Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T12:21:26.575Z Has data issue: false hasContentIssue false
  • English
  • Français

A Dual Mixed Formulation for Non-isothermal Oldroyd–StokesProblem

Published online by Cambridge University Press:  10 August 2011

M. Farhloul  and
A. Zine
M. Farhloul
Affiliation:
Département de Mathématiques et de Statistique, Université de Moncton Moncton, N.B., E1A 3E9, Canada
A. Zine*
Affiliation:
Université de Lyon, Ecole Centrale de Lyon, CNRS UMR 5208 Institut Camille Jordan 36, rue Guy de Collongue, 69134 Ecully, France
*
⋆ Corresponding author. E-mail: Abdel-Malek.Zine@ec-lyon.fr
Get access

Abstract

We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the bothextra stress and the heat flux’s vector are considered. Based on such a formulation, adual mixed finite element is constructed and analyzed. This finite element method enablesus to obtain precise approximations of the dual variable which are, for the non-isothermalfluid flow problems, the viscous and polymeric components of the extra-stress tensor, aswell as the heat flux. Furthermore, it has properties analogous to the finite volumemethods, namely, the local conservation of the momentum and the mass.

Keywords

Oldroyd–Stokes problemnon–isothermdual mixed formulation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 130 - 156
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Brezzi, F., Douglas, J.. PEERS: A new mixed finite element for plane elasticity. Japan J. Appl. Math., 1 (1984), 347–367. CrossRefGoogle Scholar
Baranger, J., Sandri, D.. A formulation of Stokes’s problem and the linear elasticity equations suggested by the Oldroyd model for viscoelastic flow. M2AN, 26 (1992), 331–345. CrossRefGoogle Scholar
F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin, 1991.
P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, 1978.
Clément, P.. Approximation by finite element functions using local regularization. RAIRO Anal. Numer., 2 (1975), 77–84. Google Scholar
Cox, C., Lee, H., Szurley, D.. Finite element approximation of the non–isothermal Stokes–Oldroyd equations. Int. J. Numer. Anal. Mod., 4 (2007), 425–440. Google Scholar
Damak Besbes, S., Guillopé, C.. Non-isothermal flows of viscoelastic incompressible fluids. Nonlinear Analysis, 44 (2001), 919–942. CrossRefGoogle Scholar
Douglas, J. Jr., Roberts, J.E.. Global estimates for mixed methods for second order elliptic equations. Math. Comp., 44 (1985), 39–52. CrossRefGoogle Scholar
L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1999.
Farhloul, M., Fortin, M.. A new mixed finite element for the Stokes and elasticity problems. SIAM J. Numer. Anal., 30 (1993), 971–990. CrossRefGoogle Scholar
Farhloul, M., Fortin, M.. Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math., 76 (1997), 419–440. CrossRefGoogle Scholar
Farhloul, M., Zine, A.M.. A new mixed finite element method for the Stokes problem. J. Math. Anal. Appl., 276 (2002), 329–342. CrossRefGoogle Scholar
Grisvard, P.. Problèmes aux limites dans les polygones, mode d’emploi. EDF Bull. Direction Etudes Rech. Sér. C Math. Inform., 1 (1986), 21–59. Google Scholar
Nedelec, J.C.. Mixed finite elements in ℝ3. Numer. Math., 35 (1980), 315–341. CrossRefGoogle Scholar
Peters, G.W.M., Baaijens, F.T.O.. Modelling of non-isothermal viscoelastic flow. J. Non-Newtonian Fluid Mech., 68 (1997), 205–224. CrossRefGoogle Scholar
P.A. Raviart, J.M. Thomas. A mixed finite element method for 2nd order elliptic problems, Lecture Notes in Mathematics, Vol. 606, Springer-Verlag, New-York, 1977, pp. 292-315.
J.E. Roberts, J.M. Thomas. Mixed and hybrid finite element methods, Handbook of Numerical Analysis, vol. II, Finite Element Methods (part I), P.G Ciarlet, J.L. Lions (Eds.), North-Holland, 1989.