Existence result for a mixture of non newtonian flows with stress diffusion using the Cahn-Hilliard formulation

Discrete and Continuous Dynamical Systems - Series B

Volumn 3, Issue 1, 2003, Pages 45-68

  Chupin, L ^a  

^a UNIVERSITÉ BORDEAUX 1   (France)

Author keywords

Cahn Hilliard equation; Navier Stokes equation; Oldroyd model

Indexed keywords

EID: 0042762786     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2003.3.45     Document Type: Article

References (24)

1. M. Berger, B. Gostiaux, Géométrie différentielle: variétés, courbes et surfaces, P U F (1987)
2. F. Boyer, Mathematical study of multiphase flow under shear through order parameter formulation, Asymptotic Analysis, Vol. 20 n.2, pp 175-212 (1999)
3. F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Computers and fluids, Vol. 31, pp 41-68 (2001)
4. J.W. Cahn, J.E. Hilliard, Free energy of non-uniform system. I. Interfacial free energy, J. Chem. Phys. 28, pp 258-267 (1958)
5. R. Chella, J. Vinals, Mixing of two-phase fluids by a cavity flow, Physical Review E 53, pp 3832-3840 (1996)
6. P. Constantin, C. Foias, Navier-Stokes Equations, University of Chicago Press (1989)
7. M. Doi, Dynamics of domains and textures, Theoretical Challenges in the dynamics of complex fluids, pp 293-314 (1997)
8. A.W. El-Kareh, L.G. Leal, Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion, J. Non-Newtonian Fluid Mech. 33, No.3, pp 257-287 (1989)
9. C. M. Elliott, H. Garcke, On the Cahn-Hilliard Equation with degenerate mobility, Siam J. Math. Anal. 27, n 1, pp 404-423 (1996)
10. P. Germain, Cours de mécanique des milieux continus, Tome 1, Masson (1973)
11. C. Guillopé, J.C. Saut, Résultats d'existence pour des fluides viscoélastiques à la loi de comportement de type différentiel, C. R. Acad. Sci. Paris, t.305, Série I, pp 489-492 (1987)
12. D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer (1990)
13. I. Kukavica, Interior gradient bounds for the SD Navier-Stokes system, Discrete and Continuous Dynamical Systems, vol. 7, pp 873-882 (2001)
14. O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach (1969)
15. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod (1969)
16. P.L. Lions, N. Masmoudi, Global solutions for some Oldroyd models of non Newtonian flows, Chin. Ann. Math., Ser. B 21, No.2, pp 131-146 (2000)
17. J.S. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. Royal Soc. of London, Serie A, vol. 454, pp 2617-2654 (1998)
18. A. Miranville, Upper bound on the dimension of the attractor for the shear-layer flow in space dimension 3, In Dynamical Systems (Stockholm, 1992), pp 61-74. World Sci. Publishing, River Edge, N J (1993)
19. J.G. Oldroyd, On the formation of rheological equations of state, Proc. R. Soc. Lond. A200, pp 523-541 (1950)
20. P.D. Olmsted, C.Y. David Lu, Phase separation of Rigid-Rod Suspensions in Shear Flow, Physical Review E 60, pp 4397-4415 (1999)
21. A. Onuki, Phase transitions of fluids in shear flow, J. Phys. Condens. Matter 9, pp 6119-6157 (1997)
22. P(0,T;B), Ann. Mat. Pura Appl. (4) 146, pp 65-96 (1987)
23. R. Temam, Navier Stokes Equations, North-Holland (1977)
24. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematial Science 68, Springer-Verlag (1997)