화학공학소재연구정보센터
홈 로그인 로그아웃 연락처 사이트맵
Korea-Australia Rheology Journal, Vol.14, No.4, 143-159, December, 2002
Export Citation
Computing transient flows with high elasticity
Roger I. Tanner†, and S-C Xue
  • School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006, Australia
E-mail:
Although much progress has been made in the computation of Eulerian steady flows with high viscoelasticity, less work has been done for the case of transient flows. Because of their importance in injection moulding, blow moulding and other forming processes, as well as their intrinsic interest, we believe more attention should be focussed in this area. Hence in this paper we review progress in unsteady flow computations with high elasticity, and show some new results in this area.
Keywords:viscoelasticity;computation;transient or unsteady flow
  1. Aboubacar M, Webster MF, J. Non-Newton. Fluid Mech., 98(2-3), 83 (2001) 
  2. Alves MA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 93(2-3), 287 (2000) 
  3. Alves MA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 97(2-3), 207 (2001) 
  4. Atalik K, Keunings R, J. Non-Newton. Fluid Mech., 102(2), 299 (2002) 
  5. Baaijens FPT, J. Non-Newton. Fluid Mech., 51(2), 141 (1994) 
  6. Baaijens FPT, J. Non-Newton. Fluid Mech., 79(2-3), 361 (1998) 
  7. Baloch A, Townsend P, Webster MF, J. Non-Newton. Fluid Mech., 65(2-3), 133 (1996) 
  8. Becker LE, Mckinley GH, Rasmussen HK, Hassager O, J. Rheol., 38(2), 377 (1994) 
  9. Beris AN, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 22, 29 (1987)
  10. Beris AN, Avgousti M, Souvaliotis A, J. Non-Newton. Fluid Mech., 44, 197 (1992) 
  11. Betlers P, Int. J. Numer. Methods Eng., 7, 306 (1981) 
  12. Bodart C, Crochet MJ, J. Non-Newton. Fluid Mech., 54, 303 (1994) 
  13. Bisgaard JV, J. Non-Newton. Fluid Mech., 12, 283 (1983) 
  14. Bishko GB, Harlen OG, McLeish TCB, Nicholson TM, J. Non-Newton. Fluid Mech., 82(2-3), 255 (1999) 
  15. Brooks AN, Hughes TJR, Comput. Meth. Appl. Mech. Eng., 32, 199 (1982) 
  16. Butcher JC, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. John Wiley and Sons, Chichester, England (1987)
  17. Carew EOA, Townsend P, Webster MF, J. Non-Newton. Fluid Mech., 50, 253 (1993) 
  18. Choi HC, Song JH, Yoo JY, J. Non-Newton. Fluid Mech., 29, 347 (1988) 
  19. Chorin AJ, J. Comput. Phys., 2, 12 (1967) 
  20. Coates PJ, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 42, 141 (1992) 
  21. Courant R, Methods of Mathematical Physics, Vol. 2, Interscience, New York (1972)
  22. Criminale WO, Ericksen JL, Filbey GL, Arch. Rat. Mech. Anal., 1, 410 (1958)
  23. Crochet MJ, Davies AR, Walters K, Numerical Simulation of Non-Newtonian Flow, Elsevier, Amsterdam (1984)
  24. Crochet MJ, Keunungs R, J. Non-Newton. Fluid Mech., 7, 199 (1980) 
  25. Crochet MJ, Keunungs R, J. Non-Newton. Fluid Mech., 10, 339 (1982) 
  26. Darwish MS, Whiteman JR, Bevis MJ, J. Non-Newton. Fluid Mech., 45, 311 (1992) 
  27. Dimitropoulos CD, Sureshkumar R, Beris AN, J. Non-Newton. Fluid Mech., 79(2-3), 433 (1998) 
  28. Donea J, Int. J. Numer. Methods Eng., 20, 101 (1984) 
  29. Fan YR, Comput. Meth. Appl. Mech. Eng., 141, 47 (1997) 
  30. Fan YR, Tanner RI, Phan-Thien N, J. Non-Newton. Fluid Mech., 84(2-3), 233 (1999) 
  31. Ferziger JH, Peric M, Computational Methods for Fluid Dynamics (3rd., Rev. Ed.), Springer-Verlag, Berlin (2002)
  32. Finlayson BA, The Method of Weighted Residuals and Variational Principles, Academic Press, New York (1972)
  33. Fortin M, Fortin A, J. Non-Newton. Fluid Mech., 32, 295 (1989) 
  34. Fortin M, Fortin A, J. Non-Newton. Fluid Mech., 41, 277 (1990)
  35. Fortin A, Zine A, J. Non-Newton. Fluid Mech., 42, 1 (1992) 
  36. Gaskell PH, Lau AKC, Int. J. Numer. Methods Fluids, 8, 617 (1988) 
  37. Gear CW, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Inc., Englewood Cliffs., New Jersey, U.S.A. (1971)
  38. Gervang B, Larsen PS, J. Non-Newton. Fluid Mech., 39, 217 (1991) 
  39. Glowinski R, Pironneau O, Annu. Rev. Fluid Mech., 24, 167 (1992) 
  40. Gottlieb D, Orszag SA, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia (1977)
  41. Guenette R, Fortin M, J. Non-Newton. Fluid Mech., 60(1), 27 (1995) 
  42. Harlen OG, Rallison JM, Szabo P, J. Non-Newton. Fluid Mech., 60(1), 81 (1995) 
  43. Harlow FH, Welch JE, Phys. Fluids, 8, 12 (1965) 
  44. Harlow FM, Welch JE, Phys. Fluids, 8, 2182 (1965) 
  45. Hassager O, Bisgaard C, J. Non-Newton. Fluid Mech., 12, 153 (1983) 
  46. Hu HH, Joseph DD, J. Non-Newton. Fluid Mech., 37, 347 (1990) 
  47. Huang X, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 64(1), 71 (1996) 
  48. Huilgol RR, PhanThien N, Fluid Mechanics of Viscoelasticity, Elsevier, Amsterdam (1997)
  49. Issa RI, J. Comput. Phys., 62, 40 (1985) 
  50. Jin H, PhanThien N, Tanner RI, Comput. Mech., 13, 443 (1994) 
  51. Kabanemi KK, Bertrand F, Tanguy PA, Aitkadi A, J. Non-Newton. Fluid Mech., 55(3), 283 (1994) 
  52. Kawahara M, Takeuchi N, Comput. Fluids, 5, 33 (1977) 
  53. Keunungs R, J. Comput. Phys., 62, 199 (1986) 
  54. Keunings R, Tucker, III, C.L. (ed.), Fundamentals of Computer Modeling of Polymer Processing, Hanser Publisher, Munich (1989)
  55. King RC, Apelian MR, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 29, 147 (1988) 
  56. Larson RG, Shaqfeh ESG, Muller SJ, J. Fluid Mech., 218, 573 (1990) 
  57. Leal LG, Laminar Flow and Convective Transport Processes, Prentice-Hall, Englewood Cliffs, N.J. (1992)
  58. Lesaint P, Raviart PA, On a Finite Element Method for Solving the Neutron Transport Equation, Academic Press, New York (1974)
  59. Luo XL, J. Non-Newton. Fluid Mech., 63(2-3), 121 (1996) 
  60. Luo XL, J. Non-Newton. Fluid Mech., 79(1), 57 (1998) 
  61. Luo XL, Tanner RI, J. Non-Newton. Fluid Mech., 31, 143 (1989) 
  62. Marchal JM, Crochet MJ, J. Non-Newton. Fluid Mech., 26, 77 (1987) 
  63. McKinley GH, Pakdel P, Oztekin A, J. Non-Newton. Fluid Mech., 67(1-3), 19 (1996)
  64. McLeish TCB, Larson RG, J. Rheol., 42(1), 81 (1998) 
  65. Mompean G, Deville M, J. Non-Newton. Fluid Mech., 72(2-3), 253 (1997) 
  66. Na Y, Yoo JY, Comput. Mech., 8, 43 (1991) 
  67. Northey PJ, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 36, 109 (1990) 
  68. Northey PJ, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 42, 117 (1992) 
  69. Oliveira PJ, J. Non-Newton. Fluid Mech., 101(1-3), 113 (2001) 
  70. Oliveira PJ, Pinho FT, J. Non-Newton. Fluid Mech., 88(1-2), 63 (1999) 
  71. Oliveira PJ, Pinho FT, Pinto GA, J. Non-Newton. Fluid Mech., 79(1), 1 (1998) 
  72. Olsson F, J. Non-Newton. Fluid Mech., 51(3), 309 (1994) 
  73. Patankar SV, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York (1980)
  74. Peyret R, Taylor TD, Computational Methods for Fluid Flow, Springer-Verlag, Berlin (1983)
  75. Perera MGN, Walters K, J. Non-Newton. Fluid Mech., 2, 49 (1977) 
  76. Petera J, J. Non-Newton. Fluid Mech., 103(1), 1 (2002) 
  77. Peters EAJF, Hulsen MA, van den Brule BHAA, J. Non-Newton. Fluid Mech., 89(1-2), 209 (2000) 
  78. Phelan FR, Malone MF, Winter HH, J. Non-Newton. Fluid Mech., 32, 197 (1989) 
  79. Phillips TN, Williams AJ, J. Non-Newton. Fluid Mech., 87(2-3), 215 (1999) 
  80. Pilitsis S, Beris AN, J. Non-Newton. Fluid Mech., 31, 231 (1989) 
  81. Press WH, Flannery BP, Teukolsky SA, Vetterling WT, Numerical Recipes in Fortran, The Art of Scientific Computing, Cambridge University Press, Cambridge (1986)
  82. Rajagopalan D, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 36, 159 (1990) 
  83. Rasmussen HK, J. Non-Newton. Fluid Mech., 84(2-3), 217 (1999) 
  84. Rasmussen HK, 13th Int. Congress on Rheology (Cambridge), 2, 178 (2000)
  85. Rasmussen HK, Gottsche S, Kjaer EM, Proc. 13th Int. Congress on Rheology (Cambridge), 2, 333 (2000)
  86. Rasmussen HK, Hassager O, J. Non-Newton. Fluid Mech., 46, 298 (1993)
  87. Rasmussen HK, Hassager O, J. Non-Newton. Fluid Mech., 56(1), 65 (1995) 
  88. Rasmussen HK, Hassager O, J. Non-Newton. Fluid Mech., 82(2-3), 189 (1999) 
  89. Rasmussen HK, Hassager O, Proc. 13th Int. Congress on Rheology (Cambridge), 2, 202 (2000)
  90. Renardy M, Zeit. Angew. Math. Mech., 65, 449 (1985)
  91. Renardy M, J. Non-Newton. Fluid Mech., 36, 419 (1990) 
  92. Roache PJ, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, U.S.A. (1998)
  93. Saad Y, Schultz MH, SIAM J. Sci. Statist. Comput., 41, 856 (1986) 
  94. Saramito P, Mathematical Modelling Numer. Anal., 28, 1 (1994)
  95. Saramito P, J. Non-Newton. Fluid Mech., 60(2-3), 199 (1995) 
  96. Saramito P, Piau JM, J. Non-Newton. Fluid Mech., 52(2), 263 (1994) 
  97. Sasmal GP, J. Non-Newton. Fluid Mech., 56(1), 15 (1995) 
  98. Sato T, Richardson SM, J. Non-Newton. Fluid Mech., 51(3), 249 (1994) 
  99. Sato T, Richardson SM, Int. J. Numer. Methods Fluids, 20, 555 (1994)
  100. Serdakowski JA, Caswell B, J. Non-Newton. Fluid Mech., 29, 217 (1988) 
  101. Singh P, Leal LG, Theor. Comput. Fluid Dynamics, 5, 107 (1993) 
  102. Somasi M, Khomami B, J. Non-Newton. Fluid Mech., 93(2-3), 339 (2000) 
  103. Stoker JJ, Water Waves, The Mathematical Theory with Applications, Interscience Publishers Inc., New York (1957)
  104. Stokes JR, Graham LJW, Lawson NJ, Boger DV, J. Fluid Mech., 429, 117 (2001) 
  105. Sun J, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 65(1), 75 (1996) 
  106. Sureshkumar R, Smith MD, Armstrong RC, Brown RA, J. Non-Newton. Fluid Mech., 82(1), 57 (1999) 
  107. Talwar KK, Ganpule HK, Khomami B, J. Non-Newton. Fluid Mech., 52(3), 293 (1994) 
  108. Talwar KK, Khomami B, J. Non-Newton. Fluid Mech., 59(1), 49 (1995) 
  109. Tanner RI, Engineering Rheology, 2nd Ed., Oxford University Press (2000)
  110. Tanner RI, Walters K, Rheology: An Historical Perspective, Elsevier, Amsterdam (1998)
  111. Tome MF, Duffy B, McKee S, J. Non-Newton. Fluid Mech., 62(1), 9 (1996) 
  112. Tsai CC, Liu TJ, J. Non-Newton. Fluid Mech., 60(2-3), 155 (1995) 
  113. VanKan J, SIAM J. Sci. Stat. Comput., 7, 870 (1986) 
  114. VanKemenade V, Deville MO, J. Non-Newton. Fluid Mech., 51(3), 277 (1994) 
  115. Verbeeten WMH, Computational Polymer Melt Rheology, Ph.D. Thesis, Technische Universiteit Eindhoven, The Netherlands (2001)
  116. Warichet V, Legat V, J. Non-Newton. Fluid Mech., 73(1-2), 95 (1997) 
  117. Wapperom P, Webster MF, J. Non-Newton. Fluid Mech., 79(2-3), 405 (1998) 
  118. Wapperom MF, Webster MF, Comput. Meth. Appl. Mech. Eng., 180, 281 (1999) 
  119. Wapperom P, Keunings R, J. Non-Newton. Fluid Mech., 95(1), 67 (2000) 
  120. Wapperom P, Keunings R, J. Non-Newton. Fluid Mech., 97(2-3), 267 (2001) 
  121. Wapperom P, Keunings R, Legat V, J. Non-Newton. Fluid Mech., 91(2-3), 273 (2000) 
  122. Wood WL, Practical Time Stepping Schemes, Clarendon Press, Oxford (1990)
  123. Xue SC, Three-Dimensional Finite Volume Numerical Modelling and Numerical Simulations of Viscoelastic Flows, Ph.D. Thesis, The University of Sydney, Australia (1997)
  124. Xue SC, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 59(2-3), 191 (1995) 
  125. Xue SC, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 74(1-3), 195 (1998) 
  126. Xue SC, Phan-Thien N, Tanner RI, Rheol. Acta, 37(2), 158 (1998) 
  127. Xue SC, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 87(2-3), 337 (1999) 
  128. Xue SC, Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., in press (2002)
  129. Xue SC, Tanner RI, Phan-Thien N, Comput. Meth. Appl. Mech. Eng., 180, 305 (1999) 
  130. Xue SC, Tanner RI, Phan-Thien N, 12th Inter. Workshop Numer. Meth. Non-Newt. Flows, Monterey Bay, California, U.S.A. (2001)
  131. Xue SC, Tanner RI, Phan-Thien N, J. Non-Newton. Fluid Mech., in press (2002)
  132. Yamaguchi H, Mishima A, Yasumoto T, Ishikawa T, J. Non-Newton. Fluid Mech., 89(3), 251 (2000) 
  133. Yoo JY, Na Y, J. Non-Newton. Fluid Mech., 39, 89 (1991)