Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

Journal of Non-Newtonian Fluid Mechanics

Volumn 166, Issue 3-4, 2011, Pages 165-179

  Oishi, C M ^a   Martins, F P ^a   Tome M F ^b   Cuminato, J A ^b   McKee, S ^c  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

^c UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

Extrudate swell; Finite difference method; Free surface flows; Implicit techniques; Pom Pom model; Viscoelastic fluids

Indexed keywords

EXTRUDATE SWELL; FINITE DIFFERENCE; FREE-SURFACE FLOW; IMPLICIT TECHNIQUES; POM-POM MODELS; VISCO-ELASTIC FLUID;

CHANNEL FLOW; FINITE DIFFERENCE METHOD; FLUIDS; NUMERICAL METHODS; REYNOLDS NUMBER; SURFACES; VISCOELASTICITY;

TWO DIMENSIONAL;

EID: 78651456832     PISSN: 03770257     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jnnfm.2010.11.001     Document Type: Article

References (71)

1. Aboubacar M., Matallah H., Tamaddon-Jahromi H.R., Webster M.F. Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows. J. Non-Newton. Fluid Mech. 2002, 103:65-103.
2. Aboubacar M., Tamaddon-Jahromi H.R., Webster M.F. Time-dependent algorithms for viscoelastic flow: bridge between finite-volume and finite-element methodology. Second MIT Conference on Computational Fluid and Solid Mechanics 2003, 815-818.
3. Aboubacar M., Aguayo J.P., Phillips P.M., Phillips T.N., Tamaddon-Jahromi H.R., Snigerev B.A., Webster M.F. Modelling Pom-Pom type models with high-order finite volume schemes. J. Non-Newton. Fluid Mech. 2005, 126:207-220.
4. Afonso A., Oliveira P.J., Pinho F.T., Alves M.A. The log-conformation tensor approach in the finite-volume method framework. J. Non-Newton. Fluid Mech. 2009, 157:55-65.
5. Aguayo J.P., Tamaddon-Jahromi H.R., Webster M.F. Extensional response of the Pom-Pom model through planar contraction flows for branched polymer melts. J. Non-Newton. Fluid Mech. 2006, 134:105-126.
6. Aguayo J.P., Phillips P.M., Phillips T.N., Tamaddon-Jahromi H.R., Snigerev B.A., Webster M.F. The numerical prediction of planar viscoelastic contraction flows using the Pom-Pom model and higher-order finite volume schemes. J. Comput. Phys. 2007, 220:586-611.
7. Alves M.A., Oliveira P.J., Pinho F.T. A convergent and universally bounded interpolation scheme for the treatment of advection. Int. J. Numer. Meth. Fluids 2003, 41:47-75.
8. Amsden A.A., Harlow F.H. A simplified MAC technique for incompressible fluid flow calculations. J. Comput. Phys. 1970, 6:332-335.
9. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge, 1970.
10. Bird R.B., Armstrong R.C., Hassager O. Dynamics of Polymeric Liquids, vol. I 1987, Willey.
11. Bishko G.B., Harlen O.G., McLeish T.C.B., Nicholson T.M. Numerical simulation of the transient flow of branched polymer melts through a planar contraction using the Pom-Pom model. J. Non-Newton. Fluid Mech. 1999, 82:255-273.
12. Bonito A., Picasso M., Laso M. Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys. 2006, 215:691-716.
13. Brown D.L., Cortez R., Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 2001, 168:464-499.
14. Buscaglia G.C. A finite element analysis of rubber coextrusion using a power-law model. Int. J. Numer. Meth. Eng. 1993, 36:2143-2156.
15. Cai Z., Westphal C.R. An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type. J. Non-Newton. Fluid Mech. 2009, 159:72-80.
16. Choi H.C., Song J.H., Yoo J.Y. Numerical-simulation of the planar contraction flow of a Giesekus fluid. J. Non-Newton. Fluid Mech. 1988, 29:347-379.
17. Chorin A.J. Numerical solution of the Navier-Stokes equations. J. Comput. Phys. 1968, 2:745-762.
18. Clemeur N., Rutgers R.P.G., Debbaut B. On the evaluation of some differential formulations for the Pom-Pom constitutive model. Rheol. Acta 2003, 42:217-231.
19. Cloitre M., Hall T., Mata C., Joseph D.D. Delayed-die swell and sedimentation of elongated particles in wormlike micellar solutions. J. Non-Newton. Fluid Mech. 1998, 79:157-171.
20. Crochet M.J., Keunings R. Finite element analysis of die-swell of a highly elastic fluid. J. Non-Newton. Fluid Mech. 1982, 10:339-356.
21. Crochet M.J., Davies A.R., Walters K. Numerical Simulation of Non-Newtonian Flow 1984, Elsevier, Amsterdam.
22. Demir H. Numerical modelling of viscoelastic cavity driven flow using finite difference simulations. Appl. Math. Comput. 2005, 166:64-83.
23. Denaro F.M. On the applications of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions. Int. J. Numer. Meth. Fluids 2003, 43:43-69.
24. Fan Y., Tanner R.I., Phan-Thien N. Galerkin/least-square finite-element methods for steady viscoelastic flows. J. Non-Newton. Fluid Mech. 1995, 84:233-256.
25. Fang J., Owens R.G., Tacher L., Parriaux A. A numerical study of the SPH method for simulating transient viscoelastic free surface flows. J. Non-Newton. Fluid Mech. 2006, 139:68-84.
26. Ganvir V., Lele A., Thaokar R., Gautham B.P. Simulation of viscoelastic flows of polymer solutions in abrupt contractions using an arbitrary Lagrangian Eulerian (ALE) based finite element method. J. Non-Newton. Fluid Mech. 2007, 143:157-169.
27. Guermond J.L., Minev P., Shen J. An overview of projection methods for incompressible flows. Comput. Method Appl. M 2006, 195:6011-6045.
28. Harlow F.H., Welch J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 1965, 8:2182-2189.
29. Hori I., Okubo S. On the normal stress effect and the Barus effect of polymer melts. J. Rheol. 1980, 24:39-53.
30. Inkson N.J., McLeish T.C.B., Harlen O.G., Groves D.J. Predicting low density polyethylene melt rheology in elongational and shear flows with " Pom-Pom" constitutive equations. J. Rheol. 1999, 43:873-896.
31. Inkson N.J., Phillips T.N., van Os R.G.M. Numerical simulation of flow past a cylinder using models of XPP type. J. Non-Newton. Fluid Mech. 2009, 156:7-20.
32. Keunings R., Crochet M.J. Numerical simulation of the flow of a viscoelastic fluid through an abrupt contraction. J. Non-Newton. Fluid Mech. 1984, 14:279-299.
33. Kim J.M., Kim C., Kim J.H., Chung C., Ahn K.H., Lee S.J. High-resolution finite element simulation of 4:1 planar contraction flow of viscoelastic fluid. J. Non-Newton. Fluid Mech. 2005, 129:23-37.
34. Luo X.L., Tanner R.I. A streamline element scheme for solving viscoelastic flow problems. Differential constitutive-equations. J. Non-Newton. Fluid Mech. 1989, 21:179-199.
35. McLeish T.C.B., Larson R.G. Molecular constitutive equations for a class of branched polymers: the Pom-Pom polymer. J. Rheol. 1998, 42:81-110.
36. Mompean G., Deville M. Unsteady finite volume simulation of Oldroyd-B fluid through a three-dimensional planar contraction. J. Non-Newton. Fluid Mech. 1997, 72:253-279.
37. Mangiavacchi N., Castelo A., Tomé M.F., Cuminato J.A., de Oliveira M.L.B., McKee S. An effective implementation of surface tension using Marker and Cell method for axisymmetric and planar flows. SIAM J. Sci. Comput. 2005, 26:1340-1368.
38. Oishi C.M., Cuminato J.A., Ferreira V.G., Tomé M.F., Castelo A., Mangiavacchi N., McKee S. A stable semi-implicit method for free surface flows. Trans. ASME J. Appl. Mech. 2006, 73:940-947.
39. Oishi C.M., Tomé M.F., Cuminato J.A., McKee S. An implicit technique for solving 3D low Reynolds number moving free surface flows. J. Comput. Phys. 2008, 227:7446-7468.
40. Oishi C.M., Cuminato J.A., Yuan J.Y., McKee S. Stability of numerical schemes on staggered grids. Numer. Lin. Alg. Appl. 2008, 15:945-967.
41. Oliveira P.J., Pinho F.T., Pinto G.A. Numerical simulation of non-linear elastic flows with a general collocated finite-volume method. J. Non-Newton. Fluid Mech. 1998, 79:1-43.
42. Owens R.G., Phillips T.N. Computational Rheology 2002, World Scientific Publishing Company.
43. Pasquali M., Scriven L.E. Free surface flows of polymer solutions with models based on the conformation tensor. J. Non-Newton. Fluid Mech. 2002, 108:363-409.
44. de Paulo G.S., Tomé M.F., McKee S. A marker-and-cell approach to viscoelastic free surface flows using the PTT model. J. Non-Newton. Fluid Mech. 2007, 147:149-174.
45. Perot B., Nallapati R. A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows. J. Comput. Phys. 2003, 184:192-214.
46. Phan-Thien N., Tanner R.I. Viscoelastic finite volume method. Rheol. Ser. 1999, 8:331-359.
47. Phillips T.N., Willians A.J. Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method. J. Non-Newton. Fluid Mech. 1999, 87:215-246.
48. Rajagopalan D., Amstrong R., Brown R. Finite element methods for calculation of steady viscoelastic flow using constitutive equations with newtonian viscosity. J. Non-Newton. Fluid Mech. 1990, 36:159-192.
49. Rubio P., Wagner M.H. LDPE melt rheology and the Pom-Pom model. J. Non-Newton. Fluid Mech. 2000, 92:245-259.
50. Russo G., Phillips T.N. Numerical prediction of extrudate swell of branched polymer melts. Rheol. Acta 2010, 49:657-676.
51. Sato T., Richardson S.M. Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method. J. Non-Newton. Fluid Mech. 1994, 51:249-275.
52. Sirakov I., Ainser A., Haouche M., Guillet J. Three-dimensional numerical simulation of viscoelastic contraction flows using the Pom-Pom differential constitutive model. J. Non-Newton. Fluid Mech. 2005, 126:163-173.
53. Tanner R.I. A theory of die-swell. J. Polym. Sci. 1970, 8:2067-2078.
54. Tanner R.I. On the congruence of some network and Pom-Pom models. Koren-Aust. Rheol. J. 2006, 18:9-14.
55. Tomé M.F., Doricio J.L., Castelo A., Cuminato J.A., McKee S. Solving viscoelastic free surface flows of a second-order-fluid using a marker-and-cell approach. Int. J. Numer. Meth. Fluids 2007, 53:599-627.
56. Tomé M.F., McKee S. GENSMAC: a computational marker-and-cell method for free surface flows in general domains. J. Comput. Phys. 1994, 110:171-186.
57. Tomé M.F., Castelo A., Cuminato J.A., Murakami J., Minghim R., Oliveira C.F., McKee S. Numerical simulation of axisymmetric free surface flows. J. Comput. Phys. 2000, 157:441-472.
58. Tomé M., Mangiavacchi N., Cuminato J.A., Castelo A., McKee S. A finite difference technique for simulating unsteady viscoelastic free surface flows. J. Non-Newton. Fluid Mech. 2002, 106:61-106.
59. Tomé M.F., Grossi L., Castelo A., Cuminato J.A., McKee S., Walters K. Die-swell, splashing drop and a numerical technique for solving the Oldroyd-B model for axisymmetric free surface flows. J. Non-Newton. Fluid Mech. 2007, 141:148-166.
60. Tomé M.F., Castelo A., Ferreira V.G., McKee S. A finite difference technique for solving the Oldroyd-B model for 3D-unsteady free surface flows. J. Non-Newton. Fluid Mech. 2008, 154:179-206.
61. Tomé M.F., Silva R.A., Oishi C.M., McKee S. Numerical solution of the Upper-Convected Maxwell model for three-dimensional free surface flows. Commun. Comput. Phys. 2009, 6:367-395.
62. Trebotich D., Colella P., Miller G.H. A stable and convergent scheme for viscoelastic flow in contraction channels. J. Comput. Phys. 2005, 205:315-342.
63. van Os R.G.M., Phillips T.N. Efficient and stable spectral element methods for predicting the flow of an XPP fluid past a cylinder. J. Non-Newton. Fluid Mech. 2005, 129:143-162.
64. Verbeeten W.M.H., Peters G.W.M., Baaijens F.P.T. Differential constitutive equations for polymer melts: the eXtended Pom-Pom model. J. Rheol. 2001, 45:823-844.
65. Verbeeten W.M.H., Peters G.W.M., Baaijens F.P.T. Viscoelastic analysis of complex polymer melt flows using the eXtended Pom-Pom model. J. Non-Newton. Fluid Mech. 2002, 108:301-326.
66. Verbeeten W.M.H., Peters G.W.M., Baaijens F.P.T. Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model. J. Non-Newton. Fluid Mech. 2004, 117:73-84.
67. Xue S.-C., Phan-Thien N., Tanner R.I. Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method. J. Non-Newton. Fluid Mech. 1995, 59:191-213.
68. Yang B., Prosperetti A. A second-order boundary-fitted projection method for free-surface flow computations. J. Comput. Phys. 2006, 231:574-590.
69. Wapperom P., Webster M.F. A second-order hybrid finite-element/volume method for viscoelastic flows. J. Non-Newton. Fluid Mech. 1998, 79:405-431.
70. Walters K., Webster M.F. The distinctive CFD challenges of computational rheology. Int. J. Numer. Meth. Fluids 2003, 43:577-596.
71. Walters K., Webster M.F., Tamaddon-Jahromi H.R. The numerical simulation of some contraction flows of highly elastic liquids and their impact on the relevance of the Couette correction in extensional rheology. Chem. Eng. Sci. 2009, 64:4632-4639.