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Local existence for the FENE-dumbbell model of polymeric fluids. (English) Zbl 1095.76004
Summary: We study the well-posedness of a multi-scale model of polymeric fluids. The microscopic model is the kinetic theory of the finitely extensible nonlinear elastic (FENE) dumbbell model. The macroscopic model describes incompressible non-Newtonian fluids with polymer stress computed via Kramers expression. The boundary condition of the FENE-type Fokker-Planck equation is proved to be unnecessary due to the singularity on the boundary. Other results are the local existence, uniqueness and regularity theorems for the FENE model in certain parameter range.

76A10 Viscoelastic fluids
35Q35 PDEs in connection with fluid mechanics
82D60 Statistical mechanical studies of polymers
Full Text: DOI
[1] Barrett J.W., Schwab C., Süli E.: Existence of global weak solutions for sone polymeric flow models, preprint
[2] Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids, 2nd Edn., Vol 1 & 2. John Wiley, New York, 1987
[3] Chauvière C., Lozinski A.: Simulation of complex viscoelastic flows using Fokker-Planck equation: 3D FENE model. Submitted to J. Non-Newtonian Fluid Mech.
[4] Doi M., Edwards S.F.: The Theory of Polymer Dynamics. Oxford University Press, Oxford, 1986
[5] Du Qiang., Liu Chun., Yu Peng.: FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. In press · Zbl 1108.76006
[6] Li T., E. W.; Zhang, P., No article title, Acta Math Appl Sin. Eng. Ser., 18, 529, (2002) · Zbl 1137.76313
[7] Li, E. W.T.; Zhang, P., No article title, Comm. Math. Phys., 248, 409, (2003)
[8] Jourdain, B.; Lelievre, T.; Le Bris, C., No article title, Math. Models Methods App. Sci., 12, 1205, (2002) · Zbl 1041.76003
[9] Jourdain, B.; Lelievre, T.; Le Bris, C., Existence of solution for a micro-macro model of polymeric fluid: the FENE model, J. Funct. Anal., 209, 162-193, (2004) · Zbl 1047.76004
[10] Jourdain B., Lelievre T.: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. Preprint, CERMICS 2002-225 · Zbl 1081.76006
[11] Kröger, M., No article title, Phys. Rep., 390, 451, (2004)
[12] Lin F.H., Liu C., Zhang P.: On a micro-macro model for polymeric fluids near equilibrium. Preprint · Zbl 1113.76017
[13] Li, T. J.; Zhang, H.; Zhang, P. W., No article title, Comm. Partial Differential Equations, 29, 903, (2004) · Zbl 1058.76010
[14] Lozinski, A.; Chauvière, C., No article title, J. Comp. Phys., 189, 607, (2003) · Zbl 1060.82525
[15] Pravia J.: Numerical methods for viscoelastic fluids. Doctoral Thesis, 2002
[16] Renardy, M., No article title, SIAM J. Math. Anal., 21, 1369, (1990) · Zbl 0734.73010
[17] theories, No article title, Math. Anal., 22, 313, (1991)
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