# A Dual Mixed Formulation for Non-isothermal Oldroyd–Stokes Problem

Mathematical Modelling of Natural Phenomena (2011)

- Volume: 6, Issue: 5, page 130-156
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topFarhloul, M., and Zine, A.. "A Dual Mixed Formulation for Non-isothermal Oldroyd–Stokes Problem." Mathematical Modelling of Natural Phenomena 6.5 (2011): 130-156. <http://eudml.org/doc/222246>.

@article{Farhloul2011,

abstract = {We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the both
extra stress and the heat flux’s vector are considered. Based on such a formulation, a
dual mixed finite element is constructed and analyzed. This finite element method enables
us to obtain precise approximations of the dual variable which are, for the non-isothermal
fluid flow problems, the viscous and polymeric components of the extra-stress tensor, as
well as the heat flux. Furthermore, it has properties analogous to the finite volume
methods, namely, the local conservation of the momentum and the mass.},

author = {Farhloul, M., Zine, A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Oldroyd–Stokes problem; non–isotherm; dual mixed formulation; Oldroyd-Stokes problem; non-isotherm},

language = {eng},

month = {8},

number = {5},

pages = {130-156},

publisher = {EDP Sciences},

title = {A Dual Mixed Formulation for Non-isothermal Oldroyd–Stokes Problem},

url = {http://eudml.org/doc/222246},

volume = {6},

year = {2011},

}

TY - JOUR

AU - Farhloul, M.

AU - Zine, A.

TI - A Dual Mixed Formulation for Non-isothermal Oldroyd–Stokes Problem

JO - Mathematical Modelling of Natural Phenomena

DA - 2011/8//

PB - EDP Sciences

VL - 6

IS - 5

SP - 130

EP - 156

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

LA - eng

KW - Oldroyd–Stokes problem; non–isotherm; dual mixed formulation; Oldroyd-Stokes problem; non-isotherm

UR - http://eudml.org/doc/222246

ER -

## References

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

## NotesEmbed ?

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