# Free-energy-dissipative schemes for the Oldroyd-B model

Sébastien Boyaval; Tony Lelièvre; Claude Mangoubi

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 3, page 523-561
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBoyaval, Sébastien, Lelièvre, Tony, and Mangoubi, Claude. "Free-energy-dissipative schemes for the Oldroyd-B model." ESAIM: Mathematical Modelling and Numerical Analysis 43.3 (2009): 523-561. <http://eudml.org/doc/250596>.

@article{Boyaval2009,

abstract = {
In this article,
we analyze the stability of various numerical schemes for differential models of viscoelastic fluids.
More precisely, we consider the prototypical Oldroyd-B model,
for which a free energy dissipation holds,
and we show under which assumptions such a dissipation is also satisfied for the numerical scheme.
Among the numerical schemes we analyze,
we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed
by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech.123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al.J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.
},

author = {Boyaval, Sébastien, Lelièvre, Tony, Mangoubi, Claude},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Viscoelastic fluids; Weissenberg number; stability; entropy; finite elements methods;
discontinuous Galerkin method; characteristic method.; finite element method},

language = {eng},

month = {4},

number = {3},

pages = {523-561},

publisher = {EDP Sciences},

title = {Free-energy-dissipative schemes for the Oldroyd-B model},

url = {http://eudml.org/doc/250596},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Boyaval, Sébastien

AU - Lelièvre, Tony

AU - Mangoubi, Claude

TI - Free-energy-dissipative schemes for the Oldroyd-B model

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/4//

PB - EDP Sciences

VL - 43

IS - 3

SP - 523

EP - 561

AB -
In this article,
we analyze the stability of various numerical schemes for differential models of viscoelastic fluids.
More precisely, we consider the prototypical Oldroyd-B model,
for which a free energy dissipation holds,
and we show under which assumptions such a dissipation is also satisfied for the numerical scheme.
Among the numerical schemes we analyze,
we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed
by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech.123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al.J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.

LA - eng

KW - Viscoelastic fluids; Weissenberg number; stability; entropy; finite elements methods;
discontinuous Galerkin method; characteristic method.; finite element method

UR - http://eudml.org/doc/250596

ER -

## References

top- D.N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations, Volume VII, R. Vichnevetsky and R.S. Steplemen Eds. (1992).
- M. Bajaj, M. Pasquali and J.R. Prakash, Coil-stretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder. J. Rheol.52 (2008) 197–223.
- J. Baranger and A. Machmoum, Existence of approximate solutions and error bounds for viscoelastic fluid flow: Characteristics method. Comput. Methods Appl. Mech. Engrg.148 (1997) 39–52.
- J.W. Barrett and S. Boyaval, Convergence of a finite element approximation to a regularized Oldroyd-B model (in preparation).
- J.W. Barrett, C. Schwab and E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Mod. Meth. Appl. Sci.15 (2005) 939–983.
- A.N. Beris and B.J. Edwards, Thermodynamics of flowing systems with internal microstructure. Oxford University Press (1994).
- J. Bonvin, M. Picasso and R. Stenberg, GLS and EVSS methods for a three fields Stokes problem arising from viscoelastic flows. Comp. Meth. Appl. Mech. Eng.190 (2001) 3893–3914.
- F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in Efficient Solution of Elliptic System, W. Hackbusch Ed. (1984) 11–19.
- F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.
- F. Brezzi, J. Douglas, Jr. and L.D. Marini, Recent results on mixed finite element methods for second order elliptic problems, in Vistas in Applied Mathematics: Numerical Analysis, Atmospheric Sciences, Immunology, A.V. Balakrishnan, A.A. Dorodnitsyn and J.L. Lions Eds. (1986) 25–43.
- R. Codina, Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comp. Meth. Appl. Mech. Engrg.156 (1998) 185–210.
- M. Crouzeix and P.A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér.3 (1973) 33–75.
- A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer Verlag, New-York (2004).
- R. Fattal and R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech.123 (2004) 281–285.
- R. Fattal and R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newtonian Fluid Mech.126 (2005) 23–37.
- A. Fattal, O.H. Hald, G. Katriel and R. Kupferman, Global stability of equilibrium manifolds, and “peaking" behavior in quadratic differential systems related to viscoelastic models. J. Non-Newtonian Fluid Mech.144 (2007) 30–41.
- E. Fernández-Cara, F. Guillén and R.R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of Numerical Analysis, Vol. 8, P.G. Ciarlet et al. Eds., Elsevier (2002) 543–661.
- J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN33 (1999) 1293–1316.
- C. Guillopé and J.C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlin. Anal. TMA15 (1990) 849–869.
- D. Hu and T. Lelièvre, New entropy estimates for the Oldroyd-B model, and related models. Commun. Math. Sci.5 (2007) 906–916.
- T.J.R. Hughes and L.P. Franca, A new finite element formulation for CFD: VII the Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces. Comp. Meth. App. Mech. Eng.65 (1987) 85–96.
- M.A. Hulsen, A sufficient condition for a positive definite configuration tensor in differential models. J. Non-Newtonian Fluid Mech.38 (1990) 93–100.
- M.A. Hulsen, R. Fattal and R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech.127 (2005) 27–39.
- B. Jourdain, C. Le Bris, T. Lelièvre and F. Otto, Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal.181 (2006) 97–148.
- N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput.58 (1992) 1–10.
- R.A. Keiller, Numerical instability of time-dependent flows. J. Non-Newtonian Fluid Mech.43 (1992) 229–246.
- R. Keunings, Simulation of viscoelastic fluid flow, in Fundamentals of Computer Modeling for Polymer Processing, C. Tucker Ed., Hanse (1989) 402–470.
- R. Keunings, A survey of computational rheology, in Proc. 13th Int. Congr. on Rheology, D.M. Binding et al Eds., British Society of Rheology (2000) 7–14.
- R. Kupferman, C. Mangoubi and E. Titi, A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime. Comm. Math. Sci.6 (2008) 235–256.
- Y. Kwon, Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations. Korea-Australia Rheology Journal16 (2004) 183–191.
- Y. Kwon and A.V. Leonov, Stability constraints in the formulation of viscoelastic constitutive equations. J. Non-Newtonian Fluid Mech.58 (1995) 25–46.
- Y. Lee and J. Xu, New formulations positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg.195 (2006) 1180–1206.
- A.I. Leonov, Analysis of simple constitutive equations for viscoelastic liquids. J. Non-Newton. Fluid Mech.42 (1992) 323–350.
- F.-H. Lin, C. Liu and P.W. Zhang, On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math.58 (2005) 1437–1471.
- P.L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B21 (2000) 131–146.
- A. Lozinski and R.G. Owens, An energy estimate for the Oldroyd-B model: theory and applications. J. Non-Newtonian Fluid Mech.112 (2003) 161–176.
- R. Mneimne and F. Testard, Introduction à la théorie des groupes de Lie classiques. Hermann (1986).
- K.W. Morton, A. Priestley and E. Süli, Convergence analysis of the Lagrange-Galerkin method with non-exact integration. RAIRO Modél. Math. Anal. Numér.22 (1988) 625–653.
- H.C. Öttinger, Beyond Equilibrium Thermodynamics. Wiley (2005).
- O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math.3 (1982) 309–332.
- J.M. Rallison and E.J. Hinch, Do we understand the physics in the constitutive equation? J. Non-Newtonian Fluid Mech.29 (1988) 37–55.
- D. Sandri, Non integrable extra stress tensor solution for a flow in a bounded domain of an Oldroyd fluid. Acta Mech.135 (1999) 95–99.
- L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér.19 (1985) 111–143.
- E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math.53 (1988) 459–483.
- R. Temam, Sur l'approximation des équations de Navier-Stokes. C. R. Acad. Sci. Paris, Sér. A262 (1966) 219–221.
- B. Thomases and M. Shelley, Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids19 (2007) 103103.
- P. Wapperom and M.A. Hulsen, Thermodynamics of viscoelastic fluids: the temperature equation. J. Rheol.42 (1998) 999–1019.
- P. Wapperom, R. Keunings and V. Legat, The backward-tracking lagrangian particle method for transient viscoelastic flows. J. Non-Newtonian Fluid Mech.91 (2000) 273–295.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.