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Free-energy-dissipative schemes for the Oldroyd-B model. (English) Zbl 1167.76018
Summary: 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 holds also for the numerical scheme. Among the analyzed numerical schemes, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by R. Fattal and R. Kupferman in [J. Non-Newton. Fluid Mech. 123, No. 2–3, 281–285 (2004; Zbl 1084.76005)], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [M. A. Hulsen et al., J. Non-Newton. Fluid Mech. 127, 27–39 (2005)]). Our analysis gives some tracks to understand these numerical observations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76A10 Viscoelastic fluids
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References:
[1] 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).
[2] 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.
[3] 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. Zbl0923.76098 · Zbl 0923.76098
[4] J.W. Barrett and S. Boyaval, Convergence of a finite element approximation to a regularized Oldroyd-B model (in preparation). · Zbl 1256.35048
[5] 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. · Zbl 1161.76453
[6] A.N. Beris and B.J. Edwards, Thermodynamics of flowing systems with internal microstructure. Oxford University Press (1994).
[7] 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. · Zbl 1014.76043
[8] 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.
[9] 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. · Zbl 0599.65072
[10] 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. · Zbl 0611.65071
[11] R. Codina, Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comp. Meth. Appl. Mech. Engrg.156 (1998) 185-210. · Zbl 0959.76040
[12] 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. · Zbl 0302.65087
[13] A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer Verlag, New-York (2004). · Zbl 1059.65103
[14] R. Fattal and R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech.123 (2004) 281-285. · Zbl 1084.76005
[15] 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. · Zbl 1099.76044
[16] 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. · Zbl 1195.76191
[17] 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. · Zbl 1024.76003
[18] J.-L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM: M2AN33 (1999) 1293-1316. · Zbl 0946.65112
[19] 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. Zbl0729.76006 · Zbl 0729.76006
[20] D. Hu and T. Lelièvre, New entropy estimates for the Oldroyd-B model, and related models. Commun. Math. Sci.5 (2007) 906-916. · Zbl 1137.35318
[21] 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. Zbl0635.76067 · Zbl 0635.76067
[22] M.A. Hulsen, A sufficient condition for a positive definite configuration tensor in differential models. J. Non-Newtonian Fluid Mech.38 (1990) 93-100.
[23] 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. · Zbl 1187.76615
[24] 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. · Zbl 1089.76006
[25] N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput.58 (1992) 1-10. · Zbl 0738.76040
[26] R.A. Keiller, Numerical instability of time-dependent flows. J. Non-Newtonian Fluid Mech.43 (1992) 229-246. Zbl0771.76018 · Zbl 0771.76018
[27] R. Keunings, Simulation of viscoelastic fluid flow, in Fundamentals of Computer Modeling for Polymer Processing, C. Tucker Ed., Hanse (1989) 402-470.
[28] 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.
[29] 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. Zbl1140.35345 · Zbl 1140.35345
[30] 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.
[31] Y. Kwon and A.V. Leonov, Stability constraints in the formulation of viscoelastic constitutive equations. J. Non-Newtonian Fluid Mech.58 (1995) 25-46.
[32] 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. Zbl1176.76068 · Zbl 1176.76068
[33] A.I. Leonov, Analysis of simple constitutive equations for viscoelastic liquids. J. Non-Newton. Fluid Mech.42 (1992) 323-350. · Zbl 0747.76017
[34] F.-H. Lin, C. Liu and P.W. Zhang, On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math.58 (2005) 1437-1471. · Zbl 1076.76006
[35] P.L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B21 (2000) 131-146. Zbl0957.35109 · Zbl 0957.35109
[36] 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. · Zbl 1065.76018
[37] R. Mneimne and F. Testard, Introduction à la théorie des groupes de Lie classiques. Hermann (1986). · Zbl 0598.22001
[38] 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. · Zbl 0661.65114
[39] H.C. Öttinger, Beyond Equilibrium Thermodynamics. Wiley (2005).
[40] O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math.3 (1982) 309-332. Zbl0505.76100 · Zbl 0505.76100
[41] 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.
[42] 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. Zbl0931.76007 · Zbl 0931.76007
[43] 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. Zbl0608.65013 · Zbl 0608.65013
[44] E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math.53 (1988) 459-483. Zbl0637.76024 · Zbl 0637.76024
[45] R. Temam, Sur l’approximation des équations de Navier-Stokes. C. R. Acad. Sci. Paris, Sér. A262 (1966) 219-221. Zbl0173.11902 · Zbl 0173.11902
[46] B. Thomases and M. Shelley, Emergence of singular structures in Oldroyd-B fluids. Phys. Fluids19 (2007) 103103. Zbl1182.76758 · Zbl 1182.76758
[47] P. Wapperom and M.A. Hulsen, Thermodynamics of viscoelastic fluids: the temperature equation. J. Rheol.42 (1998) 999-1019.
[48] 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. Zbl0972.76081 · Zbl 0972.76081
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