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Annual Review of Fluid Mechanics

Volume 53, 2021

Numerical Methods for Viscoelastic Fluid Flows

Abstract

Complex fluids exist in nature and are continually engineered for specific applications involving the addition of macromolecules to a solvent, among other means. This imparts viscoelasticity to the fluid, a property responsible for various flow instabilities and major modifications to the fluid dynamics. Recent developments in the numerical methods for the simulation of viscoelastic fluid flows, described by continuum-level differential constitutive equations, are surveyed, with a particular emphasis on the finite-volume method. This method is briefly described, and the main benchmark flows currently used in computational rheology to assess the performance of numerical methods are presented. Outstanding issues in numerical methods and novel and challenging applications of viscoelastic fluids, some of which require further developments in numerical methods, are discussed.

Keyword(s): benchmark flows, computational rheology, finite-volume method, high–Weissenberg number problem, numerical stabilization methods, viscoelastic flows
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2021-01-05
2024-04-26
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Literature Cited

  1. Aboubacar M, Webster MF. 2001. A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows. J. Non-Newton. Fluid Mech. 98:83–106
     [Google Scholar]
  2. Afonso AM, Oliveira MSN, Oliveira PJ, Alves MA, Pinho FT 2012a. The finite volume method in computational rheology. Finite Volume M ethod: Powerful Means of Engineering Design R Petrova 141–70 London: InTech Open
     [Google Scholar]
  3. Afonso AM, Oliveira PJ, Pinho FT, Alves MA 2009. The log-conformation tensor approach in the finite volume method framework. J. Non-Newton. Fluid Mech. 157:55–65
     [Google Scholar]
  4. Afonso AM, Oliveira PJ, Pinho FT, Alves MA 2011. Dynamics of high-Deborah-number entry flows: a numerical study. J. Fluid Mech. 677:272–304
     [Google Scholar]
  5. Afonso AM, Pinho FT, Alves MA 2012b. Electro-osmosis of viscoelastic fluids and prediction of electro-elastic instabilities in a cross slot using a finite volume method. J. Non-Newton. Fluid Mech. 179–180:55–68
     [Google Scholar]
  6. Afonso AM, Pinho FT, Alves MA 2012c. The kernel-conformation constitutive laws. J. Non-Newton. Fluid Mech. 167–168:30–37
     [Google Scholar]
  7. Alves MA, Oliveira PJ, Pinho FT 2003a. A convergent and universally bounded interpolation scheme for the treatment of advection. Int. J. Numer. Methods Fluids 41:47–75
     [Google Scholar]
  8. Alves MA, Oliveira PJ, Pinho FT 2003b. Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newton. Fluid Mech. 110:45–75
     [Google Scholar]
  9. Alves MA, Oliveira PJ, Pinho FT 2004. On the effect of contraction ratio in viscoelastic flow through abrupt contractions. J. Non-Newton. Fluid Mech. 122:117–30
     [Google Scholar]
  10. Alves MA, Pinho FT, Oliveira PJ 2000. Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows. J. Non-Newton. Fluid Mech. 93:287–314
     [Google Scholar]
  11. Alves MA, Pinho FT, Oliveira PJ 2001. The flow of viscoelastic fluids past a cylinder: finite-volume high-resolution methods. J. Non-Newton. Fluid Mech. 97:207–232
     [Google Scholar]
  12. Amoreira LJ, Oliveira PJ. 2010. Comparison of different formulations for the numerical calculation of unsteady incompressible viscoelastic fluid flow. Adv. Appl. Math. Mech. 2:483–502
     [Google Scholar]
  13. Anna SL. 2016. Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48:285–309
     [Google Scholar]
  14. Armstrong RC, Brown RA, Caswell B 1984. Papers from the Third International Workshop on Numerical Simulation of Viscoelastic Flows. J. Non-Newton. Fluid Mech. 16:1–2
     [Google Scholar]
  15. Arratia PE, Thomas CC, Diorio J, Gollub JP 2006. Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett. 96:144502
     [Google Scholar]
  16. Ascher UM, Ruuth SJ, Wetton BTR 1995. Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32:797–823
     [Google Scholar]
  17. Baaijens FPT. 1998. Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newton. Fluid Mech. 79:361–285
     [Google Scholar]
  18. Balci N, Thomases B, Renardy M, Doering CR 2011. Symmetric factorization of the conformation tensor in viscoelastic fluid models. J. Non-Newton. Fluid Mech. 166:546–53
     [Google Scholar]
  19. Baird DG, Renardy M. 1992. Report on the VIIth International Workshop on Numerical Methods in Non-Newtonian Flow. J. Non-Newton. Fluid Mech. 43:383–85
     [Google Scholar]
  20. Bajaj M, Pasquali M, Prakash JR 2008. Coil-stretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder. J. Rheol. 52:197–223
     [Google Scholar]
  21. Bird RB, Armstrong RC, Hassager O 1987a. Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics New York: Wiley. , 2nd. ed.
     [Google Scholar]
  22. Bird RB, Curtiss CF, Armstrong RC, Hassager O 1987b. Dynamics of Polymeric Liquids. Vol. 2: Kinetic Theory New York: Wiley. , 2nd. ed.
     [Google Scholar]
  23. Bird RB, Dotson PJ, Johnson NL 1980. Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non-Newton. Fluid Mech. 7:213–35
     [Google Scholar]
  24. Bird RB, Wiest JM. 1995. Constitutive equations for polymeric liquids. Annu. Rev. Fluid Mech. 27:169–93
     [Google Scholar]
  25. Brown RA, McKinley GH. 1994. Report on the VIIIth International Workshop on Numerical Methods in Viscoelastic Flows. J. Non-Newton. Fluid Mech. 52:407–13
     [Google Scholar]
  26. Carrozza MA, Hulsen MA, Hütter M, Anderson PD 2019. Viscoelastic fluid flow simulation using the contravariant deformation formulation. J. Non-Newton. Fluid Mech. 270:23–35
     [Google Scholar]
  27. Castillo E, Codina R. 2014. Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem. Comput. Meth. Appl. Mech. Eng. 279:579–605
     [Google Scholar]
  28. Castillo E, Codina R. 2015. First, second and third order fractional step methods for the three-field viscoelastic flow problem. J. Comput. Phy. 296:113–37
     [Google Scholar]
  29. Chilcott DM, Rallison JM. 1988. Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newton. Fluid Mech. 29:381–432
     [Google Scholar]
  30. Chorin AJ. 1968. Numerical solution of the Navier-Stokes. Math. Comput. 22:745–62
     [Google Scholar]
  31. Claus S, Phillips TN. 2013. Viscoelastic flow around a confined cylinder using spectral/hp element methods. J. Non-Newton. Fluid Mech. 200:131–46
     [Google Scholar]
  32. Comminal R, Hattel JH, Alves MA, Spangenberg J 2016. Vortex behavior of the Oldroyd-B fluid in the 4-1 planar contraction simulated with the streamfunction–log-conformation formulation. J. Non-Newton. Fluid Mech. 237:1–15
     [Google Scholar]
  33. Comminal R, Pimenta F, Hattel JH, Alves MA, Spangenberg J 2018. Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods. J. Non-Newton. Fluid Mech. 252:1–18
     [Google Scholar]
  34. Coronado O, Arora D, Behr M, Pasquali M 2007. A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation. J. Non-Newton. Fluid Mech. 147:189–99
     [Google Scholar]
  35. Coventry KD, Mackley MR. 2008. Cross-slot extensional flow birefringence observations of polymer melts using a multi-pass rheometer. J. Rheol. 52:401–15
     [Google Scholar]
  36. Crochet M. 1986. Foreword to the Proceedings of the Fourth Workshop on Numerical Methods in Viscoelastic Flow. J. Non-Newton. Fluid Mech. 20:1–9
     [Google Scholar]
  37. Crochet MJ, Davies AR, Walters K 1984. Numerical Simulation on Non-Newtonian Flow Amsterdam: Elsevier
  38. Crochet MJ, Walters K. 1983. Numerical methods in non-Newtonian fluid mechanics. Annu. Rev. Fluid Mech. 15:241–60
     [Google Scholar]
  39. Cruz FA, Poole RJ, Afonso AM, Pinho FT, Oliveira PJ, Alves MA 2014. A new viscoelastic benchmark flow: stationary bifurcation in a cross-slot. J. Non-Newton. Fluid Mech. 214:57–68
     [Google Scholar]
  40. Damanik H, Hron J, Ouazzi A, Turek S 2010. A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems. J. Non-Newton. Fluid Mech. 165:1105–13
     [Google Scholar]
  41. Darwish MS, Whiteman JR, Bevis MJ 1992. Numerical modelling of viscoelastic liquids using a finite volume method. J. Non-Newton. Fluid Mech. 45:311–37
     [Google Scholar]
  42. D'Avino G, Hulsen MA. 2010. Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution. J. Non-Newton. Fluid Mech. 165:1602–12
     [Google Scholar]
  43. D'Avino G, Hulsen MA, Maffettone PL 2012. Decoupled transient schemes for viscoelastic fluid flow with inertia. Comput. Fluids 66:183–93
     [Google Scholar]
  44. D'Avino G, Maffettone PL. 2015. Particle dynamics in viscoelastic liquids. J. Non-Newton. Fluid Mech. 215:80–104
     [Google Scholar]
  45. Duarte ASR, Miranda AIP, Oliveira PJ 2008. Numerical and analytical modeling of unsteady viscoelastic flows: the start-up and pulsating test case problems. J. Non-Newton. Fluid Mech. 154:153–69
     [Google Scholar]
  46. Dubief Y, Terrapon VE, White CM, Shaqfeh ESG, Moin P, Lele SK 2005. New advances on the interaction between polymers and vortices in turbulent flows. Flow Turb. Comb. 74:311–29
     [Google Scholar]
  47. Dupret F, Marchal JM. 1986. Sur le signe des valeurs propres du tenseur des extra-constraints dans un écoulement de fluide de Maxwell. J. Méc. Théor. Appl. 5:403–27
     [Google Scholar]
  48. Edussuriya SS, Williams AJ, Bailey C 2004. A cell-centred finite volume method for modelling viscoelastic flow. J. Non-Newton. Fluid Mech. 117:47–61
     [Google Scholar]
  49. El-Kareh AW, Leal GL. 1989. Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion. J. Non-Newton. Fluid Mech. 33:257–87
     [Google Scholar]
  50. Ellero M, Español P. 2018. Everything you always wanted to know about SDPD (but were afraid to ask). Appl. Math. Mech. 39:103–24
     [Google Scholar]
  51. Ellero M, Kröger M, Hess S 2002. Viscoelastic flows studied by smoothed particle dynamics. J. Non-Newton. Fluid Mech. 105:35–51
     [Google Scholar]
  52. Español P, Warren P. 2017. Perspective: dissipative particle dynamics. J. Chem. Phys. 146:150901
     [Google Scholar]
  53. Evans JD, Oishi CM. 2017. Transient computations using the natural stress formulation for solving sharp corner flows. J. Non-Newton. Fluid Mech. 249:48–52
     [Google Scholar]
  54. Evans RE, Walters K. 1986. Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids. J. Non-Newton. Fluid Mech. 20:11–29
     [Google Scholar]
  55. Fan Y, Tanner R, Phan-Thien N 1999. Galerkin/least-square finite-element methods for steady viscoelastic flows. J. Non-Newton. Fluid Mech. 84:233–56
     [Google Scholar]
  56. Fan Y, Yang H, Tanner RI 2005. Stress boundary layers in the viscoelastic flow past a cylinder in a channel: limiting solutions. Acta Mech. Sin. 21:311–21
     [Google Scholar]
  57. Fattal R, Kupferman R. 2004. Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newton. Fluid Mech. 123:281–85
     [Google Scholar]
  58. Fattal R, Kupferman R. 2005. Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newton. Fluid Mech. 126:23–37
     [Google Scholar]
  59. Favero J, Secchi A, Cardozo N, Jasak H 2010. Viscoelastic fluid analysis in internal and in free surface flows using the software OpenFoam. Comput. Chem. Eng. 34:1984–93
     [Google Scholar]
  60. Fernandes C, Vukčević V, Uroić T, Simões R, Carneiro OS et al. 2019. A coupled finite volume flow solver for the solution of incompressible viscoelastic flows. J. Non-Newton. Fluid Mech. 265:99–115
     [Google Scholar]
  61. Ferreira PO, Pinho FT, da Silva CB 2016. Large-eddy simulations of forced isotropic turbulence with viscoelastic fluids described by the FENE-P model. Phys. Fluids 28:125104
     [Google Scholar]
  62. Ferziger JH, Perić M. 2002. Computational Methods for Fluid Dynamics Berlin: Springer-Verlag. , 3rd. ed.
  63. Fiétier N, Deville MO. 2003. Time-dependent algorithms for the simulation of viscoelastic flows with spectral element methods: applications and stability. J. Comput. Phys. 186:93–121
     [Google Scholar]
  64. Gaskell PH, Lau AKC. 1988. Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm. Int. J. Numer. Methods Fluids 8:617–41
     [Google Scholar]
  65. Giesekus H. 1982. A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newton. Fluid Mech. 11:69–109
     [Google Scholar]
  66. Groisman A, Steinberg V. 2000. Elastic turbulence in a polymer solution flow. Nature 405:53–55
     [Google Scholar]
  67. Guénette R, Fortin M. 1995. A new mixed finite element method for computing viscoelastic flows. J. Non-Newton. Fluid Mech. 60:27–52
     [Google Scholar]
  68. Gupta A, Vincenzi D. 2018. Effect of polymer stress diffusion in the numerical simulation of elastic turbulence. J. Fluid Mech. 870:405–18
     [Google Scholar]
  69. Habla F, Marschall H, Hinrichsen O, Dietsche L, Jasak H, Favero JL 2011. Numerical simulation of viscoelastic two-phase flows using openFOAM. Chem. Eng. Sci. 66:5487–96
     [Google Scholar]
  70. Habla F, Woitalka A, Neuner S, Hinrichsen O 2012. Development of a methodology for numerical simulation of non-isothermal viscoelastic fluid flows with application to axisymmetric 4:1 contraction flows. Chem. Eng. J. 207–208:772–84
     [Google Scholar]
  71. Halin P, Lielens G, Keunings R, Legat V 1998. The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations. J. Non-Newton. Fluid Mech. 79:387–403
     [Google Scholar]
  72. Haward SJ, Oliveira MSN, Alves MA, McKinley GH 2012. Optimized cross-slot flow geometry for microfluidic extensional rheometry. Phys. Rev. Lett. 109:128301
     [Google Scholar]
  73. Hinch MA. 1993. The flow of an Oldroyd fluid around a sharp corner. J. Non-Newton. Fluid Mech. 50:161–71
     [Google Scholar]
  74. Housiadas KD, Beris AN. 2003. Polymer-induced drag reduction: effects of the variations in elasticity and inertia in turbulent viscoelastic channel flow. Phys. Fluids 15:2369–84
     [Google Scholar]
  75. Housiadas KD, Wang L, Beris AN 2010. A new method preserving the positive definiteness of a second order tensor variable in flow simulations with application to viscoelastic turbulence. Comput. Fluids 39:225–41
     [Google Scholar]
  76. Hulsen MA. 1990. A sufficient condition for a positive definite configuration tensor in differential models. J. Non-Newton. Fluid Mech. 38:93–100
     [Google Scholar]
  77. Hulsen MA, Fattal R, Kupferman R 2005. Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms. J. Non-Newton. Fluid Mech. 127:27–39
     [Google Scholar]
  78. Hulsen MA, van Heel APG, van den Brule BHAA 1997. Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newton. Fluid Mech. 70:79–101
     [Google Scholar]
  79. James DF. 2009. Boger fluids. Annu. Rev. Fluid Mech. 41:129–42
     [Google Scholar]
  80. Joseph DD. 1990. Fluid Dynamics of Viscoelastic Liquids Berlin: Springer-Verlag
  81. Kalb A, Villasmil-Urdaneta LA, Cromer M 2018. Elastic instability and secondary flow in cross-slot flow of wormlike micellar solutions. J. Non-Newton. Fluid Mech. 262:79–91
     [Google Scholar]
  82. Kenney S, Poper K, Chapagain G, Christopher GF 2013. Large Deborah number flows around confined microfluidic cylinders. Rheol. Acta 52:485–97
     [Google Scholar]
  83. Keshtiban IJ, Belblidia F, Webster MF 2004. Numerical simulation of compressible viscoelastic liquids. J. Non-Newton. Fluid Mech. 122:131–46
     [Google Scholar]
  84. Keunings R. 2003. Finite element methods for integral viscoelastic fluids. Rheol. Rev. 1:167–95
     [Google Scholar]
  85. Keunings R. 2004. Micro-macro methods for the multi-scale simulation of viscoelastc flow using molecular models of kinetic theory. Rheol. Rev. 2:67–98
     [Google Scholar]
  86. Kim J, Moin P. 1985. Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59:308–23
     [Google Scholar]
  87. King RC, Apelian MR, Armstrong RC, Brown RA 1988. Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries. J. Non-Newton. Fluid Mech. 29:147–216
     [Google Scholar]
  88. Knechtges P, Behr M, Elgeti M 2014. Fully-implicit log-conformation formulation of constitutive laws. J. Non-Newton. Fluid Mech. 214:78–87
     [Google Scholar]
  89. Kurganov A, Tadmor E. 2000. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160:241–82
     [Google Scholar]
  90. Larson RG. 1988. Constitutive Equations for Polymer Melts and Solutions Stoneham, MA: Butterworth
  91. Larson RG, Desai PS. 2015. Modeling the rheology of polymer melts and solutions. Annu. Rev. Fluid Mech. 47:47–65
     [Google Scholar]
  92. Leal LG, Denn MM, Keunings R 1988. Lake Arrowhead Workshop special issue papers—introduction. J. Non-Newton. Fluid Mech. 29:1–8
     [Google Scholar]
  93. Likhtman AE, Graham RS. 2003. Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. J. Non-Newton. Fluid Mech. 114:1–12
     [Google Scholar]
  94. Litvinov S, Xie QG, Hu XY, Adams NA, Ellero M 2016. Simulation of individual polymer chains and polymer solutions with smoothed dissipative particle dynamics. Fluids 1:7
     [Google Scholar]
  95. López-Aguilar JE, Tamaddon-Jahromi HR, Webster MF, Walters K 2016. Numerical versus experimental pressure drops for Boger fluids in sharp-corner contraction flow. Phys. Fluids 28:103104
     [Google Scholar]
  96. López-Herrera JM, Popinet S, Castrejón-Pita AA 2019. An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of weakly viscoelastic droplets. J. Non-Newton. Fluid Mech. 264:144–58
     [Google Scholar]
  97. Lozinski A, Owens RG. 2003. An energy estimate for the Oldroyd-B model: theory and applications. J. Non-Newton. Fluid Mech. 112:161–76
     [Google Scholar]
  98. Malaspinas O, Fiétier N, Deville M 2010. Lattice Boltzmann method for the simulation of viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 165:1637–53
     [Google Scholar]
  99. Marchal JM, Crochet MJ. 1987. A new mixed finite element for calculating viscoelastic flow. J. Non-Newton. Fluid Mech. 26:77–114
     [Google Scholar]
  100. Masoudian M, Kim K, Pinho FT, Sureshkumar R 2015. A Reynolds stress model for turbulent flow of homogeneous polymer solutions. Int. J. Heat Fluid Flow 54:220–35
     [Google Scholar]
  101. Matallah H, Townsend P, Webster MF 1998. Recovery and stress-splitting schemes for viscoelastic flows. J. Non-Newton. Fluid Mech. 75:139–66
     [Google Scholar]
  102. McKinley GH, Raiford WP, Brown RA, Armstrong RC 1991. Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions. J. Fluid Mech. 223:411–56
     [Google Scholar]
  103. McLeish TCB, Larson RG. 1998. Molecular constitutive equations for a class of branched polymers: the pom-pom polymer. J. Rheol. 42:81–110
     [Google Scholar]
  104. Meng S, Li XK, Evans G 2002. Numerical simulation of Oldroyd-B fluid in a contraction channel. J. Supercomput. 22:29–43
     [Google Scholar]
  105. Mirzakhalili E, Nejat A. 2015. High-order solution of viscoelastic fluids using the discontinuous Galerkin method. J. Fluids Eng. 137:031205
     [Google Scholar]
  106. Mompean G, Deville M. 1997. Unsteady finite volume simulations of Oldroyd-B fluid through a three-dimensional planar contraction. J. Non-Newton. Fluid Mech. 72:253–79
     [Google Scholar]
  107. Niethammer M, Marschall H, Kunkelmann C, Bothe D 2018. A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes. Int. J. Numer. Methods Fluids 86:131–66
     [Google Scholar]
  108. Niethammer M, Marschall H, Bothe D 2019. Robust direct numerical simulation of viscoelastic flows. Chem. Ing. Tech. 91:522–28
     [Google Scholar]
  109. Nigen S, Walters K. 2002. Viscoelastic contraction flows: comparison of axisymmetric and planar configurations. J. Non-Newton. Fluid Mech. 102:343–59
     [Google Scholar]
  110. Oldroyd JG. 1950. On the formulation of rheological equations of state. Proc. R. Soc. A 200:523–41
     [Google Scholar]
  111. Oliveira MSN, Oliveira PJ, Pinho FT, Alves MA 2007. Effect of contraction ratio upon viscoelastic flow in contractions: the axisymmetric case. J. Non-Newton. Fluid Mech. 147:92–108
     [Google Scholar]
  112. Oliveira PJ. 2000. A traceless stress tensor formulation for viscoelastic fluid flow. J. Non-Newton. Fluid Mech. 95:55–65
     [Google Scholar]
  113. Oliveira PJ. 2001. Method for time-dependent simulations of viscoelastic flows: vortex shedding behind cylinder. J. Non-Newton. Fluid Mech. 101:113–37
     [Google Scholar]
  114. Oliveira PJ. 2009. Alternative derivation of differential constitutive equations of the Oldroyd-B type. J. Non-Newton. Fluid Mech. 160:40–46
     [Google Scholar]
  115. Oliveira PJ. 2017. Reduced stress method for efficient computation of time-dependent viscoelastic flow with stress equations of FENE-P type. J. Non-Newton. Fluid Mech. 248:74–91
     [Google Scholar]
  116. Oliveira PJ, Pinho FT, Pinto GA 1998. Numerical simulation of non-linear elastic flows with a general collocated finite-volume method. J. Non-Newton. Fluid Mech. 79:1–43
     [Google Scholar]
  117. Omowunmi SC, Yuan X-F. 2013. Time-dependent non-linear dynamics of polymer solutions in microfluidic contraction flow—a numerical study on the role of elongational viscosity. Rheol. Acta 52:337–54
     [Google Scholar]
  118. Öttinger H-C, Laso M. 1992. Smart polymers in finite-volume calculations. Theoretical and Applied Rheology 1 P Moldenaers, R Keunings 286–88 Amsterdam: Elsevier
     [Google Scholar]
  119. Owens RG, Chauvière C, Philips TN 2002. A locally-upwinded spectral technique (LUST) for viscoelastic flows. J. Non-Newton. Fluid Mech. 108:49–71
     [Google Scholar]
  120. Owens RG, Phillips TN. 2002. Computational Rheology London: Imp. Coll. Press
  121. Palhares IL Jr., Oishi CM, Afonso AM, Alves MA, Pinho FT. 2016. Numerical study of the square-root conformation tensor formulation for confined and free-surface viscoelastic fluid flows. Adv. Model. Simul. Eng. Sci. 3:2
     [Google Scholar]
  122. Park HM, Lim JY. 2010. A new numerical algorithm for viscoelastic fluid flows: the grid-by-grid inversion method. J. Non-Newton. Fluid Mech. 165:238–46
     [Google Scholar]
  123. Perera MG, Walters K. 1977. Long-range memory effects in flows involving abrupt changes in geometry. J. Non-Newton. Fluid Mech. 2:49–81
     [Google Scholar]
  124. Perot JB. 1993. An analysis of the fractional step method. J. Comput. Phys. 108:51–58
     [Google Scholar]
  125. Phan-Thien N, Tanner RI. 1977. A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech. 2:353–65
     [Google Scholar]
  126. Phillips TN, Williams AJ. 1999. Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method. J. Non-Newton. Fluid Mech. 87:215–46
     [Google Scholar]
  127. Pimenta F, Alves MA. 2016. rheoTool OpenFOAM Toolbox, accessed April 30, 2020. https://github.com/fppimenta/rheoTool
  128. Pimenta F, Alves MA. 2017. Stabilization of an open-source finite-volume solver for viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 239:85–104
     [Google Scholar]
  129. Pimenta F, Alves MA. 2019. A coupled finite-volume solver for numerical simulation of electrically-driven flows. Comput. Fluids 193:104279
     [Google Scholar]
  130. Poole RJ, Alves MA, Oliveira PJ 2007. Purely elastic flow asymmetries. Phys. Rev. Lett. 99:164503
     [Google Scholar]
  131. Popinet S. 2020. Basilisk Flow Solver, accessed Jan. 20. http://basilisk.fr/
  132. Rajagopalan D, Armstrong RC, Brown RA 1990. Finite element methods for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity. J. Non-Newton. Fluid Mech. 36:159–92
     [Google Scholar]
  133. Renardy M. 1985. Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. Z. Angew. Math. Mech. 65:449–51
     [Google Scholar]
  134. Renardy M. 1994. How to integrate the upper convected Maxwell (UCM) stresses near a singularity (and maybe elsewhere, too). J. Non-Newton. Fluid Mech. 52:91–95
     [Google Scholar]
  135. Renardy M. 2000. Current issues in non-Newtonian flows: a mathematical perspective. J. Non-Newton. Fluid Mech. 90:243–59
     [Google Scholar]
  136. Ribeiro VM, Coelho PM, Pinho FT, Alves MA 2014. Viscoelastic fluid flow past a confined cylinder: three-dimensional effects and stability. Chem. Eng. Sci. 111:364–80
     [Google Scholar]
  137. Richter D, Iaccarino G, Shaqfeh ESG 2010. Simulations of three-dimensional viscoelastic flows past a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 651:415–42
     [Google Scholar]
  138. Roache PJ. 1972. Computational Fluid Dynamics Albuquerque, NM: Hermosa
  139. Sahin M, Wilson HJ. 2007. A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements. J. Non-Newton. Fluid Mech. 147:79–91
     [Google Scholar]
  140. Saramito P. 2014. On a modified non-singular log-conformation formulation for Johnson-Segalman viscoelastic fluids. J. Non-Newton. Fluid Mech. 211:16–30
     [Google Scholar]
  141. Saramito P, Wachs A. 2017. Progress in numerical simulation of yield stress fluid flows. Rheol. Acta 56:211–30
     [Google Scholar]
  142. Sasmal GP. 1995. A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction. J. Non-Newton. Fluid Mech. 56:15–47
     [Google Scholar]
  143. Sato T, Richardson SM. 1994. Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method. J. Non-Newton. Fluid Mech. 51:249–75
     [Google Scholar]
  144. Sommerfeld M. 2017. Numerical methods for dispersed multiphase flows. Particles in Flows. Advances in Mathematical Fluid Mechanics T Bodnár, G Galdi, Š Nečasová 327–96 Cham, Switz: Springer Int.
     [Google Scholar]
  145. Sousa PC, Pinho FT, Alves MA 2018. Purely-elastic flow instabilities and elastic turbulence in microfluidic cross-slot devices. Soft Matter 14:1344–54
     [Google Scholar]
  146. Sousa PC, Pinho FT, Oliveira MSN, Alves MA 2015. Purely elastic flow instabilities in microscale cross-slot devices. Soft Matter 11:8856–62
     [Google Scholar]
  147. Sousa RG, Poole RJ, Afonso AM, Pinho FT, Oliveira PJ et al. 2016. Lid-driven cavity flow of viscoelastic liquids. J. Non-Newton. Fluid Mech. 234:129–38
     [Google Scholar]
  148. Spanjaards MMA, Hulsen MA, Anderson PD 2019. Transient 3D finite element method for predicting extrudate swell of domains containing sharp edges. J. Non-Newton. Fluid Mech. 270:79–95
     [Google Scholar]
  149. Su J, Ouyang J, Wang X, Yang B, Zhou W 2013. Lattice Boltzmann method for the simulation of viscoelastic fluid flows over a large range of Weissenberg numbers. J. Non-Newton. Fluid Mech. 194:42–59
     [Google Scholar]
  150. Sun J, Phan-Thien N, Tanner RI 1996. An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG. J. Non-Newton. Fluid Mech. 65:75–91
     [Google Scholar]
  151. Sun J, Smith MD, Armstrong RC, Brown RA 1999. Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG. J. Non-Newton. Fluid Mech. 86:281–307
     [Google Scholar]
  152. Sureshkumar R, Beris AN. 1995. Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newton. Fluid Mech. 60:53–80
     [Google Scholar]
  153. Sureshkumar R, Beris AN, Handler RA 1997. Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9:743–55
     [Google Scholar]
  154. Syrakos A, Dimakopoulos Y, Tsamopoulos J 2020. A finite volume method for the simulation of elastoviscoplastic flows and its application to the lid-driven cavity case. J. Non-Newton. Fluid Mech. 275:104216
     [Google Scholar]
  155. Tang D, Marchesini FH, Cardon L, D'hooge DR 2019. Three-dimensional flow simulations for polymer extrudate swell out of slit dies from low to high aspect ratios. Phys. Fluids 31:93103
     [Google Scholar]
  156. ten Bosch BIM. 1999. On an extension of dissipative particle dynamics for viscoelastic flow modelling. J. Non-Newton. Fluid Mech. 83:231–48
     [Google Scholar]
  157. Thomases B. 2011. An analysis of the effect of stress diffusion on the dynamics of creeping viscoelastic flow. J. Non-Newton. Fluid Mech. 166:1221–28
     [Google Scholar]
  158. Tomé MF, Araújo MSB, Alves MA, Pinho FT 2008. Numerical simulation of viscoelastic flows using integral constitutive equations: a finite difference approach. J. Comput. Phys. 227:4207–43
     [Google Scholar]
  159. Vaithianathan T, Collins LR. 2003. Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. J. Comput. Phys. 187:1–21
     [Google Scholar]
  160. Vaithianathan T, Robert A, Brasseur JG, Collins LR 2006. An improved algorithm for simulating three-dimensional, viscoelastic turbulence. J. Non-Newton. Fluid Mech. 140:3–22
     [Google Scholar]
  161. Vaithianathan T, Robert A, Brasseur JG, Collins LR 2007. Polymer mixing in shear-driven turbulence. J. Fluid Mech. 585:487–97
     [Google Scholar]
  162. Valente PC, da Silva CB, Pinho FT 2014. The effect of viscoelasticity on the turbulent kinetic energy cascade. J. Fluid Mech. 760:39–62
     [Google Scholar]
  163. van Os RGM, Phillips TN 2004. Spectral element methods for transient viscoelastic flow problems. J. Comput. Phys. 201:286–314
     [Google Scholar]
  164. Varchanis S, Hopkins CC, Shen AQ, Tsamopoulos J, Haward SJ 2019a. Asymmetric flows of complex fluids past confined cylinders: a comprehensive numerical study with experimental validation. Phys. Fluids 32:053103
     [Google Scholar]
  165. Varchanis S, Makrigiorgos G, Moschopoulos P, Dimakopoulos Y, Tsamopoulos J 2019b. Modeling the rheology of thixotropic elasto-visco-plastic materials. J. Rheol. 63:609–39
     [Google Scholar]
  166. Vázquez-Quesada A, Ellero M, Español P 2012. An-SPH based particle model for computational microrheology. Microfluid. Nanofluid. 13:249–60
     [Google Scholar]
  167. Verbeeten WMH, Peters GWM, Baaijens FPT 2001. Differential constitutive equations for polymer melts: the extended Pom-Pom model. J. Rheol 45:823–843Erratum. 2001 J. Rheol 45:1489
     [Google Scholar]
  168. Walters K. 1982. Special issue on “Numerical simulation in non-Newtonian fluid mechanics.”. J. Non-Newton. Fluid Mech. 10:1
     [Google Scholar]
  169. Waters ND, King MJ. 1970. Unsteady flow of an elastico-viscous liquid. Rheol. Acta 9:345–55
     [Google Scholar]
  170. Xue SC, Phan-Thien N, Tanner RI 1998. Three dimensional numerical simulations of viscoelastic flows through planar contractions. J. Non-Newton. Fluid Mech. 74:195–245
     [Google Scholar]
  171. Xue SC, Tanner RI, Phan-Thien N 2004. Numerical modelling of transient viscoelastic flows. J. Non-Newton. Fluid Mech. 123:33–58
     [Google Scholar]
  172. Yoo JY, Na Y. 1991. A numerical study of the planar contraction flow of viscoelastic fluids using the SIMPLER algorithm. J. Non-Newton. Fluid Mech. 39:89–106
     [Google Scholar]
  173. Zografos K, Burshtein N, Shen AQ, Haward SJ, Poole RJ 2018. Elastic modifications of an inertial instability in a 3D cross-slot. J. Non-Newton. Fluid Mech. 262:12–24
     [Google Scholar]
/content/journals/10.1146/annurev-fluid-010719-060107
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Numerical Methods for Viscoelastic Fluid Flows
Annual Review of Fluid Mechanics 53, 509 (2021); https://doi.org/10.1146/annurev-fluid-010719-060107
/content/journals/10.1146/annurev-fluid-010719-060107
/content/journals/10.1146/annurev-fluid-010719-060107
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