# zbMATH — the first resource for mathematics

Numerical modeling of viscoelastic flows using equal low-order finite elements. (English) Zbl 1227.76033
Summary: A mixed finite element scheme abbreviated as I_PS_DEVSS_CNBS scheme for modeling viscoelastic flow problems is presented. The finite incremental calculus (FIC) pressure stabilization process and the discrete elastic-viscous stress-splitting method (DEVSS) are introduced into the general framework of the iterative version of the fractional step algorithm with the use of the Crank-Nicolson-based splitting. Inconsistent streamline upwinding method (SU) is employed to spatially discretize the constitutive equation of viscoelastic fluids. Equal low-order finite elements which violate the LBB compatibility conditions are successfully used in the proposed scheme. In addition, the Oldroyd-B and the PTT models have been integrated into the proposed scheme to solve the 4:1 sudden contraction flow problem. The numerical results demonstrate prominent stability and accuracy of both pressure and stress distributions over the flow domains provided by the proposed scheme within the Weissenberg number range studied in the present work, as compared with the reference solutions reported in the literatures.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76A10 Viscoelastic fluids
Full Text:
##### References:
 [1] Baaijens, F.P.T., Mixed finite element methods for viscoelastic flow analysis: a review, J. non-Newtonian fluid mech., 79, 361-385, (1998) · Zbl 0957.76024 [2] Crochet, M.J.; Keunings, R., Die swell of a Maxwell fluid: numerical prediction, J. non-Newtonian fluid mech., 7, 199-212, (1980) [3] Crochet, M.J.; Keunings, R., On numerical die swell calculations, J. non-Newtonian fluid mech., 10, 85-94, (1982) · Zbl 0482.76004 [4] Burdette, S.R.; Coates, P.J.; Armstrong, R.C.; Brown, R.A., Calculations of viscoelastic flow through an axisymmetric corrugated tube using the explicitly elliptic momentum equation formulation (EEME), J. non-Newtonian fluid mech., 33, 1-23, (1989) · Zbl 0679.76012 [5] Rajagopalan, D.; Armstrong, R.C.; Brown, R.A., Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity, J. non-Newtonian fluid mech., 36, 159-192, (1990) · Zbl 0709.76011 [6] Beris, A.N.; Armstrong, R.C.; Brown, R.A., Finite element calculation of viscoelastic flow in a journal bearing. I. small eccentricities, J. non-Newtonian fluid mech., 16, 141-172, (1984) · Zbl 0559.76015 [7] Beris, A.N.; Armstrong, R.C.; Brown, R.A., Finite element calculation of viscoelastic flow in a journal bearing. II. moderate eccentricities, J. non-Newtonian fluid mech., 19, 323-347, (1986) · Zbl 0598.76014 [8] Guenette, R.; Fortin, M., A new mixed finite element method for computing viscoelastic flows, J. non-Newtonian fluid mech., 60, 27-52, (1995) [9] Trebotich, D.; Colella, P.; Miller, G.H., A stable and convergent scheme for viscoelastic flow in contraction channels, J. comput. phys., 205, 315-342, (2005) · Zbl 1087.76005 [10] Nithiarasu, P., A fully explicit characteristic based split (CBS) scheme for viscoelastic flow calculations, Int J. numer. methods engrg., 60, 949-978, (2004) · Zbl 1060.76628 [11] Coronado, O.M.; Arora, D.; Behr, M.; Pasquali, M., Four-field Galerkin/least-squares formulation for viscoelastic fluids, J. non-Newtonian fluid mech., 140, 132-144, (2006) · Zbl 1143.76323 [12] Behr, M.; Arora, D.; Coronado, O.M.; Pasquali, M., GLS-type finite element methods for viscoelastic fluid flow simulation, Comput. fluid solid mech., 135-308, (2005) [13] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comput., 22, 742-762, (1968) · Zbl 0198.50103 [14] Temam, R., Sur 1’ approximation de la solution des équations de navier – stokes par la méthode des pas fractionnaries II, Arch. ration. mech. anal., 33, 377-385, (1969) · Zbl 0207.16904 [15] Comini, G.; Del Guidice, S., Finite element solution of incompressible navier – stokes equations, Numer. heat transfer, part A, 5, 463-478, (1972) [16] Donea, J.; Giuliani, S.; Laval, H.; Quartapelle, L., Finite element solution of unsteady navier – stokes equations by a fractional step method, Comput. methods appl. mech. engrg., 33, 53-73, (1982) · Zbl 0481.76037 [17] Li, X.K.; Han, X.H., An iterative stabilized fractional step algorithm for numerical solution of incompressible N-S equations, Int. J. numer. methods fluid, 49, 395-416, (2005) · Zbl 1185.76501 [18] Han, X.H.; Li, X.K., An iterative stabilized CNBS-CG scheme for incompressible non-isothermal non-Newtonian fluid flow, Int. J. heat mass transfer, 50, 847-856, (2007) · Zbl 1160.76386 [19] Li, X.K.; Duan, Q.L., Meshfree iterative stabilized taylor – galerkin and characteristic-based split (CBS) algorithms for incompressible N-S equations, Comput. methods appl. mech. engrg., 195, 6125-6145, (2006) · Zbl 1123.76052 [20] Guermond, J.L.; Quartapelle, L., On stability and convergence of projection methods based on pressure Poisson equation, Int J. numer. methods fluid, 26, 1039-1053, (1998) · Zbl 0912.76054 [21] Codina, R., Pressure stability in fractional step finite element methods for incompressible flows, J. comput. phys., 170, 112-140, (2001) · Zbl 1002.76063 [22] Onate, E., A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation, Comput. methods appl. mech. engrg., 182, 355-370, (2000) · Zbl 0977.76050 [23] Joseph, D.D., Change of type and loss of evolution in flow of viscoelastic fluids, J. non-Newtonian fluid mech., 20, 117-141, (1986) · Zbl 0626.76009 [24] King, R.C.; Apelian, M.R.; Armstrong, R.C.; Brown, R.A., Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries, J. non-Newtonian fluid mech., 29, 147-216, (1988) · Zbl 0666.76034 [25] Marchal, J.M.; Crochet, M.J., A new mixed finite element for calculating viscoelastic flow, J. non-Newtonian fluid mech., 26, 77-114, (1987) · Zbl 0637.76009 [26] Baaijens, F.P.T., Application of low-order discontinuous Galerkin methods to the analysis of viscoelastic flows, J. non-Newtonian fluid mech., 52, 37-57, (1994) [27] Fortin, M.; Fortin, A., A new approach for the FEM simulation of viscoelastic flows, J. non-Newtonian fluid mech., 32, 295-310, (1989) · Zbl 0672.76010 [28] Walters, K.; Webster, M.F., The distinctive CFD challenges of computational rheology, Int. J. numer. methods fluid, 43, 577-596, (2003) · Zbl 1032.76606 [29] White, S.A.; Gotsis, A.D.; Baird, D.G., Review of the entry flow problem: experimental and numerical, J. non-Newtonian fluid mech., 24, 121-160, (1987) [30] Evans, R.E.; Walters, K., Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows, J. non-Newtonian fluid mech., 32, 95-105, (1989) [31] Bogaerds, A.C.B.; Verbeeten, W.M.H.; Perters, G.W.M.; Baaijens, F.P.T., 3D viscoelastic analysis of a polymer solution in a complex flow, Comput. methods appl. mech. engrg., 180, 413-430, (1999) [32] Phan-Thien, N.; Tanner, R.I., A new constitutive equation derived from network theory, J. non-Newtonian fluid mech., 2, 353-365, (1977) · Zbl 0361.76011 [33] Codina, R.; Blasco, J., A finite element formulation for the Stokes problem allowing equal velocity – pressure interpolation, Comput. methods appl. mech. engrg., 143, 373-391, (1997) · Zbl 0893.76040 [34] Hawken, D.M.; Tamaddon-Jahromi, H.R.; Townsend, P.; Webster, M.F., A taylor – galerkin- based algorithm for viscous incompressible flow, Int J. numer. methods fluid, 10, 327-351, (1990) · Zbl 0686.76019 [35] Codina, R.; Folch, A., A stabilized finite element predictor – corrector scheme for the incompressible navier – stokes equations using a nodal-based implementation, Int. J. numer. methods fluid, 44, 483-503, (2004) · Zbl 1085.76035 [36] Blasco, J.; Codina, R.; Huerta, A., A fractional step method for the incompressible navier – stokes equations related to a predictor – multicorrector algorithm, Int. J. numer. methods fluid, 28, 1391-1419, (1998) · Zbl 0935.76041 [37] Matallah, H.; Townsend, P.; Webster, M.F., Recovery and stress-splitting schemes for viscoelastic flows, J. non-Newtonian fluid mech., 75, 139-166, (1998) · Zbl 0960.76047 [38] Oliveira, P.J.; Pinho, F.T., Analytical solutions for fully developed channel and pipe flow of phan – thien – tanner fluids, J. fluid mech., 387, 271-280, (1999) · Zbl 0938.76004 [39] Cruz, D.O.A.; Pinho, F.T.; Oliveira, P.J., Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution, J. non-Newtonian fluid mech., 132, 28-35, (2005) · Zbl 1195.76086 [40] 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-Newtonian fluid mech., 126, 207-220, (2005) · Zbl 1088.76553 [41] 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-Newtonian fluid mech., 134, 105-126, (2006) · Zbl 1123.76307 [42] Sato, T.; Richardson, S.M., Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method, J. non-Newtonian fluid mech., 51, 249-275, (1994) [43] Phillips, T.N.; Williams, A.J., Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. non-Newtonian fluid mech., 87, 215-246, (1999) · Zbl 0945.76052 [44] Alves, M.A.; Oliveira, P.J.; Pinho, F.T., Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions, J. non-Newtonian fluid mech., 110, 45-75, (2003) · Zbl 1027.76003 [45] Aboubacar, M.; Matallah, H.; 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-Newtonian fluid mech., 103, 65-103, (2002) · Zbl 1143.76492
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.