Stable finite element schemes are developed for the solution of the equations modeling the flow of viscoelastic fluids. In contrast with classical statements of these equations, which introduce the stress as a primary variable, these schemes explicitly involve the deformation tensor and elastic energy. Energy estimates and existence of solutions to the discrete problem are established for schemes of arbitrary order without any restrictions on the time step, mesh size, or Weissenberg number. Convergence to smooth solutions is established for the classical Oldroyd–B fluid. Numerical experiments for two classical benchmark problems verify the robustness of this approach.
Accepté le :
DOI : 10.1051/m2an/2016053
Mots clés : Viscoelastic fluid, Oldroyd–B, high weissenberg number problem
@article{M2AN_2017__51_3_1119_0,
author = {Perrotti, Louis and Walkington, Noel J. and Wang, Daren},
title = {Numerical approximation of viscoelastic fluids},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1119--1144},
publisher = {EDP-Sciences},
volume = {51},
number = {3},
year = {2017},
doi = {10.1051/m2an/2016053},
zbl = {1398.76122},
mrnumber = {3666659},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016053/}
}
TY - JOUR AU - Perrotti, Louis AU - Walkington, Noel J. AU - Wang, Daren TI - Numerical approximation of viscoelastic fluids JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1119 EP - 1144 VL - 51 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016053/ DO - 10.1051/m2an/2016053 LA - en ID - M2AN_2017__51_3_1119_0 ER -
%0 Journal Article %A Perrotti, Louis %A Walkington, Noel J. %A Wang, Daren %T Numerical approximation of viscoelastic fluids %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1119-1144 %V 51 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016053/ %R 10.1051/m2an/2016053 %G en %F M2AN_2017__51_3_1119_0
Perrotti, Louis; Walkington, Noel J.; Wang, Daren. Numerical approximation of viscoelastic fluids. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1119-1144. doi : 10.1051/m2an/2016053. http://archive.numdam.org/articles/10.1051/m2an/2016053/
, , and , The log-conformation tensor approach in the finite-volume method framework. J. Non-Newtonian Fluid Mechanics 157 (2009) 55–65. | DOI | Zbl
, and , Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newtonian Fluid Mechanics 110 (2003) 45–75. | DOI | Zbl
, , and , Symmetric factorization of the conformation tensor in viscoelastic fluid models. In: XVIth International Workshop on Numerical Methods for Non-Newtonian Flows. J. Non-Newtonian Fluid Mechanics 166 (2011) 546–553. | DOI | Zbl
and , Existence and approximation of a (regularized) Oldroyd-B model. Math. Models Methods Appl. Sci. 21 (2011) 1783–1837. | DOI | MR | Zbl
S. Boyaval, Lid-driven-cavity simulations of Oldroyd-B models using free-energy-dissipative schemes, in The eight European Conference on Numerical Mathematics and Advanced Applications. Springer Verlag (2010) 191–198. | Zbl
, and , Free-energy-dissipative schemes for the Oldroyd-B model. ESAIM: M2AN 43 (2009) 523–561. | DOI | Numdam | MR | Zbl
, On the inertial and extensional effects on the corner and lip vortices in a circular 4:1 abrupt contraction. J. Non-Newtonian Fluid Mech. 37 (1990) 281–296. | DOI
and , Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation. Numer. Methods Partial Differ. Eqs. 20 (2004) 248–283. | DOI | MR | Zbl
and , Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation. Numer. Methods Partial Differ. Eqs. 20 (2004) 248–283. | DOI | MR | Zbl
and , Defect correction method for viscoelastic fluid flows at high Weissenberg number. Numer. Methods Partial Differ. Eqs. 22 (2006) 145–164. | DOI | MR | Zbl
and , Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal. 41 (2003) 457–486. | DOI | MR | Zbl
, and , Flow of viscoelastic fluids past a cylinder at high weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech. 127 (2005) s27–39. | DOI | Zbl
R. Fattal and R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. (2004) 281–285. | Zbl
R. Fattal and R. Kupferman, Time-dependent simulation of viscoelastic flows at high weissenberg number using the log-conformation representations. J. Non-Newtonian Fluid Mech. (2005) 23–37. | Zbl
C. Guillopé and J.-C. Saut, Mathematical problems arising in differential models for viscoelastic fluids, in Mathematical topics in fluid mechanics (Lisbon, 1991). Vol. 274 of Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow (1992) 64–92. | MR | Zbl
M.E. Gurtin, An introduction to continuum mechanics, vol. 158 of Mathematics in Science and Engineering. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1981). | MR | Zbl
E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, in Structure-preserving algorithms for ordinary differential equations, vol. 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2002). | MR | Zbl
L. He and P. Zhang, L2 decay of solutions to micro-macro model for polymeric fluids near equilibrium. SIAM J. Math. Anal. (2009) 1905–1922. | MR | Zbl
W.J. Hrusa, M. Renardy and J.A. Nohel, Mathematical problems in viscoelasticity, vol. 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow (1987). | MR | Zbl
, The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. London 102 (1922) 161–179. | JFM
and , A model for viscoelastic fluid behavior which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2 (1977) 255–270. | DOI | Zbl
Y.-J. Lee, J. Xu and C.-S. Zhang, Stable Finite Element Discretizations for Viscoelastic Flow Models, in Numerical Methods for Non-Newtonian Fluids, edited by R. Glowinski and J. Xu. Vol. 16 of Handbook of Numerical Analysis. Elsevier (2011) 371–432. | Zbl
, and , On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58 (2005) 1437–1471. | DOI | MR | Zbl
and , Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B 21 (2000) 131–146. | DOI | MR | Zbl
and , An energy estimate for the Oldroyd B model: theory and applications. J. Non-Newtonian Fluid Mech. 112 (2003) 161–176. | DOI | Zbl
R.G. Owens and T.N. Phillips, Computational rheology. Imperial College Press, London (2002). | MR | Zbl
M. Renardy, Mathematical analysis of viscoelastic flows. Vol. 73 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000). | MR | Zbl
R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. American Mathematical Society, Providence, RI (1997). Available at: http://www.ams.org/online˙bks/surv49/ | MR | Zbl
, , and , Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG. J. Non-Newtonian Fluid Mech. 86 (1999) 281–307. | DOI | Zbl
V. Thomee, Galerkin Finite Element Methods for Parabolic Problems. Vol. 1054 of Lect. Notes Math. Springer (1984). | MR | Zbl
, Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations. ESAIM: M2AN 45 (2011) 523–540. | DOI | Numdam | MR | Zbl
K. Yosida, Functional Analysis. Vol. 123 of Grundlehren Math. Wiss., Springer Verlag (1980). | MR | Zbl
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