1999, “Dynamics of Viscoelastic Jets of Polymeric Liquid Extrudate,” J. Non-Newton. Fluid, 81(1–2), pp. 105–132. [CrossRef]
Joseph, D. D., Matta, J. E., and Chen, K. P., 1987, “Delayed Die Swell,” J. Non-Newton Fluid, 24(1), pp. 31–65. [CrossRef]
Mitsoulis, E., Vlachopoulos, J., and Mirza, F. A., 1984, “Numerical-Simulation of Entry and Exit Flows in Slit Dies,” Polym. Eng. Sci., 24(9), pp. 707–715. [CrossRef]
Mitsoulis, E., 1986, “The Numerical-Simulation of Boger Fluids: A Viscometric Approximation Approach,” Polym. Eng. Sci., 26(22), pp. 1552–1562. [CrossRef]
Mitsoulis, E., 1999, “Three-Dimensional Non-Newtonian Computations of Extrudate Swell With the Finite Element Method,” Comput. Method. Appl. M., 180(3–4), pp. 333–344. [CrossRef]
Gast, L., and Ellingson, W., 1999, “Die Swell Measurements of Second-Order Fluids: Numerical Experiments,” Int. J. Numer. Meth. Fl., 29(1), pp. 1–18. [CrossRef]
Ahn, Y. C., and Ryan, M. E., 1991, “A Finite-Difference Analysis of the Extrudate Swell Problem,” Int. J. Numer. Meth. Fl., 13(10), pp. 1289–1310. [CrossRef]
Tome, M. F., Doricio, J. L., Castelo, A., Cuminato, J. A., and McKee, S., 2007, “Solving Viscoelastic Free Surface Flows of a Second-Order Fluid Using a Marker-and-Cell Approach,” Int. J. Numer. Meth. Fl., 53(4), pp. 599–627. [CrossRef]
De Paulo, G. S., Tome, M. F., and McKee, S., 2007, “A Marker-and-Cell Approach to Viscoelastic Free Surface Flows Using the PTT Model,” J. Non-Newton. Fluid., 147, pp. 149–174. [CrossRef]
Mitsoulis, E., Vlachopoulos, J., and Mirza, F. A., 1985, “Simulation of Vortex Growth in Planar Entry Flow of a Viscoelastic Fluid,” J. Appl. Polym. Sci., 30(4), pp. 1379–1391. [CrossRef]
McKee, S., Tome, M. F., Ferreira, V. G., Cuminato, J. A., Castelo, A., Sousa, F. S., and Mangiavacchi, N., 2008, “Review the MAC Method,” Comput. Fluid., 37, pp. 907–930. [CrossRef]
Gingold, R. A., and Monaghan, J. J., 1977, “Smooth Particle Hydrodynamics: Theory and Application to Non-Spherical Stars,” Mon. Not. R. Astron., 181, pp. 375–389.
Rook, R., Yildiz, M., and Dost, S., 2007, “Modeling Transient Heat Transfer Using SPH and Implicit Time Integration,” Numer. Heat Tr B-Fund, 51(1), pp. 1–23. [CrossRef]
Monaghan, J. J., and Kocharyan, A., 1995, “SPH Simulation of Multi-Phase Flow,” Comput. Phys. Comm., 87, pp. 225–235. [CrossRef]
Monaghan, J. J., Huppert, H. E., and Worster, M. G., 2005, “Solidification Using Smoothed Particle Hydrodynamics,” J. Comput. Phys., 206(2), pp. 684–705. [CrossRef]
Rook, R., and Dost, S., 2007, “The Use of Smoothed Particle Hydrodynamics for Simulating Crystal Growth From Solution,” Int. J. Eng. Sci., 45(1), pp. 75–93. [CrossRef]
Seoa, S., and Min, O., 2006, “Axisymmetric SPH Simulation of Elasto-Plastic Contact in the Low Velocity Impact,” Comput. Phys. Commun., 175, pp. 583–603. [CrossRef]
Shao, S. D., and Lo, E. Y. M., 2003, “Incompressible SPH Method for Simulating Newtonian and Non-Newtonian Flows With a Free Surface,” Adv. Water Resour., 26(7), pp. 787–800. [CrossRef]
Fang, J. N., Owens, R. G., Tacher, L., and Parriaux, A., 2006, “A Numerical Study of the SPH Method for Simulating Transient Viscoelastic Free Surface Flows,” J. Non-Newton Fluid, 139(1–2), pp. 68–84. [CrossRef]
Fang, J. N., Parriaux, A., Rentschler, M., and Ancey, C., 2009, “Improved SPH Methods for Simulating Free Surface Flows of Viscous Fluids,” App. Num. Math., 59(2), pp. 251–271. [CrossRef]
Morris, J. P., Fox, P. J., and Zhu, Y., 1997, “Modeling Low Reynolds Number Incompressible Flows Using SPH,” J. Comput. Phys., 136(1), pp. 214–226. [CrossRef]
Ellero, M., Kroger, M., and Hess, S., 2002, “Viscoelastic Flows Studied by Smoothed Particle Dynamics,” J. Non-Newton Fluid, 105(1), pp. 35–51. [CrossRef]
Ellero, M., and Tanner, R. I., 2005, “SPH Simulations of Transient Viscoelastic Flows at Low Reynolds Number,” J. Non-Newton Fluid, 132(1–3), pp. 61–72. [CrossRef]
Rafiee, A., Manzari, M. T., and Hosseini, M., 2007, “An Incompressible SPH Method for Simulation of Unsteady Viscoelastic Free-Surface Flows,” Int. J. Non-Linear Mech., 42(10), pp. 1210–1223. [CrossRef]
Massoudi, M., and Vaidya, A., 2008, “On Some Generalizations of the Second Grade Fluid Model,” Nonlinear Anal. Real., 9(3), pp. 1169–1183. [CrossRef]
Labropulu, F., Xu, X., and Chinichian, M., 2000, “Unsteady Stagnation Point Flow of a Non-Newtonian Second-Grade Fluid,” Math. Subj. Class., 60, pp. 3797–3807.
Osswald, T. A., and Hernández-Ortiz, J. P., 2006, Polymer Processing: Modeling and Simulation, Hanser Gardner Publications, Inc., München, Germany.
Dunn, J. E., and Fosdick, R. L., 1974, “Thermodynamics, Stability and Boundedness of Fluids of Complexity 2 and Fluids of Second Grade,” Arch. Ration. Mech. Anal., 56, pp. 191–252. [CrossRef]
Han, C. D., 2007, Rheology and Processing of Polymeric Materials, Oxford University Press, Inc., New York.
Monaghan, J. J., 2005, “Smoothed Particle Hydrodynamics,” Rep. Prog. Phys., 68, pp. 1703–1759. [CrossRef]
Monaghan, J. J., 1992, “Smoothed Particle Hydrodynamics,” Annu. Rev. Astron. Astr., 30, pp. 543–574. [CrossRef]
Liu, G. R., and Liu, M. B., 2003, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing Co. Pte. Ltd., Singapore.
Yildiz, M., Rook, R. A., and Suleman, A., 2009, “SPH With the Multiple Boundary Tangent Method,” Int. J. Numer. Meth. Eng., 77(10), pp. 1416–1438. [CrossRef]
Shadloo, M. S., Zainali, A., Sadek, S. H., and Yildiz, M., 2011, “Improved Incompressible Smoothed Particle Hydrodynamics Method for Simulating Flow Around Bluff Bodies,” Comput. Method. Appl. M., 200(9–12), pp. 1008–1020. [CrossRef]
Shadloo, M. S., Zainali, A., Yildiz, M., and Suleman, A., 2012, “A Robust Weakly Compressible SPH Method and Its Comparison With an Incompressible SPH,” Int. J. Numer. Meth. Eng., 89(8), pp. 939–956. [CrossRef]
Issa, R., Lee, E. S., Violeau, D., and Laurence, D. R., 2005, “Incompressible Separated Flows Simulations With the Smoothed Particle Hydrodynamics Gridless Method,” Int. J. Numer. Meth. Fl., 47(10–11), pp. 1101–1106. [CrossRef]
Rodriguez-Paz, M., and Bonet, J., 2005, “A Corrected Smooth Particle Hydrodynamics Formulation of the Shallow-Water Equations,” Comput. Struct., 83(17–18), pp. 1396–1410. [CrossRef]
Yildiz, M., and Dost, S., 2009, Growth of Bulk SiGe Single Crystals by Liquid Phase Diffusion Method: Experimental and Computational Aspects, VDM Verlag, Germany.
Batchelor, J., Berry, J. P., and Horsfall, F., 1973, “Die Swell in Elastic and Viscous Fluids,” Polymer, 14(7), pp. 297–299. [CrossRef]
Horsfall, F., 1973, “A Theoretical Treatment of Die Swell in a Newtonian Liquid,” Polymer, 14(6), pp. 262–266. [CrossRef]
Tomé, M. F., Grossi, L., Castelo, A., Cuminato, J. A., McKee, S., and Walters, K., 2007, “Die-Swell, Splashing Drop and a Numerical Technique for Solving the Oldroyd-B Model for Axisymmetric Free Surface Flows,” J. Non-Newton Fluid, 141(2–3), pp. 148–166. [CrossRef]

Figures

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Fig. 1

The simulation domain in particle representation

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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

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Fig. 3

A schematic for enforcing zero pressure condition on the free surface

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Fig. 4

Velocity profile at the middle of the channel

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Fig. 5

Fully developed non-Newtonian second-order periodic flow

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Fig. 6

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

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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.

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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)

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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.

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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)

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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.

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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.

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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)

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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.

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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.

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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

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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.

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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.

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